the cusp-cusp bifurcationbpeckham/export/cctalk.pdf · 2007. 10. 18. · the cusp-cusp bifurcation...

The Cusp-cusp Bifurcationfor noninvertible maps of the plane

Bruce PeckhamDept. of Mathematics and Statistics

University of Minnesota, DuluthDuluth, Minnesota 55812

bpeckham@d.umn.edu

Coauthors: Bernd Krauskopf and Hinke OsingaUniversity of Bristol

J0

J0

J1

W

f(W )

z

y

x

C1

C0

S

.

.

Midwest Dynamical Systems Conference, October 21, 2007, The University of Michigan, Ann Arbor

1

Talk Outline

1. Background on noninvertible maps of the plane: some features and bifurca-tions

2. The codimension-two cusp-cusp bifurcation

(a) The two cusps

(b) The normal form of the model

(c) Definition of equivalence

(d) Analysis of the model: five codimension-one bifurcations, eight genericcases

3. An adaptive control application

4. Summary

2

Some research groups in noninvertible maps of the plane

• Gumowski and Mira 1980 book Recurrences and Discrete Dynamic Sys-

tems

(x, y) 7→ (ax + y, b + x2)

• Christian Mira “group”: Laura Gardini, Millerioux, Barugola, Cathala, Car-casses, Ralph Abraham. Several books and lots of articles.

• Yannis Kevrekidis “group”: Christos Frouzakis, Ray Adomaitis, Rafael deLlave, Rico-Martinez, BP

• Dick McGehee group: Evelyn Sander, Josh Nien, Rick Wicklin, ...

• Lorenz (1989 paper):

(x, y) 7→ ((1 − aτ )x − τxy, (1 − τ )y + τx2)

• Bristol Group: Bernd Krauskopf, Hinke Osinge, James England, BP, ...

• V. Maistrenko and Y. Maistrenko

A good introductory reference to Noninvertible maps of the plane: On some

properties of invariant sets of two-dimensional noninvertible maps, Frouza-kis, Gardini, Kevrekidis, Millerioux, and Mira, IJBC, Vol 7, No. 6, (1997),1167-1194.

3

Features and bifurcationsunique to

NONINVERTIBLE maps of the plane

4

Some features unique to noninvertible maps of the plane

• Critical curves: J0(≈ LC−1), J1(≈ LC)J0 = {x : det(DF (x)) = 0}; J1 = F (J0)

• Folding of phase space: Z0 −Z2, Z1 < Z3, ... . Figs from FGKMM (1997).

5

Some features unique to noninvertible maps of the plane (cont.)

• Interaction of critical curves with

– fixed/periodic points (eigenvalue zero)

– Unstable manifolds (outsets): self-intersections, loops, cusps

– Stable manifolds (insets): disconnected, allowing disconnected and mul-tiply connected basins of attraction

– Invariant circles

• Chaotic attractors

6

Example: Lorenz (1989)(x, y) 7→ ((1 − aτ )x − τxy, (1 − τ )y + τx2)

0.25 0.45 0.65x

−0.2

0.0

0.2

0.4

0.6

y

0.4 0.5 0.6 0.70.1

0.2

0.3

0.4

0.5

0.6

y

0.2 0.4 0.6 0.8x

−0.3

−0.1

0.1

0.3

0.5

0.7

0.25 0.35 0.45 0.55 0.65 0.75−0.2

0.0

0.2

0.4

0.6

0.8

(a) (b)

(c) (d)

Γ

Γ

OO

O O

Frouzakis, Kevrekidis, P, 2003

0.25 0.35 0.45 0.55 0.65 0.75x

−0.2

0.0

0.2

0.4

0.6

y

(b)

(c)

J0

J1

J1

C

L(C)

L(C)

7

Some bifurcations unique to noninvertible maps of the plane

• Codimension-one Interactions of J0 with

1. fixed/periodic points - transition from orientation preserving to orienta-tion reversing,

2. creating self-intersections of W u via tangency (Mira: contact bifurca-

tion) or loop formation

3. creating disconnected or multiply connected basin boundaries

4. breakup of an invariant circle (loops on what “used to be” the smoothinvariant curve - Lorenz)

5. creating intersections of basin boundaries with J0

• Codimension-two

1. (0,1) eigenvalues (Josh Nien thesis with Dick McGehee - 1997) (Unfold-ings not finitely determined.)

