On the maximal parameter range of global stability for a nonlocal thermostat model

Guidotti, Patrick; Merino, Sandro

doi:10.1007/s00028-020-00641-7

Cited by 1 publication

(24 citation statements)

References 12 publications

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“…1 and that motivates the approach described in Ref. 2 and the analysis performed in the present article. The main results are valid for

L < \infty

only.…”

Section: Introductionsupporting

confidence: 52%

“…In this section, we will adapt the stability analysis presented in Ref. 2 (Section 4) that builds on results in Ref. 22 to the integral equation obtained from the nonlinear thermostat problem

{false(2 false)}_{L}

with Dirichlet boundary conditions.…”

Section: Asymptotic Stability For Solutions Of the Nonlinear Volterramentioning

confidence: 99%

“…In Section 4 of Ref. 2, the interpretation of () as a feedback system is discussed in detail. The discussion shows how the well‐known Nyquist and Popov stability criteria for feedback systems can be applied heuristically.…”

Section: Introductionmentioning

confidence: 99%

“…However, we stress that the informal and motivational discussion in Section 4 of Ref. 2 is not used for the development of the rigorous proofs given there not for the ones in the present paper. Here we address the open question of the stability of the periodic solutions of () that are produced by the Hopf bifurcation and study a problem analogous to () on the full real line and on a bounded interval with Dirichlet boundary; see () below.…”

Section: Introductionmentioning

confidence: 99%

“…The discussion in Section 4 of Ref. 2 shows that time‐wandering sources lead to Volterra integral equations whose kernel is no longer in convolution form

a (t, τ) = a (t - τ)

. This makes a theoretical analysis more challenging.…”

Section: Introductionmentioning

confidence: 99%

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On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

Guidotti

Merino

2021

Stud Appl Math

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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.

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“…1 and that motivates the approach described in Ref. 2 and the analysis performed in the present article. The main results are valid for

L < \infty

only.…”

Section: Introductionsupporting

confidence: 52%

“…In this section, we will adapt the stability analysis presented in Ref. 2 (Section 4) that builds on results in Ref. 22 to the integral equation obtained from the nonlinear thermostat problem

{false(2 false)}_{L}

with Dirichlet boundary conditions.…”

Section: Asymptotic Stability For Solutions Of the Nonlinear Volterramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The discussion in Section 4 of Ref. 2 shows that time‐wandering sources lead to Volterra integral equations whose kernel is no longer in convolution form