On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation

TL;DR

This paper analyzes how parameter-dependent perturbations affect the spectrum of a scalar Laplacian and applies these findings to demonstrate the existence and stability of periodic oscillations in a nonlinear heat equation with feedback, using Popov's criterion.

Contribution

It provides a detailed spectral analysis of nonlocal, non-self-adjoint perturbations and establishes Hopf bifurcation and stability results for a nonlinear heat equation with feedback boundary control.

Findings

Spectral perturbation characterized for Dirichlet and real line cases.

Existence of periodic self-oscillations from Hopf bifurcation proved.

Stability analyzed via Popov criterion for integral equations.

Abstract

A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that e.g. can be interpreted as a thermostat device or in the context of an agent based price formation model for a market. The existence and the stability of periodic self-oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation is proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. The bifurcation and stability results follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only it can be extended naturally to arbitrary euclidean dimension and to manifolds.