标量Heat Equation的Wiener强振动、Popov曲线和Hopf超临界分岔
On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation
摘要：
研究了一类非局部非自伴秩一扰动下标量Laplacian谱的参数依赖扰动。扰动谱的详细描述得到的Dirichlet边界条件上的有界区间，以及为全实数线上的问题。将所得结果应用于一个相关参数依赖的非线性非局部抛物型方程的研究。该等式对反馈系统进行建模，该反馈系统例如可以被解释为恒温器设备或者在用于市场的基于代理的价格形成模型的上下文中。证明了一类非线性非局部热传导方程的周期自振荡的存在性和稳定性。研究了一类具有Dirichlet边界条件的非线性抛物型方程和一类具有反馈边界控制的非线性Neumann边界条件的相关问题的分歧和稳定性.将非线性抛物型方程的稳定性分析归结为对一个相关的非线性Volterra积分方程的研究，然后利用积分方程的Popov准则得到了分支和稳定性结果.虽然这个问题是在标量的情况下研究，但它可以自然地推广到任意欧氏维数和流形。
英文摘要：
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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