Research ArticleStability NeimarkndashSacker Bifurcation and Approximation of theInvariant Curve of Certain Homogeneous Second-OrderFractional Difference Equation

Mirela Garic-Demirovic Samra Moranjkic Mehmed Nurkanovic and Zehra Nurkanovic

University of Tuzla Department of Mathematics Tuzla Tuzla 75000 Bosnia and Herzegovina

Correspondence should be addressed to Mehmed Nurkanovic nurkanmyahoocom

Received 4 May 2020 Accepted 10 June 2020 Published 5 August 2020

Academic Editor Abdul Qadeer Khan

Copyright copy 2020 Mirela Garic-Demirovic et al +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

We investigate the local and global character of the unique equilibrium point and boundedness of the solutions of certainhomogeneous fractional difference equation with quadratic terms Also we consider NeimarkndashSacker bifurcations and give theasymptotic approximation of the invariant curve

1 Introduction and Preliminaries

In this paper the subject of our consideration is the fol-lowing difference equation

un+1 Au2

n

au2n + bununminus 1 + cu2

nminus 1 n 0 1 2 (1)

with positive parameters A a b and c where initial con-ditions uminus 1 u0 are positive numbers By substituting xn

1un (n 0 1 2 ) equation (1) reduces to the followingequation

xn+1 ax2

nminus 1 + bxnxnminus 1 + cx2n

Ax2nminus 1

n 0 1 2 (2)

Equations (1) and (2) are the special cases of the fol-lowing homogeneous rational difference equation

xn+1 Ax2

n + Bxnxnminus 1 + Cx2nminus 1

ax2n + bxnxnminus 1 + cx2

nminus 1 n 0 1 2 (3)

Equation (3) is a general homogeneous rational differ-ence equation with quadratic terms and it is very compli-cated for investigation in many special cases +e functionassociated with the right side of equation (3) is of the form

f(u v) (Au2 + Buv + Cv2)(au2 + buv + cv2) with thefollowing monotonicity property it is either monotonicallydecreasing in the first variable and monotonically increasingin the second one or monotonically increasing in the firstvariable and monotonically decreasing in the second vari-able Using the theory of monotone maps it was possible toinvestigate the global dynamics of some special cases ofequation (3) [1ndash4] in the situations when correspondingfunction f is monotonically decreasing in the first variableand monotonically increasing in the second variable

In [4] the authors investigated the local and globalcharacter of the unique equilibrium point and the existenceof NeimarkndashSacker and period-doubling bifurcations ofequation (3) +ey have also studied the local and globalstability of the minimal period-two solutions for somespecial cases of the parameters +e special case when A

B 0 and C 1 ie

xn+1 x2

nminus 1ax2

n + bxnxnminus 1 + cx2nminus 1

n 0 1 2 (4)

was considered in [2] It was shown that equation (4) ischaracterized by three types of global behavior with respectto the existence of a unique positive equilibrium and

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 6254013 12 pageshttpsdoiorg10115520206254013

existence of one or twominimal period-two solutions one ofwhich is locally asymptotically stable and the other is asaddle point An important feature of this equation is thecoexistence of an equilibrium and the minimal period-twosolution which are both locally asymptotically stable Alsothe basins of attraction of these solutions are described indetail

In [3] a special case of equation (3) was considered whenB c 0 ie

xn+1 Ax2

n + Cx2nminus 1

ax2n + bxnxnminus 1

n 0 1 2 (5)

It was shown that (5) exhibits period-two bifurcationand that stable manifolds of the minimal period-two solu-tions represent the boundaries of the basin of attraction oflocally stable equilibrium point and the basins of attractionof points (0infin) and (infin 0) Further in the situations whenequilibrium is a saddle point corresponding stable manifoldseparates the basins of attraction of the points (0infin) and(infin 0)

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder since by now there isno any general result for this type of monotonicity Someinitial steps regarding this were taken in [5] where theconsidered case was A p C a 1 B b c 0 ie

xn+1 p +xn

xnminus 11113888 1113889

2

n 0 1 2 (6)

In [5] the authors have successfully used the embeddingmethod to demonstrate the boundedness of the solutionsand then they determined the invariant interval of equation(6) +at was a crucial idea for proving that local asymptoticstability (which holds when pgt 1) implies global asymptoticstability of the unique positive equilibrium point whenpgt

2

radic Also the existence of NeimarkndashSacker bifurcation is

shown and asymptotic approximation of the invariant curveis computed

+e investigation of the dynamics of equation (4) and itsspecial cases has been the subject of many research studiesfor the last ten years Some of these cases as we have seenhave been successfully realized However only the ideasfrom [5] finally made the problem of investigating the be-havior of equation (1) or equation (2) solvable Namely notethat equation (2) has the form

xn+1 p +xn

xnminus 1+ q

xn

xnminus 11113888 1113889

2

n 0 1 2 (7)

where (aA) p (bA) 1 and (cA) q +erefore wewill consider equation (7) instead of equation (1) Howeverthe application of the embedding method on equation (7) issignificantly harder compared with application on equation(6) (because the form is more complicated) We will in-vestigate local and global stability of a unique equilibriumpoint and boundedness of the solutions of equation (7) andexamine the existence of NeimarkndashSacker bifurcation Also

in the situation when NeimarkndashSacker bifurcation appearswe will give the asymptotic approximation of the invariantcurve

+e special case of equation (3) when C c 1 andB b 0 was considered in [1] In the region of parameterswhere altA corresponding function f is monotonicallyincreasing by first variable and monotonically decreasing bysecond variable and through very complicated calculationsusing the so-called ldquoM-mrdquo theorems it is shown that in someareas of parametric space of parameters A and a uniquepositive equilibrium is globally stable +e existence ofperiod-doubling bifurcation is proved in the case Alt awhen corresponding function f is monotonically decreasingby first variable and monotonically increasing by secondvariable Using the theory of monotone maps the globalstability of minimal period-two solution is shown for somespecial values of parameters

Notice that equation (3) is a special case (and probablythe most complicated equation of the form (3 3)) of thefollowing general second-order rational difference equationwith quadratic terms

xn+1 Ax2

n + Bxnxnminus 1 + Cx2nminus 1 + Dxn + Exnminus 1 + F

ax2n + bxnxnminus 1 + cx2

nminus 1 + dxn + exnminus 1 + f n 0 1 2

(8)

that has caught the attention of mathematical researchersover the last ten years ([1ndash12])

+e following lemma gives us the type of local stability ofa unique positive equilibrium point of equation (7)depending on different values of parameters p and q

Lemma 1 Equation (7) has a unique equilibrium pointx p + q + 1 which is

(a) locally asymptotically stable if pgt q

(b) nonhyperbolic if p q

(c) a repeller if plt q

Proof Denote g(u v) p + (uv) + q(u2v2) +e linear-ized equation associated with equation (7) about the equi-librium point has the form

zn+1 szn + tznminus 1 (9)

where s (zgzu)(x x) and t (zgzv)(x x) Notice

s minus t zg

zu(x x)

1 + 2q

p + 1 + qgt 0 (10)

Since

|s|lt 1 minus tlt 2 ⟺slt 1 + slt 2⟺slt 1⟺pgt q

|s| |1 minus t|or (t minus 1and sle 2) ⟺ s 1 + sor (s 1and sle 2)

⟺s 1⟺p q

|s|lt|1 minus t|and |t|gt 1 ⟺ slt 1 + sand sgt 1 ⟺plt q

(11)

the conclusion follows

2 Discrete Dynamics in Nature and Society

+is paper is organized as follows In Section 2 using theembedding method [5 13] and the so-called ldquoM-mrdquo theo-rems [14ndash17] we prove global asymptotic stability of aunique positive equilibrium for pge

(q + 1)(2q + 1)

1113968and

conduct the semicycle analysis as well In Section 3 usingNeimarkndashSacker theorem [15 18ndash21] we give reduction tothe normal form and perform computation of the coeffi-cients of the aforementioned bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determine the asymptotic approximation of the invariantcurve and give a visual evidence

2 Global Asymptotic Stability

In this section we will show that all solutions of equation (7)are bounded and using the so-called ldquoM-mrdquo theorem wewill obtain sufficient conditions for unique positive equi-librium to be globally asymptotically stable Similar to[5 13] we apply the method of embedding First wesubstitute

xn p +xnminus 1

xnminus 2+ q

xnminus 1

xnminus 21113888 1113889

2

(12)

in equation (7) and obtain

xn+1 p +p

xnminus 1+

1xnminus 2

+ qxnminus 1

x2nminus 2

+ qp

xnminus 1+

1xnminus 2

+ qxnminus 1

x2nminus 2

1113888 1113889

2

(13)

