On Conley's Fundamental Theorem of Dynamical Systems
M. R. Razvan
本文推广了Conley指标理论中的动力系统的Conley基本定理。我们还得出结论，存在定期索引过滤的每一个莫尔斯分解。
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.
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Weak index pairs and the Conley index for discrete multivalued dynamical systems
受样本动力学重构问题的启发，我们重新审视了离散多值动力系统的Conley指数理论。我们引入了一个新的，限制较少的定义的孤立的邻里。事实证明，那么用于构建索引的主要工具，即索引对，不再有用。为了克服这一障碍，我们使用弱指数对的概念。
Linking combinatorial and classical dynamics: Conley index and Morse decompositions
我们证明了每一个定义在单纯复形的单形族上的Forman意义下的组合动力系统都能在该单纯复形的几何实现上产生一个多值动力系统F.此外，F可以以这样的方式选择，即两个动力系统的孤立不变集、Conley指标、Morse分解和Conley-Morse图一一对应。
Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.
Conley index for random dynamical systems
Conley指标理论是研究动力系统、微分方程和分歧理论的有力工具。本文试图将Conley指标推广到离散随机动力系统。本文主要采用Franks和Richeson在文[6]中给出的Conley指数。此外，我们还简单地讨论了时间连续随机动力系统与相应的时间- $h$映射之间孤立不变集的关系。对于应用，我们给出了几个例子来说明我们的结果。
The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed manifold of three dimensions without the scape region produces a semi-dynamical system. Thus, we have built a semiflow that allows us to apply the classical Conley theory. Furthermore, we use it to guarantee the existence of periodic orbits in this class of piecewise-smooth vector fields.
Discrete Conley Index for Zero-dimensional Basic Sets
建立了一个定理，给出了零维基本集的离散Conley指标的计算方法，并给出了相应的结构矩阵中包含的动力学信息.给出了零维基本集的约化同调Conley指标的Jordan实型分类。
Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the index
重新审视离散多值动力系统的康利指数理论的动机源于更广泛的实际应用的需要，特别是在采样动力学或组合动力学。文[B. Batko和M. Mrozek，{ \em SIAM J. Applied动力系统}，15（2016），pp. 1143-1162]基于弱索引对，在放松隔离性导致索引对缺失的情况下，似乎是朝着这个方向迈出的有希望的一步。本文是[B. Batko和M. Mrozek，{ \em SIAM J. Applied动力系统}，15（2016），pp. 1143-1162]中定义的指数的性质，即Wavenewski性质、可加性性质、同伦（延拓）性质和交换性性质。我们还提出了构造弱指标对的隔离块。
This is a self-contained tour of the Conley index and connection matrices. The starting point is Conley's fundamental theorem of dynamical systems. There is a short stop at the necessary topological background, before we proceed to the basic properties of the index. Then, the itinerary passes through the construction of connection matrices with a panoramic view of the applications: detect heteroclinic orbits arising in delay differential equations, and partial differential equations of parabolic type. The ride will be filled with examples and figures.
A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data-driven dynamics. In the present paper we continue the program by studying attractor-repeller pairs and Morse decompositions in this setting. In particular, we prove Morse equation and Morse inequalities.
