Entrainment of Chaos

TL;DR

This paper introduces the concept of entrainment of chaos, where irregular chaotic behavior is captured by limit cycles, with implications for various scientific fields and supported by theoretical analysis and simulations.

Contribution

It presents the novel phenomenon of entrainment of chaos, extending the understanding of chaos control and interaction with limit cycles in dynamical systems.

Findings

Chaotic cycles can emerge when chaos amplitude is small.

Strong perturbations or small limit cycle diameters lead to non-cyclic chaos.

Simulations demonstrate entrainment of chaos in Chua's oscillators.

Abstract

A new phenomenon, entrainment of chaos, which is understood as a seizure of an irregular behavior by limit cycles, is discussed. As a result, chaotic cycles appear if the chaos amplitude is small. Otherwise, the chaos is not necessarily cyclic, if the perturbations are strong and/or diameter of the limit cycle is small. Sensitivity as a main and a unique ingredient is considered and, in addition, period-doubling route to chaos is chosen for extension. The results may be of strong importance for engineering sciences, brainwaves and biomusicology phenomena as well as can be developed for hydrodynamics. Theoretical results are supported by simulations and discussions over Chua's oscillators, entrainment of chaos by toroidal attractors and controlling problems. Moreover, through an example, by means of the Lyapunov functions method, a chaotic attractor is provided.