+en by substituting

xnminus 1 p +xnminus 2

xnminus 3+ q

xnminus 2

xnminus 31113888 1113889

2

(14)

in equation (13) we have

xn+1 p +p

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 3

+ qp

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 31113888 1113889

2

(15)

Notice that the solutions of equation (15) are bounded

plexn+1 lep + 1 +1p

+q2

p2 +q

p+

q

p2 + q 1 +1p

+q2

p2 +q

p+

q

p21113888 1113889

2

nge 4

(16)

that is

plexn+1 lep +(q + 1) p + q + pq + p2 + q2( 1113857 pq + p2 + q2( 1113857

p4 nge 4

(17)

Since every solution xnprime1113864 1113865infinnminus 1 of equation (7) is also a

solution xn1113864 1113865infinnminus 2 of equation (15) with initial values

xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime ) + q(x0primexminus 1prime )2 andx0 p + (x1primex0prime) + q(x1primex0prime)

2 we see that the solutions ofequation (7) are also bounded

From (7) and (15) we getxn

xnminus 1

p

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 3 (18)

which implies

xn p +xnminus 1

xnminus 2+

q2xnminus 1

x2nminus 3

+pqxnminus 1

x2nminus 2

+qxnminus 1

xnminus 2xnminus 3 (19)

that is

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+pqxn

x2nminus 1

+qxn

xnminus 1xnminus 2 (20)

By replacing xn p + (xnminus 1xnminus 2) + q(xnminus 1xnminus 2)2 in

(20) we obtain the following equation

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+ pqp + xnminus 1xnminus 2( 1113857 + q xnminus 1xnminus 2( 1113857

2

x2nminus 1

+qxn

xnminus 1xnminus 2 (21)

that is

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+p2q

x2nminus 1

+pq2

x2nminus 2

+pq

xnminus 1xnminus 2+

qxn

xnminus 1xnminus 2

(22)

Remark 1 Note that equation (22) has the unique equi-librium point x p + q + 1 which is the same equilibriumas in equation (7) It follows from

p + 1 +q2

x+

p2q

x2 +pq2

x2 +pq

x2 +p

xminus x

(p + q minus x + 1) pq + qx + x2( 1113857

x2 0 (23)

Discrete Dynamics in Nature and Society 3

Furthermore notice that every solution xnprime1113864 1113865infinnminus 1 of

equation (7) is also a solution xn1113864 1113865infinnminus 2 of equation (22) with

initial values xminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) +

q(x0primexminus 1prime)2 and that it is of the form xn+1 f(xn xnminus 1 xnminus 2)

where

f(u v w) p +u

v+

q2u

w2 +p2q

v2+

pq2

w2 +pq

vw+

qu

vw (24)

Lemma 2 Every interval I of the form [p U] where

U p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) pgt q + 1 (25)

is an invariant interval for the function f

Proof As we know for pltU interval I [p U] is invariantfor the function f if

(forallu v w isin I)f(u v w) isin I (26)

For ple u v wleU we have that plef(u v w)

lep + (Up)+ (q2Up2) + q + (q2p) + (qp) + (qUp2) IfU satisfies

p +U

p+

q2U

p2 + q +q2

p+

q

p+

qU

p2 leU (27)

then we obtain f(u v w)leU It further implies that forevery pgt q + 1 there exists such U which means that I isinvariant for the function f whereUgep(q + pq + p2 + q2)(p minus q minus 1)(p + q) Since we canassume that pgt q + 1 and U p(q + pq + p2 + q2)(p minus q minus 1)(p + q) I is the invariant for f

Lemma 3 Interval I [p U] is an attracting interval forequation (22)

Proof It is clear that we need to show that every solution ofequation (22) must enter interval I Note that for arbitraryinitial conditions xminus 2 xminus 1prime xminus 1 x0prime andx0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 it holds

xn gtp for nge 0 (28)

If x1 x2 x3 isin I pgt q + 1 then xn isin I for ngt 3 byLemma 2 Otherwise if x3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) let us prove that there must be somekgt 3 such that xn isin I for all nge k Namely suppose thatx3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) for arbitraryinitial conditions xminus 2 xminus 1 x0 +en

xnminus 3 gtp xnminus 2 gtp for nge 3 (29)

So from equation (22) we obtain

xn p +xnminus 1

xnminus 2+

q2xnminus 1

x2nminus 3

+p2q

x2nminus 2

+pq2

x2nminus 3

+pq

xnminus 2xnminus 3+

qxnminus 1

xnminus 2xnminus 3

(30)

which implies

xn ltp +p2q

p2 +pq2

p2 +pq

p2 +xnminus 1

p+

q2xnminus 1

p2 +qxnminus 1

p2 (31)

that is

xn ltq + pq + p2 + q2

p+

p + q + q2

p2 xnminus 1 (32)

By induction we conclude

xn ltq + pq + p2 + q2

p1113888 1113889

1 minus p + q + q2( 1113857p2( 1113857nminus 3

1 minus p + q + q2( 1113857p2

+x3

p2 p + q + q2( 1113857( 1113857nminus 3 for ngt 3

(33)

Since (p2(p + q + q2))gt 1 for pgt q + 1 the right side in(33) is decreasing sequence converging to

q + pq + p2 + q2

p1113888 1113889

11 minus p + q + q2( 1113857p2

p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) U

(34)

+en from (33) we get

limn⟶infin

xn leU (35)

Since the case limn⟶infinxn U is not possible (otherwisethere would exist another positive equilibrium differentfrom x) it implies that there is some kgt 3 such that

pltxn ltU (36)

for all nge k ie every solution of equation (22) must enterthe interval I

Theorem 1 If pge(q + 1)(2q + 1)

1113968 then the equilibrium

point x of equation (7) is globally asymptotically stable

Proof It is enough to prove that x is an attractor of equation(22) Since there exists the invariant and attracting intervalI [p p(q + pq + p2 + q2)(p minus q minus 1)(p + q)] whenpgt q + 1 we need to check the conditions of +eorem A05in [14]

M f(M m m)

m f(m M M)

⎧⎪⎨

⎪⎩⟺

M p +p2q

m2 +M

m+

qM

m2 +q2M

m2 +pq2

m2 +pq

m2

m p +p2q

M2 +m

M+

qm

M2 +q2m

M2 +pq2

M2 +pq

M2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(37)

4 Discrete Dynamics in Nature and Society

ie

M f(M m m)

m f(m M M)1113896 ⟺

Mm2 pm2 + p2q + Mm + qM + q2M + pq2 + pq

mM2 pM2 + p2q + mM + qm + q2m + pq2 + pq1113896 (38)

By subtracting the second equation in (38) from the firstwe get

(M minus m) q minus Mp minus mp + q2

+ Mm1113872 1113873 0 (39)

from which m M or for mneM

(m minus p)M mp minus q minus q2 (40)

If m p then

0 p2

minus q minus q2 (41)

Since p2 minus q minus q2 gt 0 for pgt q + 1 it implies that mnep+erefore

M mp minus q minus q2

m minus p mnep (42)

By substituting (42) into (38) we obtain the followingquadratic equation

m2(p minus q minus 1)(p + q) minus q 2p minus q + 2pq + p

2minus 11113872 1113873m

+ q q + p2q + p

2+ p

3+ 2q

2+ q

31113872 1113873 0

(43)

whose discriminant has the form

D(p q) q 3q minus p2

+ 2q2

+ 11113872 1113873 q + 3p2q + 4p

3+ 3q

2+ 2q

31113872 1113873

(44)

It is clear that D(p q)lt 0 if pgt(q + 1)(2q + 1)

1113968and

that there are no real solutions for equation (43) Ifp

(q + 1)(2q + 1)

1113968 then m M p + q + 1 which is a

contradiction with assumption mneM Whenq + 1ltplt

(q + 1)(2q + 1)

1113968 equation (43) has two positive

roots mplusmn gtp it implies that Mplusmn gt 0 (for examplem 30032 M 95212 for p 3 and q 199 where(m M) is in invariant interval I) +erefore the conditionsof +eorem A05 in [14] are satisfied forpge

(q + 1)(2q + 1)

1113968and every solution of equation (22)

converges to x p + q + 1 By Remark 1 the uniqueequilibrium x p + q + 1 of equation (7) is an attractor Byusing Lemma 1 we conclude that x is globally asymptoticallystable (see Figure 1)