Conley Index of Limit Cycles in Filippov Vector Fields
该研究在多值动力系统框架下，针对平面Filippov系统的极限环，利用Mrozek提出的多值容许映射的Conley指数，计算不同类型极限环的Conley指数，扩展了相关研究范围。
In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. We show that near an isolated invariant continuum the flow is topologically equivalent to a C1 flow. We deduce that isolated invariant continua in surfaces have the shape of finite polyhedra. We also show the existence of regular isolating blocks of isolated invariant continua and we use them to compute their Conley index provided that we have some knowledge about the truncated unstable manifold. We also see that the ring structure of the cohomology index of an isol…
Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invaria…
A dynamical interpretation of the connection map of an attractor-repeller decomposition
本文在 Conley index theory 框架下，为吸引子-排斥子分解中正合序列的连接同态 $Γ$ 提供了动力学解释，阐明了排斥子发射波前经连接轨道演化至吸引子并被接收的过程。
A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector …
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。
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 Conley index for flows is a topological invariant describing the behavior around an isolated invariant set $S$. It is defined as the homotopy type of a quotient space $N/L$, where $(N,L)$ is an index pair for $S$. In the case of a discrete dynamical system, i.e., a continuous self-map $f\colon X\to X$, the definition is similar. But one needs to consider the index map $f_{(N,L)}\colon N/L\to N/L$ induced by $f$. The Conley index in this situation is defined as the homotopy class $[f_{(N,L)}]$ modulo shift equivalence. The shift equivalence relation is rarely used outside this context and n…
The random case of Conley's theorem: II. The complete Lyapunov function
Conley在\cite{Con}中构造了紧致度量空间上的流的完备Lyapunov函数，该函数在链递归集的轨道上是常数，在链递归集的轨道外是严格递减的。这表明动力学复杂性集中在链递归集上，而链递归集外的动力学行为相当简单。本文在底空间$(Ω,\mathcal F,\mathbb P)$是可分度量空间且具有概率测度的假设下，对随机动力系统得到了类似的结果.通过构造一个完备的李雅普诺夫函数，该函数在随机链回归集内的轨道上是常数，在随机链回归集外的轨道上是严格递减的，得到了Conley动力系统基本定理的随机情形.进一步，将随机动力系统的结果推广到非紧状态空间。
Conley-Morse persistence barcode: a homological signature of combinatorial bifurcations
本文提出Conley-Morse持续条形码，用于描述组合分岔的同调特征，基于拓扑持续性理论，利用Conley指数构建持续模块，并通过gentle代数证明其分解为简单区间。
Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in dynamical systems are studied: One is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the two attractors, and also obtain the possible components and their configuration. Moreover, we find that there is always a separatrix or cycle separatrix, which separates the two attractors. Finally, the biological meanings and implications of these structures are g…
The Conley index theory is a powerful topological tool for describing the basic structure of dynamical systems. One important feature of this theory is the attractor-repeller decomposition of isolated invariant sets. In this decomposition, all points in the invariant set belong to the attractor, its associated dual repeller, or a connecting region. In this connecting region, points tend towards the attractor in forwards time and the repeller in backwards time. This decomposition is also, in a certain topological sense, stable under perturbation. Conley theory is well-developed for flows and ho…
Characterizing Slow Exit and Entrance Points
几何奇异扰动理论（GSPT）和Conley指标理论是分析动力系统的两种强有力的方法。康利已经认识到，使用他的指数是更容易的奇异扰动问题。在本文中，我们将重温Conley的结果，并证明Fenichel标准形的GSPT技术可以用来简化Conley指标技术的应用，甚至进一步。我们也希望我们的研究结果能在不同领域之间架起一座更好的桥梁。此外，我们将展示如何解释康利的条件平均。用二维vanderPol方程和三维Morris-Lecar模型说明了计算结果。
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 …
Conley Index at Infinity
本文的目的是探讨Conley指标技术在研究有限不变集和无限不变集之间的异宿联系中的可能性。为此，我们提醒读者的庞加莱紧化：这种变换允许项目$n$维向量空间$X$上的$n$维单位半球的$X\times \mathbb{R}$和无穷大的$(n-1)$维赤道称为球在无穷大。在可正规化条件下，$X$上的向量场变换为Poincaré半球上的向量场，Poincaré半球的相关流使赤道不变.赤道上的动力学反映了无穷远处的动力学，但现在是有限的，可以用康利指数技术来研究。此外，我们观察到赤道附近可能发生一些非孤立行为，并引入孤立不变动力补的无穷远不变集的概念。通过构造一个扩展的相空间，并将其与一个扩展的流相结合，我们能够采用Conley指数技术，证明了这种非孤立不变集的连接的存在性.
