论文标题
关于维也纳的暴力振荡，波波夫的曲线和霍普夫的超临界分叉用于标量热方程
On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation
论文作者
论文摘要
研究了标量拉普拉斯频谱的参数依赖性扰动，用于一类非局部和非偏置级别等级的扰动。在有界的间隔以及整个真实线上的问题上，获得了扰动光谱的详细描述。扰动结果应用于相关参数依赖性非线性和非局部抛物线方程的研究。等式建模了一个反馈系统，例如可以解释为恒温器设备，也可以在基于代理的市场基于代理的价格形成模型的背景下解释。证明了由HOPF分叉产生的相关非线性和非局部热方程的周期性自我振荡的存在和稳定性。对于具有Dirichlet边界条件的非线性抛物线方程以及与非线性Neumann边界条件的相关问题，分叉和稳定性结果既可以获得模拟反馈边界控制的相关问题。分叉和稳定性结果来自POPOV的积分方程标准，在将非线性抛物线方程的稳定性分析降低到相关的非线性Volterra积分方程的研究之后。尽管在标量情况下研究了问题，但它可以自然地扩展到任意的欧几里得维度和歧管。
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.
