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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关于维纳的剧烈振荡，波波夫曲线和霍夫的标量热方程的超临界分叉

研究了一类非局部和非自伴秩一扰动的标量拉普拉斯谱的参数依赖扰动。对于有界区间上的狄利克雷边界条件以及整个实线上的问题，都获得了扰动谱的详细描述。摄动结果被用于研究相关的依赖参数的非线性和非局部抛物方程。该方程式对反馈系统进行建模，该反馈系统允许将其解释为恒温器设备，或在基于代理的市场价格形成模型的背景下进行解释。证明了由Hopf分支产生的相关非线性和非局部热方程的周期自激振动的存在性和稳定性。对于具有Dirichlet边界条件的非线性抛物方程和与建模反馈边界控制的非线性Neumann边界条件有关的问题，均获得了分叉和稳定性结果。他们在简化非线性抛物方程的稳定性分析之后，从Popov积分方程准则出发，研究了相关的非线性Volterra积分方程。虽然仅在标量情况下研究问题，但可以自然地将其扩展到任意欧几里得维数和流形。他们在简化非线性抛物方程的稳定性分析之后，从Popov积分方程准则开始研究相关的非线性Volterra积分方程。虽然仅在标量情况下研究问题，但可以自然地将其扩展到任意欧几里得维数和流形。他们在简化非线性抛物方程的稳定性分析之后，从Popov积分方程准则出发，研究了相关的非线性Volterra积分方程。虽然仅在标量情况下研究问题，但可以自然地将其扩展到任意欧几里得维数和流形。