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 language | English (US) |
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Pages (from-to) | 701-720 |
Number of pages | 20 |
Journal | Nonlinearity |
Volume | 19 |
Issue number | 3 |
DOIs | |
State | Published - Mar 1 2006 |
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics