Canards and Horseshoes in the Forced van der Pol...

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Equadiff 2003 Singular Perturbations Mini-Symposium

Canards and Horseshoesin the

Forced van der Pol EquationWarren Weckesser

Department of Mathematics

Colgate University

Warren Weckesser, Equadiff 2003 – p. 1

http://math.colgate.edu/~wweckesser/


Collaborators



Collaborators

❏ John Guckenheimer, Cornell University

❏ Kathleen Hoffman, University of Maryland, Baltimore County



Collaborators

❏ John Guckenheimer, Cornell University

❏ Kathleen Hoffman, University of Maryland, Baltimore County

❏ Ricardo Oliva, Lawrence Berkeley National Lab

❏ Undergraduates (REU Summer 2002 at Cornell)Katy Bold, Chantal Edwards, Saby Guharay, Judith Hubbard, Chris Lipa



Brief History

� � � � � � � � � � � � � � � � � � � � �

❏ 1926 van der Pol ( �): relaxation oscillations

❏ 1927 van der Pol and van der Mark: hysteresis and bistability

❏ 1940. . . Cartwright and Littlewood (chaos before “chaos”)

❏ 1949 Levinson: simplified piecewise linear model. Inspired...

❏ 1963 Smale: Horseshoe Map

❏ 1978 Levi: further simplified model, symbolic dynamics

❏ 1980’s Grasman (et al): Asymptotic analysis

❏ � � � Many more analytical and numerical studies



Rewrite in Standard Fast/Slow Form

� � � � � � � � � � � � � � � � � � � � �




� � � � � � � � � � � � � � � � � � � � �

New parameters: � � � � � , � ��

New variables:

� � � �, � � �, � � � � � � � � � � �

Then

� � � � ��

� � � � � � � � ��

� � �




� � � � � � � � � � � � � � � � � � � � �

New parameters: � � � � � , � ��

New variables:

� � � �, � � �, � � � � � � � � � � �

Then

� � � � ��

� � � � � � � � ��

� � �Symmetry: � � �, � � �, � � � � � �



Slow and Fast Subsystems

� � � � ��

� � � � � � � � ��

� � �




� � � � ��

� � � � � � � � ��

� � ��

Slow Subsystem (DAE)

� ��

� � � � � � � � ��

� � �




� � � � ��

� � � � � � � � ��

� � ��


� ��

� � � � � � � � ��

� � �

Eliminate � and desingularize

� � � � � � � � �

� � � � � � � � � �

(Time reversed for

! " ! # �.)




� � � � ��

� � � � � � � � ��

� � ��


� ��

� � � � � � � � ��

� � �


� � � � � � � � �

� � � � � � � � � �

(Time reversed for

! " ! # �.)

� � � � � � ��

� � � � � � � � � � ��

� � � �




� � � � ��

� � � � � � � � ��

� � ��


� ��

� � � � � � � � ��

� � �


� � � � � � � � �

� � � � � � � � � �

(Time reversed for

! " ! # �.)

� � � � � � ��

� � � � � � � � � � ��

� � � ��

Fast Subsystem

� � � ��

� � �,� � �




� � � � ��

� � � � � � � � ��

� � ��


� ��

� � � � � � � � ��

� � �


� � � � � � � � �

� � � � � � � � � �

(Time reversed for

! " ! # �.)

� � � � � � ��

� � � � � � � � � � ��

� � � ��

Fast Subsystem

� � � ��

� � �,� � �

Critical Manifold� $ � � � � �



Phase Space

0

0.2

0.4

0.6

0.8

1−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xθ

y



Phase Space

0

0.2

0.4

0.6

0.8

1−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xθ

y

Critical Manifold, Slow Subsystem



Phase Space

0

0.2

0.4

0.6

0.8

1−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xθ

y

Fast Subsystem



Phase Space

0

0.2

0.4

0.6

0.8

1−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

xθ

y

Example: Periodic Orbit



Phase Space

0

0.2

0.4

0.6

0.8

1−2

−1.5

−1

−0.5

0

0.5

1

1.52

−2

−1

0

1

2

x

θ

y



Phase Space

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2

0

2

x

θ

y



Phase Space

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

x



Canards at the Folded Saddle

❏ If % �, there is a pair of folded equilibria (pseudo-singular points) on eachfold. One is a saddle, the other is either a node or a spiral.