2. Sander (2000) A transverse homoclinic point with no tangle

3. The Cusp-cusp bifurcation

8

Global bifurcation diagrams. Ex. Mira 1991. Note to Kevrekidis.

9

The CUSP-CUSP bifurcation

10

Motivation from FGKMM (1997): “Interactions in the neighborhood of a cusppoint”

11

The first cusp - along the phase space foldThe Cusp point C0 and its image C1. Persists under perturbation. The greenline field denotes the direction of the zero eigenspaces at points of J0.(a) and (b) General position; (c) and (d) normal form position

(a)

U

E

J0

J0

C0

(b)

V

J1

Z1

Z3

C1

(c)

E

J0

J0

0

0

y

x

C0

(d)

J1

0

0

y

z

Z1

Z3

C1

.

.

12

The second cusp - on the image of WCusps on images of curves which cross J0 tangent to the line field. Codimension-one occurrence.

(a)

Ex0

J0

W

J1

f(W )

Z0

f(x0)

Z2

(b)

Ex0

J0

W

J1

f(W )

Z0

f(x0)

Z2

(c)

Ex0

J0

W

J1

f(W )

Z0

f(x0)

Z2

.

.

Assume W is described locally by α(t) with α(t0) = x0 and tangent vectorα′(t). At f(x0), W is described locally by f(α(t)), and has tangent vectorDf(x0).α

′(t), which implies W is still smooth at f(x0) unlees α′(t) is an eigen-vector for eigenvalue zero at x0.

13

The normal forms of the cusp map and W (cont)

F :

(x

y

)7→(

z

y

)=

(−x3 − 3xy

y

). (1)

DF

(x

y

)=

[−3x2 − 3y −3x

0 1

]

is singular along the particularly simple critical curve

J0 := {y = −x2}. (2)

The image of the parabola J0 is the standard cusp

J1 := {z = ±2(√−y)3 | y ≤ 0}. (3)

A straightforward calculation shows that it has the second pre-image

J0 := {y = −14x

2}. (4)

The zero-eigenvector line field is horizontal (independent of x ∈ J0).

Normal form (truncated) of W :

W := {y = γ(x − a)2 + b}. (5)

14

The projection of the graph of the normal forms. The cusp mapsis fixed; W = {γ(x − a)2 + b} varies. The parameters a and b are the pimaryparameters while γ is a secondary parameter. We construct bifurcation dia-grams in the (a, b) plane for fixed γ. There turn out to be eight generic cases,depending on γ.

J0

J0

J1

W

f(W )

z

y

x

C1

C0

S

.

.

15

The organizing center for different values of γ

Focus on case (i): γ = 0.5.

(e)

(d)

(c)

(b)

(a)

(i)

(h)

(g)

(f)

i

ii

iii

iv

v

vi

vii

viii

.

.

16

All possible normal form configurations1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24

25 26

30 31 32

27 28

29

.

.

17

The cusp-cusp unfolding for Case VII (γ = 0.5)

2127

28

29

30 31

−2 0 2−3

0

1

a

b

vii

x

y

x

y

x

y

x

y

x

y

x

y

21

27

28

29

30

3131

30

29

28

27

21

z

y

z

y

z

y

z

y

z

y

z

y

21 27

28

29

3031

.

.

18

The codimension-one bifurcations:(a) Cusp transition (pass thru the cusp point C) (b) Loop creation (cusp onW at critical parameter value) (c) Intersecion at tangency (topological changein intersections in range)

(c)

x z

y y

J0J0 J1

I I

(b)

x z

y y

J0J0 J1

L L

(a)

x z

y y

J0J0 J1

C C

.

.

19

The codimension-one bifurcations (continued)

(a) tangency-creation (W tangent to J0) (b) Enter-exit (W tangent to J0)

(b)

x z

y y

J0J0 J1

E E

(a)

x z

y y

J0J0 J1

T T

.

.

Definition of equivalence:Toplogical equivalence of the two curves in the image.