Conjecture 1 If pgt q then the equilibrium point x p +

q + 1 is globally asymptotically stable

3 NeimarkndashSacker Bifurcations

+e following results are obtained by applying the algorithmfrom +eorem 1 and Corollary 1 in [5] (see also [22]) If wemake a change of variable yn xn minus x we will shift theequilibrium point to the origin +en the transformedequation is given by

yn+1 yn + p + 1 + q

ynminus 1 + p + 1 + qminus 1

+ qyn + p + 1 + q( 1113857

2

ynminus 1 + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠ n 0 1

(45)

Set

un ynminus 1

vn yn

for n 0 1

(46)

and write equation (1) in the equivalent formun+1 vn

vn+1 vn + p + 1 + q

un + p + 1 + qminus 1 + q

vn + p + 1 + q( 11138572

un + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠

(47)

Let F denote the corresponding map defined by

F

u

v

⎛⎝ ⎞⎠

v

v + p + 1 + q

u + p + 1 + qminus 1 + q

(v + p + 1 + q)2

(u + p + 1 + q)2minus 11113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)

+en the Jacobian matrix of F is given by

JacF(u v)

0 1

minus(p + q + v + 1)(2q(p + q + v + 1) + p + q + u + 1)

(p + q + u + 1)32q(p + q + v + 1) + p + q + u + 1

(p + q + u + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (49)

Discrete Dynamics in Nature and Society 5

+e eigenvalues of JacF(0 0) are μ(p) and μ(p) where

μ(p) 1 + 2q + i

(1 + 2q)(3 + 2q + 4p)

1113968

2(p + q + 1)

|μ(p)|

1 + 2q

p + q + 1

1113971

(50)

Lemma 4 If p p0 q then F has equilibrium point at(0 0) and eigenvalues of Jacobian matrix of F at (0 0) are μand μ where

μ p0( 1113857 12

+i

3

radic

2 (51)

Moreover μ satisfies the following

(i) μk(p0)ne 1 for k 1 2 3 4(ii) (ddp)|μ(p)| d(p0) minus (12(1 + 2q))lt 0 at

p p0

(iii) Eigenvectors associated to the μ are

q p0( 1113857 12

minusi

3

radic

2 11113888 1113889

T

p p0( 1113857 i3

radic 16

(3 minus i3

radic)1113888 1113889

(52)

such that pA μp Aq μq and pq 1 whereA JacF(0 0)

Proof Let p p0 q +en we obtain

A JacF(0 0)

0 1

minus1 + 2q

x

1 + 2q

x

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0 1

minus 1 1⎛⎝ ⎞⎠ (53)

After straightforward calculation for p p0 q weobtain |μ(p0)| 1 and

μ p0( 1113857 12

+i

3

radic

2

μ2 p0( 1113857 minus12

+i

3

radic

2

μ3 p0( 1113857 minus 1

μ4 p0( 1113857 minus12

minusi

3

radic

2

(54)

from which it follows that μk(p0)ne 1 for k 1 2 3 4Furthermore we get

ddp

|μ(p)| minus

1 + 2q

1113968

2(p + q + 1)32

ddp

|μ(p)|1113888 1113889pp0

minus1

2(1 + 2q)lt 0

(55)

It is easy to see that p(p0)A μp(p0) andp(p0)q(p0) 1

Let p p0 + η where η is a sufficiently small parameterFrom Lemma 4 we can transform system (47) into thenormal form

F(λ x) F(λ x) + O x5

1113872 1113873 (56)

Repeller

Locally asymptotically stable

Nonhyperb

olic

0 1 2 3 40

1

2

3

4

(a)

Globally asymptotically stable

0 5 10 15 20 250

5

10

15

20

25

(b)

Figure 1 Local (a) and global (b) asymptotic stability in (p q) plane

6 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

+is paper is organized as follows In Section 2 using theembedding method [5 13] and the so-called ldquoM-mrdquo theo-rems [14ndash17] we prove global asymptotic stability of aunique positive equilibrium for pge

(q + 1)(2q + 1)

1113968and

conduct the semicycle analysis as well In Section 3 usingNeimarkndashSacker theorem [15 18ndash21] we give reduction tothe normal form and perform computation of the coeffi-cients of the aforementioned bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determine the asymptotic approximation of the invariantcurve and give a visual evidence

2 Global Asymptotic Stability

In this section we will show that all solutions of equation (7)are bounded and using the so-called ldquoM-mrdquo theorem wewill obtain sufficient conditions for unique positive equi-librium to be globally asymptotically stable Similar to[5 13] we apply the method of embedding First wesubstitute

xn p +xnminus 1

xnminus 2+ q

xnminus 1

xnminus 21113888 1113889

2

(12)

in equation (7) and obtain

xn+1 p +p

xnminus 1+

1xnminus 2

+ qxnminus 1

x2nminus 2

+ qp

xnminus 1+

1xnminus 2

+ qxnminus 1

x2nminus 2

1113888 1113889

2

(13)

+en by substituting

xnminus 1 p +xnminus 2

xnminus 3+ q

xnminus 2

xnminus 31113888 1113889

2

(14)

in equation (13) we have

xn+1 p +p

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 3

+ qp

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 31113888 1113889

2

(15)

Notice that the solutions of equation (15) are bounded

plexn+1 lep + 1 +1p

+q2

p2 +q

p+

q

p2 + q 1 +1p

+q2

p2 +q

p+

q

p21113888 1113889

2

nge 4

(16)

that is

plexn+1 lep +(q + 1) p + q + pq + p2 + q2( 1113857 pq + p2 + q2( 1113857

p4 nge 4

(17)

Since every solution xnprime1113864 1113865infinnminus 1 of equation (7) is also a

solution xn1113864 1113865infinnminus 2 of equation (15) with initial values

xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime ) + q(x0primexminus 1prime )2 andx0 p + (x1primex0prime) + q(x1primex0prime)

2 we see that the solutions ofequation (7) are also bounded

From (7) and (15) we getxn

xnminus 1

p

xnminus 1+

1xnminus 2

+q2

x2nminus 3

+pq

x2nminus 2

+q

xnminus 2xnminus 3 (18)

which implies

xn p +xnminus 1

xnminus 2+

q2xnminus 1

x2nminus 3

+pqxnminus 1

x2nminus 2

+qxnminus 1

xnminus 2xnminus 3 (19)

that is

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+pqxn

x2nminus 1

+qxn

xnminus 1xnminus 2 (20)

By replacing xn p + (xnminus 1xnminus 2) + q(xnminus 1xnminus 2)2 in

(20) we obtain the following equation

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+ pqp + xnminus 1xnminus 2( 1113857 + q xnminus 1xnminus 2( 1113857

2

x2nminus 1

+qxn

xnminus 1xnminus 2 (21)

that is

xn+1 p +xn

xnminus 1+

q2xn

x2nminus 2

+p2q

x2nminus 1

+pq2

x2nminus 2

+pq

xnminus 1xnminus 2+

qxn

xnminus 1xnminus 2

(22)

Remark 1 Note that equation (22) has the unique equi-librium point x p + q + 1 which is the same equilibriumas in equation (7) It follows from

p + 1 +q2

x+

p2q

x2 +pq2

x2 +pq

x2 +p

xminus x

(p + q minus x + 1) pq + qx + x2( 1113857

x2 0 (23)

Discrete Dynamics in Nature and Society 3

Furthermore notice that every solution xnprime1113864 1113865infinnminus 1 of

equation (7) is also a solution xn1113864 1113865infinnminus 2 of equation (22) with

initial values xminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) +

q(x0primexminus 1prime)2 and that it is of the form xn+1 f(xn xnminus 1 xnminus 2)

where

f(u v w) p +u

v+

q2u

w2 +p2q

v2+

pq2

w2 +pq

vw+

qu

vw (24)

Lemma 2 Every interval I of the form [p U] where

U p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) pgt q + 1 (25)

is an invariant interval for the function f

Proof As we know for pltU interval I [p U] is invariantfor the function f if

(forallu v w isin I)f(u v w) isin I (26)

For ple u v wleU we have that plef(u v w)

lep + (Up)+ (q2Up2) + q + (q2p) + (qp) + (qUp2) IfU satisfies

p +U

p+

q2U

p2 + q +q2

p+

q

p+

qU

p2 leU (27)

then we obtain f(u v w)leU It further implies that forevery pgt q + 1 there exists such U which means that I isinvariant for the function f whereUgep(q + pq + p2 + q2)(p minus q minus 1)(p + q) Since we canassume that pgt q + 1 and U p(q + pq + p2 + q2)(p minus q minus 1)(p + q) I is the invariant for f

