Limit Cycles and Chaos Induced by a Nonlinearity with Memory

TL;DR

This paper explores how distributed time delays in nonlinear responses can induce stable limit cycles and chaos in noise-driven mechanical oscillators, revealing new dynamics and potential applications in energy harvesting.

Contribution

It demonstrates the emergence of limit cycles and chaos due to distributed time delays in a mechanical oscillator, inspired by optical resonator observations.

Findings

Stable limit cycles occur at delay times comparable to inverse dissipation rate.

Longer delays lead to chaotic dynamics with double scroll attractors.

Time delay affects the spectrum and amplitude of oscillations.

Abstract

Inspired by the observation of a distributed time delay in the nonlinear response of an optical resonator, we investigate the effects of a similar delay on a noise-driven mechanical oscillator. For a delay time that is commensurate with the inverse dissipation rate, we find stable limit cycles. For longer delays, we discover a regime of chaotic dynamics associated with a double scroll attractor. We also analyze the effects of time delay on the spectrum and oscillation amplitude of the oscillator. Our results point to new opportunities for nonlinear energy harvesting, provided that a nonlinearity with distributed time delay can be implemented in mechanical systems.