非线性分数阶演化方程解的渐近伪周期性

中文摘要 AI 辅助

本文研究由Banach空间中的Caputo分数阶演化方程建模的非线性分数阶动力系统解的动态行为，特别是渐近周期性。我们提出了一类新的加权伪$S$-渐近$(\xi,\tau,a)$-周期函数，该函数族将标准周期函数和$S$-渐近周期函数等成熟函数类型作为其特例统一起来。随后，我们系统地推导了该新函数族的基本性质，并严格证明了其完备性、相应的卷积定理以及非线性复合规则。基于这些新建立的结果，我们证明在合理假设下，Banach空间中一大类非线性分数阶动力系统（包括无延迟和具有有限延迟的系统）的温和解具有（加权伪）$S$-渐近$(\xi,\tau,a)$-周期性，这进一步意味着这些分数阶系统中不会出现分岔或混沌行为。最后，通过描述记忆依赖的复杂动态过程的示例性实例验证了理论结论，并辅以数值模拟。此外，所提出的加权伪$S$-渐近$(\xi,\tau,a)$-周期框架可以扩展到具有第一类离散间断点的函数，从而可以通过相同的分析范式处理具有不连续非线性项的系统。

英文摘要

This work focuses on the dynamic behaviors, particularly the asymptotic periodicity, of solutions to nonlinear fractional-order dynamical systems modeled by Caputo fractional evolution equations in Banach spaces. We propose a new class of weighted pseudo $S$-asymptotically $(ξ,τ,a)$-periodic functions, which unifies well-established function types such as standard periodic functions and $S$-asymptotically periodic functions as its special instances. We then systematically derive the fundamental properties of this new function family, and rigorously prove its completeness, corresponding convolution theorems, and nonlinear composition rules. Building on these newly established results, we demonstrate that under reasonable hypotheses, the mild solutions to a broad class of nonlinear fractional-order dynamical systems --- both delay-free and with finite delay --- in Banach spaces admit the (weighted pseudo) $S$-asymptotically $(ξ,τ,a)$-periodic property, which further implies that no bifurcation or chaotic behavior emerges in these fractional-order systems. The theoretical conclusions are finally validated by illustrative examples describing memory-dependent complex dynamic processes, supplemented with supporting numerical simulations. Additionally, the presented weighted pseudo $S$-asymptotically $(ξ,τ,a)$-periodic framework can be extended to functions with first-kind discrete discontinuities, so that systems with discontinuous nonlinear terms can be handled via the same analytical paradigm.

发表机构

• Shanghai Jiao Tong University（上海交通大学）
• Shanghai Dianji University（上海电机学院）
• Morgan State University（摩根州立大学）
• Fudan University（复旦大学）

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