Properties of periodic solutions near their oscillation threshold for a class of hyperbolic partial differential equations with localized nonlinearity

TL;DR

This paper analyzes the properties of periodic solutions near their oscillation threshold in a class of hyperbolic PDEs with localized nonlinearity, extending Fourier and bifurcation analysis techniques to infinite-dimensional systems.

Contribution

It extends the harmonic balance method and Fourier analysis to hyperbolic PDEs with localized nonlinearity, providing detailed bifurcation properties and error estimates.

Findings

Identifies bifurcation point and amplitude-period dependence.

Retrieves standard Hopf bifurcation properties.

Provides error estimates for Fourier series truncation.

Abstract

The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure interaction. It also encompasses particular types of delay differential equations. These systems undergo a bifurcation with the appearance of a small amplitude periodic regime. Assuming a certain regularity of the oscillating solution, several of its properties around the bifurcation are given: bifurcation point, dependence of both the amplitude and period with respect to the bifurcation parameter, and law of decrease of the Fourier series components. All the properties of the standard Hopf bifurcation in the non-hyperbolic case are retrieved. In addition, this study is based on a Fourier domain analysis and the harmonic balance method has been extended to the class of infinite dimensional problems hereby considered. Estimates on the errors made if the Fourier series is truncated are provided.