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arXiv · 2007.04958

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

Patrick Guidotti·Sandro Merino
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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.

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Patrick Guidotti, Sandro Merino. 2020-07-09. On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation. https://arxiv.org/abs/2007.04958

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