Lemma 3 Interval I [p U] is an attracting interval forequation (22)

Proof It is clear that we need to show that every solution ofequation (22) must enter interval I Note that for arbitraryinitial conditions xminus 2 xminus 1prime xminus 1 x0prime andx0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 it holds

xn gtp for nge 0 (28)

If x1 x2 x3 isin I pgt q + 1 then xn isin I for ngt 3 byLemma 2 Otherwise if x3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) let us prove that there must be somekgt 3 such that xn isin I for all nge k Namely suppose thatx3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) for arbitraryinitial conditions xminus 2 xminus 1 x0 +en

xnminus 3 gtp xnminus 2 gtp for nge 3 (29)

So from equation (22) we obtain

xn p +xnminus 1

xnminus 2+

q2xnminus 1

x2nminus 3

+p2q

x2nminus 2

+pq2

x2nminus 3

+pq

xnminus 2xnminus 3+

qxnminus 1

xnminus 2xnminus 3

(30)

which implies

xn ltp +p2q

p2 +pq2

p2 +pq

p2 +xnminus 1

p+

q2xnminus 1

p2 +qxnminus 1

p2 (31)

that is

xn ltq + pq + p2 + q2

p+

p + q + q2

p2 xnminus 1 (32)

By induction we conclude

xn ltq + pq + p2 + q2

p1113888 1113889

1 minus p + q + q2( 1113857p2( 1113857nminus 3

1 minus p + q + q2( 1113857p2

+x3

p2 p + q + q2( 1113857( 1113857nminus 3 for ngt 3

(33)

Since (p2(p + q + q2))gt 1 for pgt q + 1 the right side in(33) is decreasing sequence converging to

q + pq + p2 + q2

p1113888 1113889

11 minus p + q + q2( 1113857p2

p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) U

(34)

+en from (33) we get

limn⟶infin

xn leU (35)

Since the case limn⟶infinxn U is not possible (otherwisethere would exist another positive equilibrium differentfrom x) it implies that there is some kgt 3 such that

pltxn ltU (36)

for all nge k ie every solution of equation (22) must enterthe interval I

Theorem 1 If pge(q + 1)(2q + 1)

1113968 then the equilibrium

point x of equation (7) is globally asymptotically stable

Proof It is enough to prove that x is an attractor of equation(22) Since there exists the invariant and attracting intervalI [p p(q + pq + p2 + q2)(p minus q minus 1)(p + q)] whenpgt q + 1 we need to check the conditions of +eorem A05in [14]

M f(M m m)

m f(m M M)

⎧⎪⎨

⎪⎩⟺

M p +p2q

m2 +M

m+

qM

m2 +q2M

m2 +pq2

m2 +pq

m2

m p +p2q

M2 +m

M+

qm

M2 +q2m

M2 +pq2

M2 +pq

M2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(37)

4 Discrete Dynamics in Nature and Society

ie

M f(M m m)

m f(m M M)1113896 ⟺

Mm2 pm2 + p2q + Mm + qM + q2M + pq2 + pq

mM2 pM2 + p2q + mM + qm + q2m + pq2 + pq1113896 (38)

By subtracting the second equation in (38) from the firstwe get

(M minus m) q minus Mp minus mp + q2

+ Mm1113872 1113873 0 (39)

from which m M or for mneM

(m minus p)M mp minus q minus q2 (40)

If m p then

0 p2

minus q minus q2 (41)

Since p2 minus q minus q2 gt 0 for pgt q + 1 it implies that mnep+erefore

M mp minus q minus q2

m minus p mnep (42)

By substituting (42) into (38) we obtain the followingquadratic equation

m2(p minus q minus 1)(p + q) minus q 2p minus q + 2pq + p

2minus 11113872 1113873m

+ q q + p2q + p

2+ p

3+ 2q

2+ q

31113872 1113873 0

(43)

whose discriminant has the form

D(p q) q 3q minus p2

+ 2q2

+ 11113872 1113873 q + 3p2q + 4p

3+ 3q

2+ 2q

31113872 1113873

(44)

It is clear that D(p q)lt 0 if pgt(q + 1)(2q + 1)

1113968and

that there are no real solutions for equation (43) Ifp

(q + 1)(2q + 1)

1113968 then m M p + q + 1 which is a

contradiction with assumption mneM Whenq + 1ltplt

(q + 1)(2q + 1)

1113968 equation (43) has two positive

roots mplusmn gtp it implies that Mplusmn gt 0 (for examplem 30032 M 95212 for p 3 and q 199 where(m M) is in invariant interval I) +erefore the conditionsof +eorem A05 in [14] are satisfied forpge

(q + 1)(2q + 1)

1113968and every solution of equation (22)

converges to x p + q + 1 By Remark 1 the uniqueequilibrium x p + q + 1 of equation (7) is an attractor Byusing Lemma 1 we conclude that x is globally asymptoticallystable (see Figure 1)

Conjecture 1 If pgt q then the equilibrium point x p +

q + 1 is globally asymptotically stable

3 NeimarkndashSacker Bifurcations

+e following results are obtained by applying the algorithmfrom +eorem 1 and Corollary 1 in [5] (see also [22]) If wemake a change of variable yn xn minus x we will shift theequilibrium point to the origin +en the transformedequation is given by

yn+1 yn + p + 1 + q

ynminus 1 + p + 1 + qminus 1

+ qyn + p + 1 + q( 1113857

2

ynminus 1 + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠ n 0 1

(45)

Set

un ynminus 1

vn yn

for n 0 1

(46)

and write equation (1) in the equivalent formun+1 vn

vn+1 vn + p + 1 + q

un + p + 1 + qminus 1 + q

vn + p + 1 + q( 11138572

un + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠

(47)

Let F denote the corresponding map defined by

F

u

v

⎛⎝ ⎞⎠

v

v + p + 1 + q

u + p + 1 + qminus 1 + q

(v + p + 1 + q)2

(u + p + 1 + q)2minus 11113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)

+en the Jacobian matrix of F is given by

JacF(u v)

0 1

minus(p + q + v + 1)(2q(p + q + v + 1) + p + q + u + 1)

(p + q + u + 1)32q(p + q + v + 1) + p + q + u + 1

(p + q + u + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (49)

Discrete Dynamics in Nature and Society 5

+e eigenvalues of JacF(0 0) are μ(p) and μ(p) where

μ(p) 1 + 2q + i

(1 + 2q)(3 + 2q + 4p)

1113968

2(p + q + 1)

|μ(p)|

1 + 2q

p + q + 1

1113971

(50)

Lemma 4 If p p0 q then F has equilibrium point at(0 0) and eigenvalues of Jacobian matrix of F at (0 0) are μand μ where

μ p0( 1113857 12

+i

3

radic

2 (51)

Moreover μ satisfies the following

(i) μk(p0)ne 1 for k 1 2 3 4(ii) (ddp)|μ(p)| d(p0) minus (12(1 + 2q))lt 0 at

p p0

(iii) Eigenvectors associated to the μ are

q p0( 1113857 12

minusi

3

radic

2 11113888 1113889

T

p p0( 1113857 i3

radic 16

(3 minus i3

radic)1113888 1113889

(52)

such that pA μp Aq μq and pq 1 whereA JacF(0 0)

Proof Let p p0 q +en we obtain

A JacF(0 0)

0 1

minus1 + 2q

x

1 + 2q

x

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0 1

minus 1 1⎛⎝ ⎞⎠ (53)

After straightforward calculation for p p0 q weobtain |μ(p0)| 1 and

μ p0( 1113857 12

+i

3

radic

2

μ2 p0( 1113857 minus12

+i

3

radic

2

μ3 p0( 1113857 minus 1

μ4 p0( 1113857 minus12

minusi

3

radic

2

(54)

from which it follows that μk(p0)ne 1 for k 1 2 3 4Furthermore we get

ddp

|μ(p)| minus

1 + 2q

1113968

2(p + q + 1)32

ddp

|μ(p)|1113888 1113889pp0

minus1

2(1 + 2q)lt 0

(55)

It is easy to see that p(p0)A μp(p0) andp(p0)q(p0) 1

Let p p0 + η where η is a sufficiently small parameterFrom Lemma 4 we can transform system (47) into thenormal form

F(λ x) F(λ x) + O x5

1113872 1113873 (56)

Repeller

Locally asymptotically stable

Nonhyperb

olic

0 1 2 3 40

1

2

3

4

(a)

