Plykin-like attractor in non-autonomous coupled oscillators

TL;DR

This paper investigates a non-autonomous coupled oscillator system, deriving an explicit Poincaré map that reveals a Plykin-type hyperbolic attractor on an invariant sphere, confirmed through numerical and analytical methods.

Contribution

It introduces a new model of coupled oscillators with a Plykin-like hyperbolic attractor and provides explicit derivation and verification of its hyperbolic nature.

Findings

Existence of a Plykin-type attractor in the system

Explicit form of the Poincaré map for the system

Numerical evidence confirming hyperbolic dynamics

Abstract

A system of two coupled non-autonomous oscillators is considered. Dynamics of complex amplitudes is governed by differential equations with periodic piecewise continuous dependence of the coefficients on time. The Poincar\'{e} map is derived explicitly. With exclusion of the overall phase, on which the evolution of other variables does not depend, the Poincar\'{e} map is reduced to 3D mapping. It possesses an attractor of Plykin type located on an invariant sphere. Computer verification of the cone criterion confirms the hyperbolic nature of the attractor in the 3D map. Some results of numerical studies of the dynamics for the coupled oscillators are presented, including the attractor portraits, Lyapunov exponents, and the power spectral density.