Canards and Horseshoes in the Forced van der Pol Equation
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1 Equadiff 3 Singular Perturbations Mini-Symposium Canards and Horseshoes in the Forced van der Pol Equation Warren Weckesser Department of Mathematics Colgate University FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
2 Collaborators FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
3 Collaborators John Guckenheimer, Cornell University Kathleen Hoffman, University of Maryland, Baltimore County FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
4 Collaborators John Guckenheimer, Cornell University Kathleen Hoffman, University of Maryland, Baltimore County Ricardo Oliva, Lawrence Berkeley National Lab Undergraduates (REU Summer at Cornell) Katy Bold, Chantal Edwards, Saby Guharay, Judith Hubbard, Chris Lipa FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
5 Brief History 96 van der Pol ( ): relaation oscillations van der Pol and van der Mark: hysteresis and bistability Cartwright and Littlewood (chaos before chaos ) Levinson: simplified piecewise linear model. Inspired... Smale: Horseshoe Map Levi: further simplified model, symbolic dynamics 98 s Grasman (et al): Asymptotic analysis Manymoreanalyticalandnumericalstudies FIRST [ ] PREV NEXT [ 4 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 3
6 Rewrite in Standard Fast/Slow Form FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 4
7 Warren Weckesser, Equadiff 3 p. 4 Rewrite in Standard Fast/Slow Form, New parameters:,, New variables: Then FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT
8 Warren Weckesser, Equadiff 3 p. 4 Rewrite in Standard Fast/Slow Form, New parameters:,, New variables: Then,, Symmetry: FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT
9 Slow and Fast Subsystems FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 5
10 Warren Weckesser, Equadiff 3 p. 5 Slow and Fast Subsystems Slow Subsystem (DAE) FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT
11 Warren Weckesser, Equadiff 3 p. 5 Slow and Fast Subsystems Slow Subsystem (DAE) and desingularize Eliminate #!! ".) (Time reversed for FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT
12 Slow and Fast Subsystems Slow Subsystem (DAE) and desingularize Eliminate #!! ".) (Time reversed for FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 5
13 Slow and Fast Subsystems Fast Subsystem Slow Subsystem (DAE), and desingularize Eliminate #!! ".) (Time reversed for FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 5
14 Slow and Fast Subsystems Fast Subsystem Slow Subsystem (DAE), and desingularize Critical Manifold Eliminate $ #!! ".) (Time reversed for FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 5
15 Phase Space.5.5 y θ FIRST [ 5 ] PREV NEXT [ 7 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 6
16 Phase Space.5 Critical Manifold, Slow Subsystem.5 y θ FIRST [ 5 ] PREV NEXT [ 7 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 6
17 Phase Space.5.5 y Fast Subsystem θ FIRST [ 5 ] PREV NEXT [ 7 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 6
18 Phase Space.5 Eample: Periodic Orbit.5 y θ FIRST [ 6 ] PREV NEXT [ 8 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 7
19 Phase Space y θ.8.5 FIRST [ 6 ] PREV NEXT [ 8 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 7
20 Phase Space.5.5 y θ FIRST [ 6 ] PREV NEXT [ 8 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 7
21 Phase Space θ FIRST [ 6 ] PREV NEXT [ 8 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 7
22 Phase Space θ FIRST [ 7 ] PREV NEXT [ 9 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 8
23 Canards at the Folded Saddle If, there is a pair of folded equilibria (pseudo-singular points) on each fold. One is a saddle, the other is either a node or a spiral. % FIRST [ 8 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p. 9
24 Canards at the Folded Saddle If, there is a pair of folded equilibria (pseudo-singular points) on each fold. One is a saddle, the other is either a node or a spiral. % Canards form at folded saddles. (Benoit 983; Mischenko, Kolesov, Kolesov, & Rhozov 994; Szmolyan & Wechselberger 3) FIRST [ 8 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p. 9
25 Canards at the Folded Saddle If, there is a pair of folded equilibria (pseudo-singular points) on each fold. One is a saddle, the other is either a node or a spiral. % Canards form at folded saddles. (Benoit 983; Mischenko, Kolesov, Kolesov, & Rhozov 994; Szmolyan & Wechselberger 3) Representative System: ' & & Critical manifold is. The origin is a folded equilibrium. FIRST [ 8 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p. 9
26 Canards at a Folded Saddle.5 y z FIRST [ 9 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
![NEXT [ ] LAST QUIT](/docs-images/87/97060771/images/27-2.jpg "Warren Weckesser,")