Globally asymptotically stable

0 5 10 15 20 250

5

10

15

20

25

(b)

Figure 1 Local (a) and global (b) asymptotic stability in (p q) plane

6 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

Furthermore notice that every solution xnprime1113864 1113865infinnminus 1 of

equation (7) is also a solution xn1113864 1113865infinnminus 2 of equation (22) with

initial values xminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) +

q(x0primexminus 1prime)2 and that it is of the form xn+1 f(xn xnminus 1 xnminus 2)

where

f(u v w) p +u

v+

q2u

w2 +p2q

v2+

pq2

w2 +pq

vw+

qu

vw (24)

Lemma 2 Every interval I of the form [p U] where

U p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) pgt q + 1 (25)

is an invariant interval for the function f

Proof As we know for pltU interval I [p U] is invariantfor the function f if

(forallu v w isin I)f(u v w) isin I (26)

For ple u v wleU we have that plef(u v w)

lep + (Up)+ (q2Up2) + q + (q2p) + (qp) + (qUp2) IfU satisfies

p +U

p+

q2U

p2 + q +q2

p+

q

p+

qU

p2 leU (27)

then we obtain f(u v w)leU It further implies that forevery pgt q + 1 there exists such U which means that I isinvariant for the function f whereUgep(q + pq + p2 + q2)(p minus q minus 1)(p + q) Since we canassume that pgt q + 1 and U p(q + pq + p2 + q2)(p minus q minus 1)(p + q) I is the invariant for f

Lemma 3 Interval I [p U] is an attracting interval forequation (22)

Proof It is clear that we need to show that every solution ofequation (22) must enter interval I Note that for arbitraryinitial conditions xminus 2 xminus 1prime xminus 1 x0prime andx0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 it holds

xn gtp for nge 0 (28)

If x1 x2 x3 isin I pgt q + 1 then xn isin I for ngt 3 byLemma 2 Otherwise if x3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) let us prove that there must be somekgt 3 such that xn isin I for all nge k Namely suppose thatx3 gt (p(q + pq + p2 + q2)(p minus q minus 1)(p + q)) for arbitraryinitial conditions xminus 2 xminus 1 x0 +en

xnminus 3 gtp xnminus 2 gtp for nge 3 (29)

So from equation (22) we obtain

xn p +xnminus 1

xnminus 2+

q2xnminus 1

x2nminus 3

+p2q

x2nminus 2

+pq2

x2nminus 3

+pq

xnminus 2xnminus 3+

qxnminus 1

xnminus 2xnminus 3

(30)

which implies

xn ltp +p2q

p2 +pq2

p2 +pq

p2 +xnminus 1

p+

q2xnminus 1

p2 +qxnminus 1

p2 (31)

that is

xn ltq + pq + p2 + q2

p+

p + q + q2

p2 xnminus 1 (32)

By induction we conclude

xn ltq + pq + p2 + q2

p1113888 1113889

1 minus p + q + q2( 1113857p2( 1113857nminus 3

1 minus p + q + q2( 1113857p2

+x3

p2 p + q + q2( 1113857( 1113857nminus 3 for ngt 3

(33)

Since (p2(p + q + q2))gt 1 for pgt q + 1 the right side in(33) is decreasing sequence converging to

q + pq + p2 + q2

p1113888 1113889

11 minus p + q + q2( 1113857p2

p q + pq + p2 + q2( 1113857

(p minus q minus 1)(p + q) U

(34)

+en from (33) we get

limn⟶infin

xn leU (35)

Since the case limn⟶infinxn U is not possible (otherwisethere would exist another positive equilibrium differentfrom x) it implies that there is some kgt 3 such that

pltxn ltU (36)

for all nge k ie every solution of equation (22) must enterthe interval I

Theorem 1 If pge(q + 1)(2q + 1)

1113968 then the equilibrium

point x of equation (7) is globally asymptotically stable

Proof It is enough to prove that x is an attractor of equation(22) Since there exists the invariant and attracting intervalI [p p(q + pq + p2 + q2)(p minus q minus 1)(p + q)] whenpgt q + 1 we need to check the conditions of +eorem A05in [14]

M f(M m m)

m f(m M M)

⎧⎪⎨

⎪⎩⟺

M p +p2q

m2 +M

m+

qM

m2 +q2M

m2 +pq2

m2 +pq

m2

m p +p2q

M2 +m

M+

qm

M2 +q2m

M2 +pq2

M2 +pq

M2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(37)

4 Discrete Dynamics in Nature and Society

ie

M f(M m m)

m f(m M M)1113896 ⟺

Mm2 pm2 + p2q + Mm + qM + q2M + pq2 + pq

mM2 pM2 + p2q + mM + qm + q2m + pq2 + pq1113896 (38)

By subtracting the second equation in (38) from the firstwe get

(M minus m) q minus Mp minus mp + q2

+ Mm1113872 1113873 0 (39)

from which m M or for mneM

(m minus p)M mp minus q minus q2 (40)

If m p then

0 p2

minus q minus q2 (41)

Since p2 minus q minus q2 gt 0 for pgt q + 1 it implies that mnep+erefore

M mp minus q minus q2

m minus p mnep (42)

By substituting (42) into (38) we obtain the followingquadratic equation

m2(p minus q minus 1)(p + q) minus q 2p minus q + 2pq + p

2minus 11113872 1113873m

+ q q + p2q + p

2+ p

3+ 2q

2+ q

31113872 1113873 0

(43)

whose discriminant has the form

D(p q) q 3q minus p2

+ 2q2

+ 11113872 1113873 q + 3p2q + 4p

3+ 3q

2+ 2q

31113872 1113873

(44)

It is clear that D(p q)lt 0 if pgt(q + 1)(2q + 1)

1113968and

that there are no real solutions for equation (43) Ifp

(q + 1)(2q + 1)

1113968 then m M p + q + 1 which is a

contradiction with assumption mneM Whenq + 1ltplt

(q + 1)(2q + 1)

1113968 equation (43) has two positive

roots mplusmn gtp it implies that Mplusmn gt 0 (for examplem 30032 M 95212 for p 3 and q 199 where(m M) is in invariant interval I) +erefore the conditionsof +eorem A05 in [14] are satisfied forpge

(q + 1)(2q + 1)

1113968and every solution of equation (22)

converges to x p + q + 1 By Remark 1 the uniqueequilibrium x p + q + 1 of equation (7) is an attractor Byusing Lemma 1 we conclude that x is globally asymptoticallystable (see Figure 1)

Conjecture 1 If pgt q then the equilibrium point x p +

q + 1 is globally asymptotically stable

3 NeimarkndashSacker Bifurcations

+e following results are obtained by applying the algorithmfrom +eorem 1 and Corollary 1 in [5] (see also [22]) If wemake a change of variable yn xn minus x we will shift theequilibrium point to the origin +en the transformedequation is given by

yn+1 yn + p + 1 + q

ynminus 1 + p + 1 + qminus 1

+ qyn + p + 1 + q( 1113857

2

ynminus 1 + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠ n 0 1

(45)

Set

un ynminus 1

vn yn

for n 0 1

(46)

and write equation (1) in the equivalent formun+1 vn

vn+1 vn + p + 1 + q

un + p + 1 + qminus 1 + q

vn + p + 1 + q( 11138572

un + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠

(47)

Let F denote the corresponding map defined by

F

u

v

⎛⎝ ⎞⎠

v

v + p + 1 + q

u + p + 1 + qminus 1 + q

(v + p + 1 + q)2

(u + p + 1 + q)2minus 11113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)

+en the Jacobian matrix of F is given by

JacF(u v)

0 1

minus(p + q + v + 1)(2q(p + q + v + 1) + p + q + u + 1)

(p + q + u + 1)32q(p + q + v + 1) + p + q + u + 1

(p + q + u + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (49)

Discrete Dynamics in Nature and Society 5

+e eigenvalues of JacF(0 0) are μ(p) and μ(p) where

μ(p) 1 + 2q + i

(1 + 2q)(3 + 2q + 4p)

1113968

2(p + q + 1)

|μ(p)|

1 + 2q

p + q + 1

1113971

(50)

Lemma 4 If p p0 q then F has equilibrium point at(0 0) and eigenvalues of Jacobian matrix of F at (0 0) are μand μ where

μ p0( 1113857 12

+i

3

radic

2 (51)

