A New Multi-wing Chaotic Attractor with Constant

Lyapunov Exponents from Sprott-C System

Zhongtang Wua,b, Minghua Liuc, Mengjiao Wanga, Jiuchao Fenga

aSouth China University of Technology, Guangzhou, ChinabShenzhen University, Shenzhen, ChinacJinggangshan University, Jian, China

Abstract

This paper initiates an approach for generating novel multi-wing chaoticattractor from Sprott-C System. The new multi-wing attractor has constantLyapunov exponents as the system parameter varies. Compared with thetraditional multi-wing Lorenz chaotic attractors, the proposed multi-wingchaotic attractors are much easier to be constructed and implemented byanalog circuits. Furthermore, a module-based circuit diagram is designed forrealizing various multi-wing attractors.

Keywords: multi-wing attractors, sawtooth wave function, constantLyapunov exponents, circuit implementation

1. Introduction

It is very important to design new chaotic attractors and enhance exist-ing chaotic attractors to generate more complex dynamics and topologicalstructure in the field of chaos theory[1-3]. Sometimes it is a key issue formany engineering applications.

Email address: fengjc@scut.edu.cn (Jiuchao Feng)1This paper is supported by the National Natural Science Foundation of China (Grant

No. 60872123, 61101014), the Joint Fund of the National Natural Science Foundation andthe Guangdong Provincial Natural Science Foundation (Grant No. U0835001), the fundfor Higher-level Talent in Guangdong Province, China (No. N9101070),science and tech-nology in Jiangxi province department of education project (GJJ13542), and JinggangshanUniversity natural science fund project (JZB1203)

Preprint submitted to The Science World Journal January 11, 2014

There are two important e!orts in general. One is to generate multi-scrollattractor by Chua’ s circuit, the other is to generate multi-wing attractor fromLorenz-like system[4]. The generalized Chua’ s circuit can generate variousmulti-scroll chaotic attractor by using many di!erent non-linear functions [5-19], such as hysteresis [5, 10, 11], saturated[6], switched[7, 13] threshold[9],sawtooth[14], multi-segment piecewise-linear [15, 16], trigonometric [17], hy-perbolic [18, 19], absolute functions [19] and so on. Up to today, there areonly several methods to generate multi-wing attractor mainly from Lorenz-like system[20-26], and the fundamental theories are still not formed. Theobjective of this paper is to present some new findings in successfully gen-erating multi-wing attractor from a Sprott-C System. It is more interestingthat the multi-wing attractor has constant Lyapunov exponents.

The organization of the paper is as follows. The new multi-wing chaoticattractor is introduced in Sect.2. Sect.3 analyzes the basic dynamical behav-iors of multi-wing chaotic attractor. The eight-wing and ten-wing attractorsare introduced in Sect.4. Circuit implementation of the eight-wing attractorsis demonstrated in Sect.5. Finally, conclusions are given in Sect.6.

2. The multi-wing chaotic attractor

The model of the new multi-wing attractor can be obtained by modifyingthe Sprott-C System with f(u) function. The Sprott-C System is defined bysystem (1). f(x) is sawtooth wave function defined by equation (3).

x = yzy = x! yz = 1! x2

(1)

x = z(y ! f(y))y = x! (y ! f(y))z = p! x2

(2)

f(x) =M!

i=!M

sign(x! 2iG) (3)

When p = 1, M = 1, G = 1, the system(2) is chaotic and its threeLyapunov exponents are L1 = 0.2472, L2 = !0.14194, L3 = !1.1052. Thephase diagrams of new attractor are shown in Figure 1.

2

−3−2

−10

12

3

−5

0

5−3

−2

−1

0

1

2

3

xy

z

(a)

−3 −2 −1 0 1 2 3−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

(b)

−5 −4 −3 −2 −1 0 1 2 3 4 5−3

−2

−1

0

1

2

3

y

z

(c)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x

z

(d)

Figure 1: Phase diagrams of system (2). (a) a three-dimensional view. (b)x! y plane. (c)x! z plane. (d) y ! z plane

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3. Dynamical behaviors of the new multi-wing chaotic attractor

3.1. Equilibrium Points

When M = 1, G = 1 and p = 1, the f(y) defined by (5) and one canobtain the equation (4). The equilibrium points can be computed to obtainas P1, P2 . . .P8. P1 = (1,!2, 0), P2 = (1, 0, 0), P3 = (1, 2, 0), P4 = (1, 4, 0),P5 = (!1,!4, 0), P6 = (!1,!2, 0), P7 = (!1, 0, 0), P8 = (!1, 2, 0)

z(y ! f(y)) = 0x! (y ! f(y)) = 01! x2 = 0

(4)

f(y) =

"

#

#

$

#

#

%

y + 3, y " !2y + 1,!2 < y " 0y ! 1, 0 < y " 2y ! 3, y > 2

(5)

By linearizing system (2)at any equilibrium point (x0, y0, z0), one obtainsthe Jacobian matrix J0

J0 =

&

'

0 z0 y0 ! f(y0)1 !1 0

!2x0 0 0

(

) . (6)

Because x0 = y0 ! f(y0), so

J0 =

&

'

0 z0 x0

1 !1 0!2x0 0 0

(

) . (7)

|!I ! J0| =

*

*

*

*

*

*

! !z0 !x0

!1 !+ 1 02x0 0 !

