Conley's fundamental theorem for a class of hybrid systems

混合系统的Conley指数理论

Conley Index Theory for Hybrid Systems

针对一类混合动力系统，通过混合悬架半流将其视为连续半流，定义了同调Conley指数，证明了该指数良定义、在延拓下不变，并满足Ważewski性质等重构结果。

On Conley's Fundamental Theorem of Dynamical Systems

In this paper, we generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of regular index filtration for every Morse decomposition.

A Conley-type Lyapunov function for the strong chain recurrent set

Let $ϕ:X\times\mathbb{R} \rightarrow X$ be a continuous flow on a compact metric space $(X,d)$. In this article we constructively prove the existence of a continuous Lyapunov function for $ϕ$ which is strictly decreasing outside $\mathcal{SCR}_d(ϕ)$. Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set.

Isolating Neighborhoods and Filippov Systems: Extending Conley Theory to Differential Inclusions

The Conley index theory is a powerful topological tool for obtaining information about invariant sets in continuous dynamical systems. A key feature of Conley theory is that the index is robust under perturbation; given a continuous family of flows $\{φ_λ\}$, the index remains constant over a range of parameter values, avoiding many of the complications associated with bifurcations. This theory is well-developed for flows and homomorphisms, and has even been extended to certain classes of semiflows. However, in recent years mathematicians and scientists have become interested in differential i…

The random case of Conley's theorem: II. The complete Lyapunov function

Conley in \cite{Con} constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behavior outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems under the assumption that the base space $(Ω,\mathcal F,\mathbb P)$ is a separable metric space endowed with a probability measure. By constructing a…

一类混合系统的吸引子格计算

Computation of Attractor Lattices for a Class of Hybrid Systems

该研究针对紧致度量空间上的一类混合系统，证明其吸引子格同构于关联半流的吸引子格，通过网格构造实现混合吸引子子格的外近似，还计算了多个系统的Conley-Morse图与吸引子格。