Moreover μ satisfies the following

(i) μk(p0)ne 1 for k 1 2 3 4(ii) (ddp)|μ(p)| d(p0) minus (12(1 + 2q))lt 0 at

p p0

(iii) Eigenvectors associated to the μ are

q p0( 1113857 12

minusi

3

radic

2 11113888 1113889

T

p p0( 1113857 i3

radic 16

(3 minus i3

radic)1113888 1113889

(52)

such that pA μp Aq μq and pq 1 whereA JacF(0 0)

Proof Let p p0 q +en we obtain

A JacF(0 0)

0 1

minus1 + 2q

x

1 + 2q

x

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0 1

minus 1 1⎛⎝ ⎞⎠ (53)

After straightforward calculation for p p0 q weobtain |μ(p0)| 1 and

μ p0( 1113857 12

+i

3

radic

2

μ2 p0( 1113857 minus12

+i

3

radic

2

μ3 p0( 1113857 minus 1

μ4 p0( 1113857 minus12

minusi

3

radic

2

(54)

from which it follows that μk(p0)ne 1 for k 1 2 3 4Furthermore we get

ddp

|μ(p)| minus

1 + 2q

1113968

2(p + q + 1)32

ddp

|μ(p)|1113888 1113889pp0

minus1

2(1 + 2q)lt 0

(55)

It is easy to see that p(p0)A μp(p0) andp(p0)q(p0) 1

Let p p0 + η where η is a sufficiently small parameterFrom Lemma 4 we can transform system (47) into thenormal form

F(λ x) F(λ x) + O x5

1113872 1113873 (56)

Repeller

Locally asymptotically stable

Nonhyperb

olic

0 1 2 3 40

1

2

3

4

(a)

Globally asymptotically stable

0 5 10 15 20 250

5

10

15

20

25

(b)

Figure 1 Local (a) and global (b) asymptotic stability in (p q) plane

6 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

ie

M f(M m m)

m f(m M M)1113896 ⟺

Mm2 pm2 + p2q + Mm + qM + q2M + pq2 + pq

mM2 pM2 + p2q + mM + qm + q2m + pq2 + pq1113896 (38)

By subtracting the second equation in (38) from the firstwe get

(M minus m) q minus Mp minus mp + q2

+ Mm1113872 1113873 0 (39)

from which m M or for mneM

(m minus p)M mp minus q minus q2 (40)

If m p then

0 p2

minus q minus q2 (41)

Since p2 minus q minus q2 gt 0 for pgt q + 1 it implies that mnep+erefore

M mp minus q minus q2

m minus p mnep (42)

By substituting (42) into (38) we obtain the followingquadratic equation

m2(p minus q minus 1)(p + q) minus q 2p minus q + 2pq + p

2minus 11113872 1113873m

+ q q + p2q + p

2+ p

3+ 2q

2+ q

31113872 1113873 0

(43)

whose discriminant has the form

D(p q) q 3q minus p2

+ 2q2

+ 11113872 1113873 q + 3p2q + 4p

3+ 3q

2+ 2q

31113872 1113873

(44)

It is clear that D(p q)lt 0 if pgt(q + 1)(2q + 1)

1113968and

that there are no real solutions for equation (43) Ifp

(q + 1)(2q + 1)

1113968 then m M p + q + 1 which is a

contradiction with assumption mneM Whenq + 1ltplt

(q + 1)(2q + 1)

1113968 equation (43) has two positive

roots mplusmn gtp it implies that Mplusmn gt 0 (for examplem 30032 M 95212 for p 3 and q 199 where(m M) is in invariant interval I) +erefore the conditionsof +eorem A05 in [14] are satisfied forpge

(q + 1)(2q + 1)

1113968and every solution of equation (22)

converges to x p + q + 1 By Remark 1 the uniqueequilibrium x p + q + 1 of equation (7) is an attractor Byusing Lemma 1 we conclude that x is globally asymptoticallystable (see Figure 1)

Conjecture 1 If pgt q then the equilibrium point x p +

q + 1 is globally asymptotically stable

3 NeimarkndashSacker Bifurcations

+e following results are obtained by applying the algorithmfrom +eorem 1 and Corollary 1 in [5] (see also [22]) If wemake a change of variable yn xn minus x we will shift theequilibrium point to the origin +en the transformedequation is given by

yn+1 yn + p + 1 + q

ynminus 1 + p + 1 + qminus 1

+ qyn + p + 1 + q( 1113857

2

ynminus 1 + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠ n 0 1

(45)

Set

un ynminus 1

vn yn

for n 0 1

(46)

and write equation (1) in the equivalent formun+1 vn

vn+1 vn + p + 1 + q

un + p + 1 + qminus 1 + q

vn + p + 1 + q( 11138572

un + p + 1 + q( 11138572 minus 1⎛⎝ ⎞⎠

(47)

Let F denote the corresponding map defined by

F

u

v

⎛⎝ ⎞⎠

v

v + p + 1 + q

u + p + 1 + qminus 1 + q

(v + p + 1 + q)2

(u + p + 1 + q)2minus 11113888 1113889

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)

+en the Jacobian matrix of F is given by

JacF(u v)

0 1

minus(p + q + v + 1)(2q(p + q + v + 1) + p + q + u + 1)

(p + q + u + 1)32q(p + q + v + 1) + p + q + u + 1

(p + q + u + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (49)

Discrete Dynamics in Nature and Society 5

+e eigenvalues of JacF(0 0) are μ(p) and μ(p) where

μ(p) 1 + 2q + i

(1 + 2q)(3 + 2q + 4p)

1113968

2(p + q + 1)

|μ(p)|

1 + 2q

p + q + 1

1113971

(50)

Lemma 4 If p p0 q then F has equilibrium point at(0 0) and eigenvalues of Jacobian matrix of F at (0 0) are μand μ where

μ p0( 1113857 12

+i

3

radic

2 (51)

Moreover μ satisfies the following

(i) μk(p0)ne 1 for k 1 2 3 4(ii) (ddp)|μ(p)| d(p0) minus (12(1 + 2q))lt 0 at

p p0

(iii) Eigenvectors associated to the μ are

q p0( 1113857 12

minusi

3

radic

2 11113888 1113889

T

p p0( 1113857 i3

radic 16

(3 minus i3

radic)1113888 1113889

(52)

such that pA μp Aq μq and pq 1 whereA JacF(0 0)

Proof Let p p0 q +en we obtain

A JacF(0 0)

0 1

minus1 + 2q

x

1 + 2q

x

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0 1

minus 1 1⎛⎝ ⎞⎠ (53)

After straightforward calculation for p p0 q weobtain |μ(p0)| 1 and

μ p0( 1113857 12

+i

3

radic

2

μ2 p0( 1113857 minus12

+i

3

radic

2

μ3 p0( 1113857 minus 1

μ4 p0( 1113857 minus12

minusi

3

radic

2

(54)

from which it follows that μk(p0)ne 1 for k 1 2 3 4Furthermore we get

ddp

|μ(p)| minus

1 + 2q

1113968

2(p + q + 1)32

ddp

|μ(p)|1113888 1113889pp0

minus1

2(1 + 2q)lt 0

(55)

It is easy to see that p(p0)A μp(p0) andp(p0)q(p0) 1

Let p p0 + η where η is a sufficiently small parameterFrom Lemma 4 we can transform system (47) into thenormal form

F(λ x) F(λ x) + O x5

1113872 1113873 (56)

Repeller

Locally asymptotically stable

Nonhyperb

olic

0 1 2 3 40

1

2

3

4

(a)

Globally asymptotically stable

0 5 10 15 20 250

5

10

15

20

25

(b)

Figure 1 Local (a) and global (b) asymptotic stability in (p q) plane

6 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

+e eigenvalues of JacF(0 0) are μ(p) and μ(p) where

μ(p) 1 + 2q + i

(1 + 2q)(3 + 2q + 4p)

1113968

2(p + q + 1)

|μ(p)|

1 + 2q

p + q + 1

1113971

(50)

Lemma 4 If p p0 q then F has equilibrium point at(0 0) and eigenvalues of Jacobian matrix of F at (0 0) are μand μ where

μ p0( 1113857 12

+i

3

radic

2 (51)

Moreover μ satisfies the following

(i) μk(p0)ne 1 for k 1 2 3 4(ii) (ddp)|μ(p)| d(p0) minus (12(1 + 2q))lt 0 at

p p0

(iii) Eigenvectors associated to the μ are

q p0( 1113857 12

minusi

3

radic

2 11113888 1113889

T

p p0( 1113857 i3

radic 16

(3 minus i3

radic)1113888 1113889

(52)

such that pA μp Aq μq and pq 1 whereA JacF(0 0)