*

*

*

*

*

*

. (8)

when x0 = 1, !1 = 1.25, !2 = !1.80 and !3 = !0.45. when x0 = !1,!1 = 1.55, !2 = !1.27 + 0.56i and !3 = !1.27 ! 0.56i. The equilibriumpoints and their stability are defined in Table 1.

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3.2. Lyapunov exponents, Lyapunov dimension, and Poincare Mappings

The three Lyapunov expoments are L1 = 0.2472, L1 = !0.14194, L1 =!1.1052 when M = 1, G = 1 and p = 1. The Lyapunov dimension of system(1) is described as

DL = j +1

|Lj+1|

j!

i=1

Li

= 2 +L1 + L2

|L3|

= 2 +0.2472! 0.14194

1.1052= 2.0952

When parameter p is varying, and the largest Lyapunov exponent alsovaries. The largest Lyapunov exponent spectrum is shown in Figure 2, andit is easy to know that the system (2) has constant Lyapunov exponents.Poincare mappings are shown in Figure 3. From the phase portraits andPoincare mappings of system (2), it can be demonstrated that the new multi-wing attractor is chaotic.

4. Ten-wing and 2N -wing Attractors

When M = 2, G = 1, p = 1, the system(2) is ten-wing chaotic attractorand the phase diagrams are shown in Figure 4. Set M = 1 + (N ! 1)/2,G = 1 and p = 1, the system (2) is 2N -wing chaotic system.

equilibrium point Eigenvalues Stability

P1 = (1,!2, 0) !1 > 0, !2 < 0, !3 < 0 stable nodeP2 = (1, 0, 0) !1 > 0, !2 < 0, !3 < 0 stable nodeP3 = (1, 2, 0) !1 > 0, !2 < 0, !3 < 0 stable nodeP4 = (1, 4, 0) !1 > 0, !2 < 0, !3 < 0 stable nodeP5 = (!1,!4, 0) !1 > 0, Re(!2) < 0, Re(!3) < 0 stable focusP6 = (!1,!2, 0) !1 > 0, Re(!2) < 0, Re(!3) < 0 stable focusP7 = (!1, 0, 0) !1 > 0, Re(!2) < 0, Re(!3) < 0 stable focusP8 = (!1, 2, 0) !1 > 0, Re(!2) < 0, Re(!3) < 0 stable focus

Table 1: Equilibria of the system and its stability

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5. Circuit Implementation

The Circuit of the new multi-wing attractor is designed by the modularmethod proposed [26] and the circuit diagram is shown in Figure 5. The cir-cuit mainly consists of two parts named as the N1 and N2. N1 is a nonlinearfunction generator, and N2 is calculus function circuit of the chaotic systems.The operational amplifier in the circuit model is TL082CP, and multiplier isAD633JN, which voltage gain is 0.1. The power voltage is ±15 v.

Figure 6 depicts the multi-wing attractor obtained from the circuit builtin figure 5. It is observed that the number of wings can be easily controlled bythe proposed sawtooth wave function. In addition, the experimental resultsare well matched with those obtained from numerical simulations as shownin figure 5(a),(b) and (c).

6. Conclusions

The multi-wing attractor is generated from Sprott-C System with saw-tooth wave function, and the new attractor has constant Lyapunov exponentswhen the parameter p is varying. This approach enables 2N additional wingsof new attractor by creating 2N additional equilibrium points. The Lyapunovexponents and Poincare mappings of the six-wing attractor are investigatedby numerical simulation. Moreover, ten-wing attractor in Sprott-C Systemwith sawtooth wave function is also revealed. These research results are veri-fied by circuit experiment. More detailed analysis will be provided elsewherein the near future.

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

p

Lyap

unov

exp

onen

ts

Figure 2: Lyapunov exponent spectrum

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−6 −4 −2 0 2 4 6−5

−4

−3

−2

−1

0

1

2

3

4

5

z

y

(a)

−3 −2 −1 0 1 2 3−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

z

(b)

−4 −3 −2 −1 0 1 2 3−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

(c)

Figure 3: Poincare Mappings. (a) x = 1. (c) y = !2. (d)z = 0

7

−3 −2 −1 0 1 2 3 4−8

−6

−4

−2

0

2

4

6

8

x

y

(a)

−8 −6 −4 −2 0 2 4 6 8−5

−4

−3

−2

−1

0

1

2

3

4

5

y

z

(b)

−3 −2 −1 0 1 2 3 4−5

−4

−3

−2

−1

0

1

2

3

4

5

x

z

(c)

−4−2

02

4

−10−5

05

10−5

0

5

xy

z

(d)

Figure 4: Phase diagrams for (a)x-y-z plane, (b) x-y plane, (c)y-z plane (d) x-z plane

8

Figure 5: Circuit model.

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