On the non-persistence of Hamiltonian identity cycles

异维环的持续性

Persistence of Heterodimensional Cycles

本文研究余指数1异维环在一般双参数分岔中的持续性，证明其产生鲁棒异维双曲集，并解决任意 $C^r$（含 $\infty,\omega$）正则性下的局部稳定化问题。

非对称加权Dowker持续性及其在动力系统中的应用

Asymmetrically Weighted Dowker Persistence and Applications in Dynamical Systems

本文提出非对称加权Dowker持续性框架，用于区分高维动力系统中的周期与非周期循环，通过构建加权有向网络并分析持续性，增强对动态结构的敏感性。

Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a peri…

非光滑微分系统中从周期积分流形分支出的极限环

Limit cycles bifurcating from periodic integral manifold in non-smooth differential systems

本文将Melnikov方法推广到具有非线性切换流形和一般周期积分流形的高维非光滑自治微分系统，研究从周期积分流形分支出的极限环，并通过实例验证了结果的适用性。

Endotactic Networks and Toric Differential Inclusions

An important dynamical property of biological interaction networks is persistence, which intuitively means that "no species goes extinct". It has been conjectured that dynamical system models of weakly reversible networks (i.e., networks for which each reaction is part of a cycle) are persistent. The property of persistence is also related to the well known global attractor conjecture. An approach for the proof of the global attractor conjecture uses an embedding of weakly reversible dynamical systems into toric differential inclusions. We show that the larger class of endotactic dynamical sys…

Swimming as a limit cycle

Steady swimming can be characterized as both periodic and stable. These characteristics are the very definition of limit cycles, and so we ask "Can we view swimming as a limit cycle?" In this paper we will find that the answer is "yes". We will define a class of dissipative systems which correspond to the passive dynamics of a body immersed in a Navier-Stokes fluid (i.e. the dynamics of a dead fish). Upon performing reduction by symmetry we will find a hyperbolically stable fixed point which corresponds to the stability of a dead fish in stagnant water. Given a periodic force on the shape of t…