Proof Let p p0 q +en we obtain

A JacF(0 0)

0 1

minus1 + 2q

x

1 + 2q

x

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0 1

minus 1 1⎛⎝ ⎞⎠ (53)

After straightforward calculation for p p0 q weobtain |μ(p0)| 1 and

μ p0( 1113857 12

+i

3

radic

2

μ2 p0( 1113857 minus12

+i

3

radic

2

μ3 p0( 1113857 minus 1

μ4 p0( 1113857 minus12

minusi

3

radic

2

(54)

from which it follows that μk(p0)ne 1 for k 1 2 3 4Furthermore we get

ddp

|μ(p)| minus

1 + 2q

1113968

2(p + q + 1)32

ddp

|μ(p)|1113888 1113889pp0

minus1

2(1 + 2q)lt 0

(55)

It is easy to see that p(p0)A μp(p0) andp(p0)q(p0) 1

Let p p0 + η where η is a sufficiently small parameterFrom Lemma 4 we can transform system (47) into thenormal form

F(λ x) F(λ x) + O x5

1113872 1113873 (56)

Repeller

Locally asymptotically stable

Nonhyperb

olic

0 1 2 3 40

1

2

3

4

(a)

Globally asymptotically stable

0 5 10 15 20 250

5

10

15

20

25

(b)

Figure 1 Local (a) and global (b) asymptotic stability in (p q) plane

6 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

and there are smooth functions a(λ) b(λ) and ω(λ) so thatin polar coordinates the function F(λ x) is given by

r

θ1113888 1113889

|μ(λ)|r minus a(λ)r3

θ + ω(λ) + b(λ)r21113888 1113889 (57)

Now we compute a(p0) following the procedure in [22]Notice that p p0 if and only if η 0 First we computeK20 K11 and K02 defined in [22] For p p0 q we have

Fu

v1113888 1113889 A

u

v1113888 1113889 + G

u

v1113888 1113889 (58)

where

Gu

v

⎛⎝ ⎞⎠ ≔

0

v + 2q + 1u + 2q + 1

minus 1 + qv + 2q + 1u + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ + u minus v

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (59)

Hence for p p0 system (47) is equivalent toun+1

vn+11113888 1113889 A

un

vn

1113888 1113889 + Gun

vn

1113888 1113889 (60)

Define the basis of R2 by Φ (q q) whereq q(p0) (12) minus (i

3

radic2) then we can represent (u v)

as

u

v

⎛⎝ ⎞⎠ Φz

z

⎛⎝ ⎞⎠ (qz + qz)

1 + i3

radic

2z +

1 minus i3

radic

2z

z + z

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(61)

Let

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠ 12

g20z2

+ 2g11zz + g02z2

1113872 1113873 + O |z|3

1113872 1113873

(62)

Now we have

G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

0

z + z + 2q + 1((1 + i

3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

+ qz + z + 2q + 1

((1 + i3

radic)2)z +((1 minus i

3

radic)2)z + 2q + 1

1113888 1113889

2

minus 1⎛⎝ ⎞⎠ minus1 minus i

3

radic

2z minus

1 + i3

radic

2z minus 1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(63)

Since

g20 z2

zz2 G Φz

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 + i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g11 z2

zz zzG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

g02 z2

z2zG Φ

z

z

⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

z0

0

minus 5q minus 2 minus i3

radicq

(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(64)then

K20 μ2I minus A1113872 1113873minus 1

g20

(1 + i3

radic)(minus 5q minus 2 + i

3

radicq)

4(2q + 1)2

5q + 2 minus i3

radicq

2(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K11 (I minus A)minus 1

g11

4q + 1(2q + 1)2

4q + 1(2q + 1)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K02 μ2I minus A1113872 1113873minus 1

g02 K20

(65)

By using K20 K11 and K02 we have

Discrete Dynamics in Nature and Society 7

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 4 6 8 10 12

2

4

6

8

10

12

(b)

Figure 2 Trajectories (a) and invariant curve (b) for p 29 and q 3

0 500 1000 1500 2000 2500

5

10

15

(a)

0 500 1000 1500 2000 2500

5

10

15

(b)

Figure 3 Orbits with initial conditions (x0 xminus 1) (21 21) (a) and (x0 xminus 1) (68 73) (b) for p 29 and q 3

2 4 6 8 10 12

2

4

6

8

10

12

(a)

2 3 4 5 62

3

4

5

6

(b)

Figure 4 Trajectories for p 32 and q 3 (a) and p 15 and q 15 (b)

8 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

g21 z3

zz2zzG Φ

z

z

⎛⎝ ⎞⎠ +12

K20z2

+ 2K11zz + K02z2

1113872 1113873⎛⎝ ⎞⎠

z0

0

minus2i

3

radicq2

(2q + 1)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (66)

145 150 155 160 165

4

6

8

10

12

14

00 05 10 15 20 25 30 350

20

40

60

80

100

28 29 30 31 324

6

8

10

12

14

060 065 070 075 080 085

1000

2000

3000

4000

5000

(a)

145 150 155 160 165

0000

0001

0002

0003

0004

00 05 10 15 20 25 30 35

ndash005

ndash004

ndash003

ndash002

ndash001

000

28 29 30 31 32ndash0014

ndash0012

ndash0010

ndash0008

ndash0006

ndash0004

ndash0002

0000

060 065 070 075 080 085ndash004

ndash003

ndash002

ndash001

000

(b)

Figure 5 Bifurcation diagrams in (p x) plane (a) and corresponding Lyapunov exponent (b) for values of parameter q 3

Discrete Dynamics in Nature and Society 9

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

Finally we get

a p0( 1113857 12Re pg21μ( 1113857 minus

q2

(2q + 1)4lt 0 (67)

If x is fixed point of F then invariant curve Γ(λ) can beapproximated by

x1

x21113888 1113889 asymp x + 2ρ0Re qe

iθ1113872 1113873 + ρ20 Re K20e

2iθ1113872 1113873 + K111113872 1113873 (68)

where

d ddη

|μ(λ)|λλ0

ρ0

minusd

aη

1113971

θ isin R

(69)

+us we prove the following result

Theorem 2 Let x p + q + 1 then there is a neighborhoodU of the equilibrium point x and δ gt 0 such that for |p minus q|lt δand x0 xminus 1 isin U ω-limit set of solution of equation (7) is theequilibrium point if pgt q and belongs to a closed invariant C1

curve Γ(p) encircling the equilibrium point if plt q Fur-thermore Γ(q) 0 and invariant curve Γ(p) can be ap-proximated by

x1

x2

⎛⎝ ⎞⎠ asymp

1 + p + q +

(q minus p)(2q + 1)3

2q2

1113971

(cos t +3

radicsin t) minus

(p minus q)(2q + 1)(8q minus (4q + 1)cos 2 t +3

radic(2q + 1)sin 2 t + 2)

4q2

1 + p + q +

2(q minus p)(2q + 1)3

q2

1113971

cos t minus(p minus q)(2q + 1)(

3

radicq sin 2 t + 8q +(5q + 2)cos 2 t + 2)

4q2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(70)

Since a(p0)ne 0 a nondegenerate NeimarkndashSacker bi-furcation occurs at the critical value p0( q) We proveda(p0)lt 0 and d(p0)lt 0 so it implies if pltp0 then anattracting closed curve exists surrounding the unstable fixedpoint when the parameter p crosses the bifurcation value p0(supercritical NeimarkndashSacker bifurcation) As p increasesthe attracting closed curve decreases in size and merges withthe fixed point at p p0 leaving a stable fixed point(subcritical NeimarkndashSacker bifurcation) All orbits startingoutside or inside the closed invariant curve except at theorigin tend to the attracting closed curve

+e asymptotic approximation of the invariant curve isshown in Figure 2 and some orbits and trajectories in the casewhere the unique equilibrium point is stable or nonhyperbolicare given in Figures 3 and 4 Using parameter value q 3 anddecreasing the dynamical parameter p the unique positiveequilibrium point loses its stability via NeimarkndashSacker bi-furcation leading to chaos as depicted in Figure 5(a) Also theLyapunov exponents corresponding to Figure 5(a) are shownin Figure 5(b) which verifies the existence of chaos after theoccurrence of NeimarkndashSacker bifurcation