❏ Canards form at folded saddles.(Benoit 1983; Mischenko, Kolesov, Kolesov, & Rhozov 1994; Szmolyan & Wechselberger

2003)





❏ Canards form at folded saddles.(Benoit 1983; Mischenko, Kolesov, Kolesov, & Rhozov 1994; Szmolyan & Wechselberger

2003)

❏ Representative System:

� � � � � � �

� � � & � ' �

� & �Critical manifold is � � �

�

.The origin is a folded equilibrium.



Canards at a Folded Saddle

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1−0.5

00.5

1−1

−0.5

0

0.5

x

z

y




−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

z

Fol

d




−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

z

Fol

d

Maximal canard




−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

z

Fol

d



Forced van der Pol Poincaré Map

0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

Canard forms




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

“Jump away” canard




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

Maximal canard




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

“Jump back” canard




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

Canard ends




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y




0

0.2

0.4

0.6

0.8

1

−2−1.5−1−0.500.511.52

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

θ

Σ1

x

Σ2

y

Symmetry:( � ( $



Horseshoe in the Forced van der Pol System

“Cartoon”

No Canards No Canards

Jump Across Canards

Maximal Canard

Jump Back Canards

κε



Horseshoe in the Forced van der Pol System

“Cartoon”

No Canards No Canards

Jump Across Canards

Maximal Canard

Jump Back Canards

κε

Numerical Computation

0.15 0.2 0.25 0.3 0.35 0.4 0.45

−0.69

−0.67

−0.65

−0.63

θ

y

ε = 0.00010, a= 1.1000, ω= 1.5500



Bifurcations of Periodic Orbits

1.05 1.1 1.15

1.594

1.596

1.598

1.6

1.602

1.604

1.606

a

L2 N

orm

of x

3Sa: nn

3Sb: bb

3Sc: aa

3Aa: bn

3Ab: an 3Ac: ab

6a: bnbb

6b: anbb 6c: anbn

12a: bnbbbnbn

12b: anbbanan

12c: anbnabab




1.05 1.1 1.15

1.594

1.596

1.598

1.6

1.602

1.604

1.606

a

L2 N

orm

of x

3Sa: nn

3Sb: bb

3Sc: aa

3Aa: bn

3Ab: an 3Ac: ab

6a: bnbb

6b: anbb 6c: anbn

12a: bnbbbnbn

12b: anbbanan

12c: anbnabab Maximal Canard




1.05 1.1 1.15

1.594

1.596

1.598

1.6

1.602

1.604

1.606

a

L2 N

orm

of x

3Sa: nn

3Sb: bb

3Sc: aa

3Aa: bn

3Ab: an 3Ac: ab

6a: bnbb

6b: anbb 6c: anbn

12a: bnbbbnbn

12b: anbbanan

12c: anbnabab Example fromfull shift onthree symbols



Example: Maximal Canard Bifurcation

Periodic orbit at the limit point has a maximal canard.

0.239 4.218

−1

1

maximal canard: crosses fold, follows unstable manifold all the wayback to fold, then jumps

θ

x



Example: Maximal Canard Bifurcation

Periodic orbit on a branch leading to the left-most limit point (symbols bb ).

0.239 4.218

−1

1

canard: crosses fold

jumps back

jumps at fold

θ

x



Example: Periodic Orbits and Symbolic Dynamics

Periodic orbit with symbol sequence anbbanan .

0.55 14.87

−1

1

θ

x

a

n

b

b

a

n

a

n



Summary



Summary

❏ A detailed numerical study combined with an understanding of the fast/slowdynamics provides a clear picture of the horseshoe map in the forced vander Pol system.



Summary

❏ A detailed numerical study combined with an understanding of the fast/slowdynamics provides a clear picture of the horseshoe map in the forced vander Pol system.

❏ Canards that form at nonhyperbolic points of the slow manifold play a keyrole in this picture.



References

☞ K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva, and W.Weckesser, The forced van der Pol equation II: canards in the reduced sys tem , to appearin SIAM J. Appl. Dyn. Sys.

☞ J. Guckenheimer, K. A. Hoffman, and W. Weckesser, The forced van der Pol equation I: theslow flow and its bifurcations , SIAM J. Appl. Dyn. Sys. 2(1) 1-35 (2003).

☞ J. Guckenheimer, K. Hoffman, and W. Weckesser, Global bifurcations of periodic orbits inthe forced van der Pol equation , in Global Analysis of Dynamical Systems, H. W. Broer, B.Krauskopf, G. Vegter, eds., Inst. of Physics Publ., Bristol, (2001).

☞ J. Guckenheimer, K. Hoffman, and W. Weckesser, Numerical computation of canards , Int. J.Bif. Chaos, 10, 2669-2687 (2000)


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