Olga Bernardi, Anna Florio, Jim Wiseman
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.
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For a continuous flow on a compact metric space, the aim of this paper is to prove a Conley-type decomposition of the strong chain recurrent set. We first discuss in details the main properties of strong chain recurrent sets. We then introduce the notion of strongly stable set as an invariant set which is the intersection of the $ω$-limits of a specific family of nested and definitively invariant neighborhoods of itself. This notion strengthens the one of stable set; moreover, any attractor results strongly stable. We then show that strongly stable sets play the role of attractors in the decom…
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…
We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index …
The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $ϕ$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(ϕ)=\bigcup [B(A)-A]$, where $\mathcal{CR}(ϕ)$ denotes the chain recurrent set of $ϕ$, $A$ stands for an attractor and $B(A)$ is the basin determined by $A$. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the $ω$-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, i…
In this paper we introduce a new compactness condition - Property (C) - for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient f…
It is known by the Conley's theorem that the chain recurrent set $CR(ϕ)$ of a deterministic flow $ϕ$ on a compact metric space is the complement of the union of sets $B(A)-A$, where $A$ varies over the collection of attractors and $B(A)$ is the basin of attraction of $A$. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called \emph{absorbing condition}. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove th…
We shall address the alternative definition of chain recurrent set for the action of a semigroup of continuous self maps, given by M. Hurley \cite {mh} in noncompact space. Following this, we shall address the characterization of chain recurrence in terms of attractors given by C. Conley in \cite {conley}.
Shadow chains and Conley chains for continuous-time semiflows
本文基于阴影链和Conley链的概念，研究连续时间半流的性质，证明两者在强紧动态下产生相同的链复发集和链图，同时改进了半流的s-均匀连续性定义。
In \cite {kl1}, we extended various notions of recurrence for the generalized semigroup analogous to their counterpart in the classical theory of dynamics. In this paper, we shall address the alternative definition of chain recurrent set in terms of attractors, given by Hurley in \cite {mh} following Conley\rq{}s characterization in \cite {conley}. We shall also discuss the notion of topological transitivity and chain transitivity in this general setting.
Singular Morse-Smale Flows on Pseudomanifolds with Spherical-Cone Singularities: Conley Theory and Intersection Homology
将Morse-Conley理论推广到具有球锥奇点的伪流形，通过引入奇异Morse-Smale流，建立了奇点Conley指数的局部动力学公式，证明了全局Lyapunov函数的存在性，并利用磨光化方法关联奇异与光滑情形，最后通过动力系统方法计算交合同调。
This short expository note provides an introduction to the concept of chain recurrence in topological dynamics and a proof of the existence complete Lyapunov functions for homeomorphisms of compact metric spaces due to Charles Conley. I have used it as supplementary material in introductory dynamics courses.
The aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict -that is strictly decreasing- outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strict…
We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally-defined "hybrid complete Lyapunov function" exists for every hybrid system in this class. Motivated by mechanics and control settings where physical or engineered events cause abrupt changes in a system's governing dynamics, our results apply to a large class of Lagrangian hybrid systems (with impacts) studied extensively in the robotics literature. Viewed formally, these results generalize those of Conley and …
We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized recurrent set. In particular, for a compact topological manifold with dimension greater or equal $2$, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient c…
In this paper we find conditions which guarantee that a given flow $Φ$ on a compact metric space $X$ admits a Lyapunov one-form $ω$ lying in a prescribed Čech cohomology class $ξ\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $ξ$ to the chain recurrent set of $Φ$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.
Recurrent and gradient dynamics of Iterated Function Systems
本文提出一种用于研究一般迭代函数系统定性动力学的框架，通过二元关系编码点的下游关系，分离周期行为与梯度行为。
Priestley duality and representations of recurrent dynamics
本文利用 Priestley 对偶建立任意动力系统全局动力学与序结构的联系，为链递归集提供 Hausdorff 紧化框架。
We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti.
Dynamical Systems: Discrete, Continuous and Hybrid
本文利用闭关系动力学统一研究离散、连续和混合动力系统，系统描述了吸引子-排斥子结构、Conley链回归关系及Lyapunov函数构造。
After relating the notion of $ω$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $ω(n) <n^{-ε}$, where $ε>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $ω$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over…
We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov stable chain recurrent class of $X$ is singular hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular hyperbolic.
Let $\g$ be a semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is tr…
Lyapunov-Guided Representation of Recurrent Neural Network Performance
本研究将RNN视为动力系统，利用Lyapunov谱分析结合自编码器（AeLLE）关联超参数与准确性，实现训练早期预测最终精度，为RNN架构变体提供新解释方法。
Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and render…
A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots
本文为广义Mazur纽结定义的Pixton微分同胚构造了具有最少临界点数的拟能量函数，并给出了Lyapunov函数临界点数量的下界估计。
We show that non-trivial chain recurrent classes for generic $C^1$ star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability. As a result, we show that $C^1$ generic star flows have only finitely many Lyapunov stable chain recurrent classes.
The Fundamental Theorem of Dynamical Systems: all at once and all in the same place
本文概述了动力系统基本定理的多种形式，发展了同时处理连续时间与离散时间拓扑动力学框架的统一方法，并首次为连续时间流和半流提供了完全 Lyapunov 函数。
The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $Λ$ contained in an optical hypersurface of a cotangent bundle, under …
Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions
该研究探究了n维欧氏空间上光滑渐近稳定向量场空间与对应Lyapunov函数空间的拓扑性质，证明了不同维度下的连通性与弱同伦型结论，拓展了微分拓扑在动力系统中的应用并给出多个相关推论。
A trichotomy for generic sectional-hyperbolic chain-recurrent classes
本文研究了 general sectional-hyperbolic 链-复发类的三元分类问题，证明了其在 C^1 意义下的一般性质。
The paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any $ A $-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of chaotic one-dimensional attractor and repeller.
Symmetry, Defects, and Diffusion in Continuous-memory Recurrent Networks
本文提出几何框架，分析连续记忆递归网络在缺陷与噪声下的对称性保持，通过等变性和切向/法向分解给出稳定性界，并验证解码器协方差目标可改善噪声记忆。
The Flow-Limit of Reflect-Reflect-Relax: Existence, Stability, and Discrete-Time Behavior
研究了反-反-松弛算法在流限制下的存在性、稳定性及离散时间行为，揭示了其作为前向欧拉离散化的流，并解释了迭代最优松弛参数的产生。
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