A geometric criterion for the existence of chaos based on periodic orbits in continuous-time autonomous systems

Xu Zhang; Guanrong Chen

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arXiv · 1906.05405

A geometric criterion for the existence of chaos based on periodic orbits in continuous-time autonomous systems

Xu Zhang·Guanrong Chen

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Abstract

A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on cross-sections of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbits of Poincar\'{e} maps.

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Xu Zhang, Guanrong Chen. 2019-06-12. A geometric criterion for the existence of chaos based on periodic orbits in continuous-time autonomous systems. https://arxiv.org/abs/1906.05405

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