Topological Horseshoe Induced by Periodic Switching Between Non-Isochronous Planar Systems

TL;DR

This paper provides a numerical criterion for chaos in planar systems created by periodic switching, based on the non-isochronicity of subsystems and the period function of Hamiltonian systems.

Contribution

It introduces a new criterion for chaos in switched planar systems using the period function and monotonicity properties of Hamiltonian systems.

Findings

Derived a checkable criterion for chaos via numerical methods.

Linked the existence of a topological horseshoe to the non-isochronicity of subsystems.

Provided explicit conditions guaranteeing chaotic dynamics in switched Hamiltonian systems.

Abstract

We establish a criterion for the existence of a topological horseshoe in a class of planar systems generated by periodic switching between two subsystems, each admitting a family of closed orbits, where the mechanism for chaos arises from the non-isochronicity of each subsystem. Exploiting the relationship between the period function of a Hamiltonian system and the rate of change of the area enclosed by its periodic orbits, we derive a criterion, which can be checked by numerical methods, for the existence of horseshoe in planar systems obtained by switching between two Hamiltonian subsystems. Furthermore, by invoking monotonicity results for the period function in Newtonian Hamiltonian systems, we obtain an explicit and computable criterion that guarantees chaotic dynamics in planar systems generated by switching between two such subsystems.