Chaotic attractors of relaxation oscillators

John Guckenheimer, Martin Wechselberger, Lai-Sang Young

Research output: Contribution to journalArticle

Abstract

We develop a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional Henon-like maps are diffeomorphisms that limit on non-invertible one-dimensional maps. Wang and Young formulated hypotheses that suffice to prove the existence of chaotic attractors in these families. Three-dimensional singularly perturbed vector fields have return maps that are also two-dimensional diffeomorphisms limiting on one-dimensional maps. We describe a generic mechanism that produces folds in these return maps and demonstrate that the Wang-Young hypotheses are satisfied. Our analysis requires a careful study of the convergence of the return maps to their singular limits in the Ck topology for k ≥ 3. The theoretical results are illustrated with a numerical study of a variant of the forced van der Pol oscillator.

Original languageEnglish (US)
Pages (from-to)701-720
Number of pages20
JournalNonlinearity
Volume19
Issue number3
DOIs
StatePublished - Mar 1 2006

Fingerprint

Relaxation oscillators
relaxation oscillators
Return Map
Chaotic Attractor
One-dimensional Maps
Diffeomorphisms
Vector Field
Noninvertible Maps
Geometric Singular Perturbation Theory
Van Der Pol Oscillator
Three-dimensional
Singular Limit
Singularly Perturbed
Numerical Study
Time Scales
Fold
Limiting
Dynamical system
Topology
Demonstrate

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Chaotic attractors of relaxation oscillators. / Guckenheimer, John; Wechselberger, Martin; Young, Lai-Sang.

In: Nonlinearity, Vol. 19, No. 3, 01.03.2006, p. 701-720.

Research output: Contribution to journalArticle

Guckenheimer, J, Wechselberger, M & Young, L-S 2006, 'Chaotic attractors of relaxation oscillators', Nonlinearity, vol. 19, no. 3, pp. 701-720. https://doi.org/10.1088/0951-7715/19/3/009
Guckenheimer, John ; Wechselberger, Martin ; Young, Lai-Sang. / Chaotic attractors of relaxation oscillators. In: Nonlinearity. 2006 ; Vol. 19, No. 3. pp. 701-720.
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