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The conley index for fast-slow systems. II: multidimensional slow variable. (English) Zbl 1107.34051

Summary: We use the Conley index theory to develop a general method to prove the existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors’ earlier work [the authors and J. F. Reineck, J. Dyn. Differ. Equations 11, 427–470 (1999; Zbl 0945.34029)] which is now extended to systems with multidimensional slow variables. The new key idea is the observation that the Conley index in fast–slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast-slow systems with one-dimensional slow flow.

MSC:

34E15 Singular perturbations for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37B30 Index theory for dynamical systems, Morse-Conley indices
37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C29 Homoclinic and heteroclinic orbits for dynamical systems

Citations:

Zbl 0945.34029

Software:

conley
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Full Text: DOI

References:

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