20

Explicit computation of bifurcation curves: all parabolain (a,b)

1. The cusp transition, denoted by C, where the curve W passes through J0

at the pre-cusp point C0, which means that F (W ) passes exactly throughthe cusp point C1 on J1; The locus of this bifurcation in the (a, b)-plane isthe parabola

b = cC(γ) a2 = −γ a2. (6)

2. The loop-creation bifurcation, denoted by L, where W crosses J0 tangentto the (horizontal) line field E ;

b = cL(γ) a2 = −a2. (7)

3. The intersection-at-tangency bifurcation, denoted by I , where F (W ) self-intersects at a tangency point with J1;

b = cI(γ) a2 = −(9γ + 4) a2. (8)

4. The tangency-creation bifurcation, denoted by T , where W is tangent toJ0;

b = cT (γ) a2 = − γ

1 + γa2. (9)

5. The enter-exit bifurcation, denoted by E, where W is tangent to J0;

b = cE(γ) a2 = − γ

1 + 4γa2. (10)

21

Classification in the γ-Coefficient spaceThe eight cases, depending on γ, the quadratic coefficient in the normal formof the curve W := {y = γ(x − a)2 + b}:

−2 0 2−6

−4

−2

0

2

4

6

−1 −

2

3−

1

2−

1

3−

1

4 1

0−1

0

−

1

2−

1

3−

1

4

−1

−

1

3−

1

4

i ii iii iv v vi vii viii

¡¡µ1

2

3

4

5

6

7

78

9

9

10

10

11

11

11

12 12

13

14

14@@R

15

15

616

17

17

18

19

¡¡µ20

20

21

2122

27

AAK23

2824 29

24 29

2425

25 30

26

31

32

cC(γ)

cE(γ)

cL(γ)

cT (γ)

cI(γ)

γ

c∗(γ)

.

.

22

Adaptive Control Application

g :

(x

y

)7→(

−xy + η

βy + px(−yx+η−1)c+x2

), (11)

Frouzakis, Adomaitis, Kevrekidis, Golden and Ydstie 1992. (β = 1.0)Searched for cusp-cusp point numerically in (p, η) plane for c = 1.2, β = 1.0???In KOP: Fixed p = 0.81, searched in (η, β) plane. !!!

−0.18 −0.14 −0.1 −0.060.99

1.00

1.01

−1.5 −1.0 −0.50.1

0.2

0.3

−0.06 −0.04 −0.02 0

0.50

0.54

0.58

−0.06 −0.05

0.49

0.50

(a)

NS

SN

SN

γ

β(b)

p0

W u(p0)

@@R

C0

J0

J0

y

z

(c)

p1

W u(p1)

J1

y

z

@@IC1

(d)

W u(p1)

J1

y

z

.

.

23

Adaptive Control Application (continued)

Case VII!

(a) p0

W u(p0)

C0J0

J0

(b) p1

W u(p1)

J1

C1

−1 0 1

−1

0

1

×10−2 γc

βc

×10−5

21

@@R27

AAK

28

29

30

3131

30

29

¢¢

28

¡¡ª27

21

31

30

29282721

z

z

z

y

y

y yy

zz

z

z

z

y

y

y.

.

2127

28

29

30 31

−2 0 2−3

0

1

a

b

vii

x

y

x

y

x

y

x

y

x

y

x

y

21

27

28

29

30

3131

30

29

28

27

21

z

y

z

y

z

y

z

y

z

y

z

y

21 27

28

29

3031

.

.

Application Normal form

24

Future plans

• Find or create a better example - with period less than 30

• Implement algorithms for all codimension-one points, as well as codimension-two points

• Identify other codimension-two points

• Connect behavior inside Arnold tongues to behavior outside (irrational ro-tation arcs for invertible maps)

25

Summary:

• Systematic analysis of a codimension-two point using singularity theory

• Definition of equivalence class allowed identification of five codimension-onebifurcations

• Key transitions can be studied with only the first iterate of the map underinvestigation.

• Normal form allowed explicit computations of bifucation curves in (a, b) pa-rameter space, and classification diagram for cases I - VIII in (γ, coefficient)space.

• Allows classification of type I - VIII in applications

• Might provide a framework for analyzing other noninvertible bifurcations

26