Topology and Homoclinic Trajectories of Discrete Dynamical Systems

Global bifurcation of homoclinic trajectories of discrete dynamical systems

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles.

Group theoretic conditions for existence of robust relative homoclinic trajectories

We consider robust relative homoclinic trajectories (RHTs) for equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these result we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.

具有广义型分段常数变元的微分方程的同宿与异宿轨迹

Homoclinic and Heteroclinic Trajectories of Differential Equations with Piecewise Constant Arguments of Generalized Type

本文研究广义型分段常数变元拟线性系统的渐近行为，利用离散时间映射和Banach不动点定理严格证明同宿与异宿解的存在性，并讨论解的双曲集合。

Cech cohomology, homoclinic trajectories and robustness of non-saddle sets

In this paper we study flows $φ:M\times\mathbb{R}\longrightarrow M$ having an isolated non-saddle set. We see that the complexity of the region of influence of an isolated non-saddle set $K$ depends on the way in which $K$ sits on the phase space at the cohomological level. We construct flows in surfaces having isolated non-saddle sets with a prescribed structure for its region of influence. We also study parametrized families of smooth flows and continuations of non-saddle sets.

部分双曲同宿类中的非双曲遍历测度与马蹄

Non-hyperbolic ergodic measures and horseshoes in partially hyperbolic homoclinic classes

针对鲁棒非双曲传递微分同胚，本文证明其同宿类中每个遍历测度（含非双曲情形）均可由双曲集在弱*拓扑与熵意义下逼近。

奇异流的Lyapunov稳定链回复类

Lyapunov stable chain recurrence classes for singular flows

本文证明对于远离同宿相切的C^1通用向量场，非平凡的Lyapunov稳定链回复类必为同宿类，方法是通过C^2向量场逼近及Gibbs F-态的收敛性论证。