On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation. Issue 3 (13th January 2021)

• Record Type: Journal Article
• Authors: Guidotti, Patrick; Merino, Sandro
• Is Part Of: Studies in applied mathematics. Volume 146:Issue 3(2021)
• Page Start: 677
• Page End: 729
• Publication Date: 2021-01-13
• Subjects: Hopf bifurcation -- nonlinear reaction diffusion systems -- nonlocal nonlinearity -- nonlinear feedback control systems -- Popov criterion -- Volterra integral equation; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12364
• Languages: English
• ISSNs: 0022-2526

Abstract

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 admits an interpretation 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 are 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. They 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.