Universal asymptotic behavior in nonlinear systems driven by a two-frequency forcing

TL;DR

This paper derives universal long-time asymptotic expressions for the response of nonlinear systems driven by two frequencies, revealing features like resonance width that are independent of system details and applicable to sub-Fourier signal processing.

Contribution

It introduces a non-perturbative method to analyze long-term behavior and identifies universal features of the asymptotic response in two-frequency driven nonlinear systems.

Findings

Resonance width is smaller than Fourier width by a factor related to the driving frequencies.

Asymptotic response features are independent of system-specific parameters.

Symmetry properties influence additional universal response features.

Abstract

We examine the time-dependent behavior of a nonlinear system driven by a two-frequency forcing. By using a non-perturbative approach, we are able to derive an asymptotic expression, valid in the long-time limit, for the time average of the output variable which describes the response of the system. We identify several universal features of the asymptotic response of the system, which are independent of the details of the model. In particular, we determine an asymptotic expression for the width of the resonance observed by keeping one frequency fixed, and varying the other one. We show that this width is smaller than the usually assumed Fourier width by a factor determined by the two driving frequencies, and independent of the model system parameters. Additional general features can also be identified depending on the specific symmetry properties of the system. Our results find direct application in the study of sub-Fourier signal processing with nonlinear systems.