Parametric Resonance in a dissipative system \'a la Kronig-Penney

TL;DR

This paper investigates how dissipation affects parametric resonance in a damped Kronig-Penney model, revealing that dissipation limits the number of resonance bands and influences the boundedness of auxiliary functions.

Contribution

It provides a detailed analysis of the interplay between parametric resonance and dissipation, introducing the use of Lewis-Reisenfeld invariant in this context.

Findings

Dissipation reduces the number of PR-bands at higher frequencies.

The auxiliary function rho(t) can be bounded or unbounded depending on initial conditions.

Dissipation suppresses higher frequency resonance bands.

Abstract

The competition between parametric resonance (PR) and dissipation is studied in the damped Kronig-Penney model, with time-dependent dissipation rate gamma(t). In the classical case, it is shown that dissipation leaves just a finite number of PR-bands at most, suppressing those at higher frequencies. An analysis of the Lewis-Reisenfeld invariant I(q,p,rho) is performed, showing that, in the PR regime, the auxiliary function rho(t) can be chosen bounded or unbounded, depending on the initial conditions.