27 Canards at a Folded Saddle FIRST [ 9 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
28 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
29 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
30 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
31 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
32 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
33 Canards at a Folded Saddle.5 Maimal canard.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
34 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
35 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
36 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
37 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
38 Canards at a Folded Saddle.5.5 z.5.5 Fold FIRST [ ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p.
39 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
40 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
41 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
42 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
43 Forced van der Pol Poincaré Map Σ Canard forms.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
44 Forced van der Pol Poincaré Map Σ Jump away canard.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
45 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
46 Forced van der Pol Poincaré Map Σ Maimal canard.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
47 Forced van der Pol Poincaré Map Σ Jump back canard.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
48 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
49 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
50 Forced van der Pol Poincaré Map Σ Canard ends.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
51 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
52 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
53 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
54 Forced van der Pol Poincaré Map Σ.5.5 y.5.5 Symmetry:.5.5 ( ( $.5 Σ θ FIRST [ ] PREV NEXT [ 3 ] LAST QUIT Warren Weckesser, Equadiff 3 p.
55 Horseshoe in the Forced van der Pol System Cartoon Jump Across Canards κ ε Maimal Canard No Canards Jump Back Canards No Canards FIRST [ ] PREV NEXT [ 4 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 3
56 Horseshoe in the Forced van der Pol System Cartoon Jump Across Canards κ ε Maimal Canard No Canards Jump Back Canards No Canards.63 Numerical Computation.65 y ε =., a=., ω= θ FIRST [ ] PREV NEXT [ 4 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 3
57 Bifurcations of Periodic Orbits.66 3Aa: bn 3Sa: nn.64 a: bnbbbnbn 6a: bnbb 3Sb: bb.6 L Norm of c: anbnabab 6b: anbb 6c: anbn b: anbbanan.596 3Ac: ab 3Ab: an.594 3Sc: aa.5..5 a FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 4
58 Bifurcations of Periodic Orbits.66 3Aa: bn 3Sa: nn.64 a: bnbbbnbn 6a: bnbb 3Sb: bb.6 L Norm of Maimal Canard c: anbnabab 6b: anbb 6c: anbn b: anbbanan.596 3Ac: ab 3Ab: an.594 3Sc: aa.5..5 a FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 4
59 Bifurcations of Periodic Orbits.66 3Aa: bn 3Sa: nn.64 a: bnbbbnbn 6a: bnbb 3Sb: bb L Norm of Ac: ab 6b: anbb 6c: anbn b: anbbanan c: anbnabab Eample from full shift on three symbols 3Ab: an.594 3Sc: aa.5..5 a FIRST [ 3 ] PREV NEXT [ 5 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 4
60 Eample: Maimal Canard Bifurcation Periodic orbit at the limit point has a maimal canard. maimal canard: crosses fold, follows unstable manifold all the way back to fold, then jumps θ FIRST [ 4 ] PREV NEXT [ 6 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 5
61 Eample: Maimal Canard Bifurcation Periodic orbit on a branch leading to the left-most limit point (symbols bb). jumps back jumps at fold canard: crosses fold θ FIRST [ 5 ] PREV NEXT [ 7 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 6
62 Eample: Periodic Orbits and Symbolic Dynamics Periodic orbit with symbol sequence anbbanan. a b a a n b n n θ FIRST [ 6 ] PREV NEXT [ 8 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 7
63 Summary FIRST [ 7 ] PREV NEXT [ 9 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 8
64 Summary A detailed numerical study combined with an understanding of the fast/slow dynamics provides a clear picture of the horseshoe map in the forced van der Pol system. FIRST [ 7 ] PREV NEXT [ 9 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 8
65 Summary A detailed numerical study combined with an understanding of the fast/slow dynamics provides a clear picture of the horseshoe map in the forced van der Pol system. Canards that form at nonhyperbolic points of the slow manifold play a key role in this picture. FIRST [ 7 ] PREV NEXT [ 9 ] LAST QUIT Warren Weckesser, Equadiff 3 p. 8
66 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 system, to appear in SIAM J. Appl. Dyn. Sys. J. Guckenheimer, K. A. Hoffman, and W. Weckesser, The forced van der Pol equation I: the slow flow and its bifurcations, SIAM J. Appl. Dyn. Sys. () -35 (3). J. Guckenheimer, K. Hoffman, and W. Weckesser, Global bifurcations of periodic orbits in the forced van der Pol equation, in Global Analysis of Dynamical Systems, H. W. Broer, B. Krauskopf, G. Vegter, eds., Inst. of Physics Publ., Bristol, (). J. Guckenheimer, K. Hoffman, and W. Weckesser, Numerical computation of canards, Int. J. Bif. Chaos,, () FIRST [ 8 ] PREV NEXT [ ] LAST QUIT Warren Weckesser, Equadiff 3 p. 9
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