4 Conclusion

+e investigation of the dynamic stability of homogeneousdifference equation (3) and all its special cases is very

complicated +e corresponding function f(u v) associatedwith the right side of equation (3) is monotonically in-creasing in the first variable andmonotonically decreasing inthe second variable or monotonically decreasing in the firstvariable and monotonically increasing in the second variableor it switches its type of monotonicity between the first andsecond case or vice versa depending on the parameterswhich appear in the function

+e theory of monotone maps (or more precisely thetheory of competitive maps) was used for determining thedynamics of equation (3) or some special case of equation(3) in the scenario when corresponding function f(u v) ismonotonically decreasing in the first variable and mono-tonically increasing in the second variable (see [2 3])

+e investigation of the special cases of equation (3)when corresponding function f is monotonically increasingin the first variable and monotonically decreasing in thesecond variable is significantly harder because there is nogeneral result for this type of monotonicity

In every situation when the corresponding equationdoes not possess minimal period-two solutions the globalstability of a unique equilibrium usually can be determinedby applying the so-called ldquoM-mrdquo theorems and finding aninvariant interval of the map f before that of courseHowever it is very often impossible or extremely compli-cated to conduct as we saw in [1] +at is the case with

10 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

equation (1) which does not possess minimal period-twosolution (since the corresponding function is monotonicallyincreasing by first and monotonically decreasing by secondvariable) So for that reason we studied equation (7) insteadof equation (1) By using the method of embedding we wereable to connect equation (7) with equation (15) Namely wehave shown that every solution xn

prime1113864 1113865infinnminus 1 of equation (7) is

also a solution xn1113864 1113865infinnminus 2 of equation (15) with initial con-

ditions xminus 3 xminus 1prime xminus 2 x0prime xminus 1 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2 and x0 p + (x1primex0prime) + q(x1primex0prime)2 Furthermore we

have shown that every solution of equation (15) is boundedwhich implies that every solution of equation (7) is alsobounded After that we linked equation (15) with equation(22) and showed that every solution xn

prime1113864 1113865infinnminus 1 of equation (7)

is also a solution of equation (22) with initial conditionsxminus 2 xminus 1prime xminus 1 x0prime and x0 p + (x0primexminus 1prime) + q(x0primexminus 1prime)

2Additionally we determined the invariant and attractinginterval for the function that is associated with the right sideof equation (22) and successfully applied ldquoM-mrdquo theorem toget conditions for parameters p and q under which theequilibrium of equation (22) and therefore of equation (7) isglobally asymptotically stable +e area of the regions in(p q) plane where unique equilibrium is locally asymp-totically stable and is not globally asymptotically stable issmall (see Figure 1) We expect to prove Conjecture 1 insome of our future studies

Finally using NeimarkndashSacker theorem [15 18ndash21] wegave reduction to the normal form and performed com-putation of the coefficients of bifurcation based on thecomputational algorithm developed in [22] Furthermorewe determined the asymptotic approximation of the in-variant curve and provided visual evidence (see Figures 2ndash5)Also based on the computational algorithm in [23] wecalculated the Lyapunov exponents corresponding toFigure 5(a) to confirm the existence of chaos after the oc-currence of NeimarkndashSacker bifurcation as in [24 25] (seeFigure 5(b))

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported in part by the Fundamental Re-search Funds of Bosnia and Herzegovina (FMON no 01-6211-1-IV19)

References

[1] M Garic-Demirovic S Hrustic and M Nurkanovic ldquoSta-bility and periodicity of certain homogeneous second-orderfractional difference equation with quadratic termsrdquo Ad-vances in Dynamical Systems and Applications vol 14 no 2pp 149ndash178 2019

[2] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoGlobal dynamics of certain homogeneous second-orderquadratic fractional difference equationsrdquo e ScientificWorld Journal vol 2013 p 10 Article ID 210846 2013

[3] M Garic-Demirovic M R S Kulenovic and M NurkanovicldquoBasins of attraction of certain homogeneous second orderquadratic fractional difference equationrdquo Journal of Concreteand Applicable Mathematics vol 13 no 1-2 pp 35ndash50 2015

[4] M Garic-Demirovic M Nurkanovic and Z NurkanovicldquoStability periodicity and Neimark-Sacker bifurcation ofcertain homogeneous fractional difference equationrdquo Inter-national Journal of Difference Equations vol 12 no 1pp 27ndash53 2017

[5] T Khyat M R S Kulenovic and E Pilav ldquo+e Naimark-Sacker bifurcation and asymptotic approximation of the in-variant curve of a certain difference equationrdquo Journal ofComputational Analysis and Applications vol 23 no 8pp 1335ndash1346 2017

[6] S Jasarevic Hrustic M R S Kulenovic and M NurkanovicldquoGlobal dynamics and bifurcations of certain second orderrational difference equation with quadratic termsrdquo Qualita-tive eory of Dynamical Systems vol 15 no 1 pp 283ndash3072016

[7] S Kalabusic M Nurkanovic and Z Nurkanovic ldquoGlobaldynamics of certain mix monotone difference equationrdquoMathematics vol 6 no 1 p 10 2018

[8] M R S Kulenovic S Moranjkic M Nurkanovic andZ Nurkanovic ldquoGlobal asymptotic stability and Neimark-Sacker bifurcation of certain mix monotone differenceequationrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 7052935 22 pages 2018

[9] M R S Kulenovic S Moranjkic and Z NurkanovicldquoNaimark-Sacker bifurcation of second order rational dif-ference equation with quadratic termsrdquo e Journal ofNonlinear Sciences and Applications vol 10 no 7pp 3477ndash3489 2017

[10] M R S Kulenovic S Moranjkic and Z Nurkanovic ldquoGlobaldynamics and bifurcation of perturbed Sigmoid Beverton-Holt difference equationrdquo Mathematical Methods in theApplied Sciences vol 39 pp 2696ndash2715 2016

[11] S Moranjkic and Z Nurkanovic ldquoLocal and global dynamicsof certain second-order rational difference equations con-taining quadratic termsrdquo Advances in Dynamical Systems andApplications vol 12 no 2 pp 123ndash157 2017

[12] H Sedaghat ldquoGlobal behaviours of rational differenceequations of orders two and three with quadratic termsrdquoJournal of Difference Equations and Applications vol 15 no 3pp 215ndash224 2009

[13] E J Janowski and M R S Kulenovic ldquoAttractivity and globalstability for linearizable difference equationsrdquo e Journal ofComputational and Applied Mathematics vol 57 no 9pp 1592ndash1607 2009

[14] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations with Open Problems andConjectures Chapman and HallCRC Boca Raton FL USA2001

[15] M R S Kulenovic and O Merino Discrete Dynamical Sys-tems and Difference Equations with Mathematica Chapmanand HallCRC Boca Raton FL USA 2000

[16] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a two dimensional linear fractional system of differenceequationsrdquo Radovi Matematicki (Sarajevo Journal of Math-ematics) vol 11 no 1 pp 59ndash78 2002

Discrete Dynamics in Nature and Society 11

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society

[17] M R S Kulenovic andM Nurkanovic ldquoAsymptotic behaviorof a system of linear fractional difference equationsrdquo Journalof Inequalities and Applications vol 2005 no 2 pp 127ndash1432005

[18] J K Hale and H Kocak ldquoDynamics and bifurcationsrdquo in Textin Applied Mathematics Vol 3 Springer-Verlag New YorkNY USA 1991

[19] Y Kuznetsov Elements of Applied Bifurcation eorySpringer New York NY USA 1998

[20] S Robinson Stability Symbolic Dynamics and Chaos CRCPress Boca Raton FL USA 1995

[21] S Wiggins Introduction to Applied Nonlinear DynamicalSystems and Chaos Text in Applied Mathematics Vol 2Springer-Verlag New York NY USA Second edition 2003

[22] KMurakami ldquo+e invariant curve caused by Neimark-Sackerbifurcationrdquo Dynamics of Continuous Discrete and ImpulsiveSystems vol 9 pp 121ndash132 2002

[23] M Sandri ldquoNumerical calculation of Lyapunov exponentsrdquoe Mathematica Journal vol 6 pp 78ndash84 1996

[24] A Al-khedhairi A E Matouk and S S Askar ldquoBifurcationsand chaos in a novel discrete economic systemrdquo Advances inMechanical Engineering vol 11 no 4 pp 1ndash15 2019

[25] A E Matouk A A Elsadany and B Xin ldquoNeimark-Sackerbifurcation analysis and complex nonlinear dynamics in aheterogeneous quadropoly game with an isoelastic demandfunctionrdquo Nonlinear Dynamics vol 89 no 4 pp 2533ndash25522017

12 Discrete Dynamics in Nature and Society