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 any cross section 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 orbit of a Poincaré map.
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Acknowledgements
The authors would like to thank Prof. Qigui Yang and Dr. Yousu Huang for carefully reading the manuscript and pointing out several mistakes, improving our presentation greatly. The authors would like to thank the reviewers for their comments and suggestions, which help significantly improve the manuscript.
Funding
This research was supported by the National Natural Science Foundation of China (No. 11701328) and Young Scholars Program of Shandong University, Weihai (No. 2017WHWLJH09).
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Zhang, X., Chen, G. A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems. J Dyn Control Syst 29, 71–93 (2023). https://doi.org/10.1007/s10883-021-09582-x
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DOI: https://doi.org/10.1007/s10883-021-09582-x