On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation
| Autor: | Guidotti, Patrick, Merino, Sandro |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 40 pages, 10 figures |
| Databáze: | arXiv |
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