Application of Floquet theory to dynamical systems with memory

TL;DR

This paper extends Floquet theory to analyze dynamical systems with infinite memory, providing bounds for Floquet exponents and demonstrating its applicability through various physical examples.

Contribution

The paper introduces a generalized Floquet theory for systems with infinite memory, establishing bounds for Floquet exponents and illustrating its usefulness with practical examples.

Findings

Established a lower asymptotic bound for Floquet exponents in systems with memory.

Applied the theory to systems like a Brownian particle and a circuit resonator.

Demonstrated the approach's effectiveness in analyzing complex dynamical systems.

Abstract

We extend the recently developed generalized Floquet theory [Phys. Rev. Lett. 110, 170602 (2013)] to systems with infinite memory. In particular, we show that a lower asymptotic bound exists for the Floquet exponents associated to such cases. As examples, we analyze the cases of an ideal 1D system, a Brownian particle, and a circuit resonator with an ideal transmission line. All these examples show the usefulness of this new approach to the study of dynamical systems with memory, which are ubiquitous in science and technology.