Chaotic discretization theorems for forced linear and nonlinear coupled oscillators

TL;DR

This paper demonstrates chaos in coupled oscillator systems through analytical proofs and numerical simulations, revealing complex dynamics including strange attractors and bifurcations in both linear and nonlinear cases.

Contribution

It provides new analytical results proving Li-Yorke chaos in coupled oscillators and extends these results to nonlinear polynomial potential systems.

Findings

Existence of Li-Yorke chaos in four-dimensional discrete systems

Identification of strange attractors through numerical simulations

Bifurcation diagrams and Lyapunov spectra analysis

Abstract

We prove the holding of chaos in the sense of Li-Yorke for a family of four-dimensional discrete dynamical systems that are naturally associated to ODE systems describing coupled oscillators subject to an external non-conservative force, also giving an example of a discrete map that is Li-Yorke chaotic but not topologically transitive. Analytical results are generalized to a modular definition of the problem and to a system of nonlinear oscillators described by polynomial potentials in one coordinate. We perform numerical simulations looking for a strange attractor of the system; furthermore, we perform a bifurcation analysis of the system presenting 1D and 2D bifurcation diagrams, together with spectra of Lyapunov exponents and basins of attraction.