Chaotic motion at the emergence of the time averaged energy decay

TL;DR

This paper investigates the transition from regular to chaotic dynamics in a conservative system with many oscillators, linking chaos onset to the decay of the system's average energy and analyzing the underlying nonlinear behavior.

Contribution

It introduces a detailed analysis of how increasing environment oscillators induces chaos and energy transfer, using Lyapunov exponents, power spectra, and Kaplan-Yorke dimension.

Findings

Chaos occurs when the number of oscillators is between 13 and 15.

Energy transfer from the system to the environment correlates with chaotic behavior.

Lyapunov exponents decrease as the environment size increases, indicating energy dissipation.

Abstract

A system plus environment conservative model is used to characterize the nonlinear dynamics when the time averaged energy for the system particle starts to decay. The system particle dynamics is regular for low values of the $N$ environment oscillators and becomes chaotic in the interval $13\le N\le15$, where the system time averaged energy starts to decay. To characterize the nonlinear motion we estimate the Lyapunov exponent (LE), determine the power spectrum and the Kaplan-Yorke dimension. For much larger values of $N$ the energy of the system particle is completely transferred to the environment and the corresponding LEs decrease. Numerical evidences show the connection between the variations of the {\it amplitude} of the particles energy time oscillation with the time averaged energy decay and trapped trajectories.