Nested set of periodic segments. (English) Zbl 1531.37013
A periodic isolating segment is an isolating block (in the sense of the Conley index theory) for the flow generated by periodic in time ODEs in the extended phase space, see [R. Srzednicki, Nonlinear Anal., Theory Methods Appl. 22, No. 6, 707–737 (1994; Zbl 0801.34041)]. The notion of nested set of periodic segments is introduced in the paper as a special configuration of periodic segments. This tool together with a fixed point index formula proved for associated Poincaré maps, enables the author to formulate sufficient conditions for chaotic dynamics. The paper is inspired by the geometric method for detecting chaotic dynamics introduced by the author and R. Srzednicki [J. Differ. Equations 135, No. 1, 66–82 (1997; Zbl 0873.58049)]. Some examples of periodic planar system with chaotic dynamics emerging from a nested set of periodic segments are provided.
Reviewer: Zdzisław Dzedzej (Gdańsk)
MSC:
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 37C60 | Nonautonomous smooth dynamical systems |
| 37B10 | Symbolic dynamics |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34A26 | Geometric methods in ordinary differential equations |
Keywords:
periodic segment; Lefschetz number; fixed point index; Ważewski retract method; symbolic dynamics; Poincaré map; periodic ODEsReferences:
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