非线性扩散问题中对称诱导分岔点的验证
Validation of Symmetry Induced Bifurcation Points in Nonlinear Diffusion Problems
浏览论文内容
中文总结 AI 辅助
本文提出一个计算机辅助证明的理论框架，用于验证非线性扩散问题中由对称性诱导的跨临界和音叉分岔点，通过扩展非线性系统的求零问题实现，并适用于一维和二维域上的抛物型系统和高阶方程。
中文摘要 AI 辅助
非线性抛物型偏微分方程经常出现在应用现象的建模中。它们通常表现出复杂的动力学，这些动力学由其相关的平衡解组织。作为所涉及的模型参数的函数，平衡点可以通过分岔分析来检测，而分岔分析又围绕对分岔点的更深入理解展开。在许多应用系统中，使用传统数学技术严格建立这些组织中心是困难的。在本文中，我们提出了一个用于计算机辅助证明非线性问题中对称诱导的跨临界分岔和音叉分岔的理论框架。这是通过将存在性问题重新表述为相关扩展非线性系统的求零问题来实现的。该方法具有通用性，允许各种底层对称性。我们展示了如何将这些结果应用于抛物型系统和高阶方程，在一维和二维域上。
英文摘要
Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.
发表机构
- CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院）
- Department of Mathematical Sciences, George Mason University（乔治梅森大学）
机构由 AI 辅助整理，请以论文原文为准。
