偏微分方程及其半离散化中混沌的系统性方法

中文摘要 AI 辅助

我们提出了一种系统性方法，用以刻画Herzog空间上齐次一维常系数偏微分方程解中的Devaney混沌。我们证明，对于最高阶时间导数缺乏混合空间导数的方程，总存在一个加权空间，使得其解在该空间中呈现混沌。这一分析被扩展至包含最高阶时间项中的混合偏导数情形。此外，我们通过识别两个上下文之间的等距同构，将Herzog空间中的混沌行为与方程的半离散化联系起来，从而也为这些半离散化上的混沌提供了刻画。本工作以两个应用作为结尾，为四阶Moore-Gibson-Thompson方程和粘性van Wijngaarden-Eringen方程的动力学提供了新的见解。

英文摘要

We present a systematic approach to characterize Devaney chaos in the solution of homogeneous one dimensional constant-coefficients PDEs on Herzog spaces. We show that for equations where the highest-order time derivative lacks mixed spatial derivatives, there always exists a weighted space in which the solution is chaotic. This analysis is extended to include mixed partial derivatives in the highest-order temporal term. Moreover, we link the chaotic behavior in Herzog spaces to the semidiscretization of the equations by identifying isometric isomorphisms between the two contexts, providing also a characterization for chaos on these semidiscretizations. The work concludes with two applications that provide new insights on the dynamics of the fourth-order Moore-Gibson-Thompson equation and the viscous van Wijngaarden-Eringen equation.

发表机构

• Universitat Politècnica de València（瓦伦西亚理工大学）

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