Conley index for multivalued maps on finite topological spaces
本文发展了有限拓扑空间上多值映射的Conley理论，为离散时间动力系统建立了孤立不变集与指标对，引入了定义良好的Conley指数并验证了Wazewski性质与延拓性质。
Random Chain Recurrent Sets for Random Dynamical Systems
Conley定理证明了紧致度量空间上的确定流$ϕ$的链递归集$CR(ϕ)$是集并$B(A)-A$的补集，其中$A$在吸引子集上变化，而$B(A)$是$A$吸引区域.最近的研究表明，对于非紧可分完全度量空间上的随机动力系统，在所谓的\emph{absorbing condition}下也有类似的分解结果。本文对随机动力系统引入了随机链回归集的概念，并在非紧可分完全度量空间\emph{without}的吸收条件下证明了随机Conley定理.
The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conl…
This paper concerns the computation and identification of the (homological) Conley index over the integers, in the context of discrete dynamical systems generated by continuous maps. We discuss the significance with respect to nonlinear dynamics of using integer, as opposed to field, coefficients. We translate the problem into the language of commutative ring theory. More precisely, we relate shift equivalence in the category of finitely generated abelian groups to the classification of $\mathbb{Z}[t]$-modules whose underlying abelian group is given. We provide tools to handle the classificati…
The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference…
The random case of Conley's theorem: III. Random semiflow case and Morse decomposition
本文第一部分将作者\cite{Liu,Liu2}的结果从随机流推广到随机半流，即得到了无穷维随机动力系统的Conley分解定理。第二部分通过引入随机半流的后向轨道，将不变随机紧集（如全局随机吸引子）分解为随机Morse集和它们之间的连接轨道，从而将Conley \cite{Con}提出的不变集的Morse分解推广到随机半流上，肯定地回答了Caraballo和Langa \cite{CL}提出的一个公开问题.
A theorem on computation of the homological Conley index of an isolated invariant set of the Poincaré map associated to a section in a rotating local dynamical system $ϕ$ is proved. Let $(N,L)$ be an index pair for a discretization $ϕ^h$ of $ϕ$, where $h>0$, and let $S$ denote the invariant part of $N\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, t…
The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of `set-difference' taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretical models of spectral, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations build order-theoretic models of dynamical systems, which are used to develop tools for computing global characteristi…
The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for d…
In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits certain attractor…
Graded differential groups, Cartan-Eilenberg systems and conjectures in Conley index theory
本文定义了偏序集上的Cartan-Eilenberg系统，证明自由P-分次微分群间的过滤链同构等价于关联系统同构，解决了关于连接矩阵唯一性的公开猜想并证明三种理论等价。
A sufficient condition for bifurcation in random dynamical systems
得到了随机Conley指标的一些性质，并给出了离散时间和连续时间随机动力系统存在抽象歧点的一个充分条件.通过几个例子证明了这种随机分岔现象。
Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions
本文针对Conley-Morse持续同调条形码计算成本高的问题，提出用更简单的块取代指数对，通过更新矩阵分解的方法实现高效计算，优化了现有算法。
The Szymczak Functor on the Category of Finite Sets and Finite Relations
针对关系型Conley理论构建需求，本文研究有限集与关系范畴上的Szymczak函子，给出Szymczak范畴同构类的可算法化分类，为相关理论构建奠定基础。
Maslov index of Hamiltonian systems
利用Conley-Zehnder指数和时间1流给出了线性自治哈密顿系统基本解的Maslov指数的计算公式.
Sharkovskiis theorem under small random perturbations
本文通过随机Conley指数，建立了小随机扰动下的一维映射随机动力系统的Sharkovskii型定理，实现了对随机周期点的精确检测。
Classical Morse theory revisited I -- Backward $λ$-Lemma and homotopy type
我们引入了两个工具，动态增厚和流选择器，以克服臭名昭著的不连续性的梯度流端点映射附近的非退化临界点。更准确地说，我们解释某些康利对$(N,L)$，建立在[2，3]，作为一个动力学增厚的稳定流形的稳定纤维化。作为第一个应用，为了说明这个概念的有效性，我们重新证明了经典莫尔斯理论的一个基本定理，Milnor的同伦细胞附着定理[1]。动力加厚导致一个概念上简单和简短的证明。
We establish new global bifurcation theorems for dynamical systems in terms of local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation $u_t+A u=f_λ(u)$ in a Banach space $X$, where $A$ is a sectorial operator with compact resolvent. Assume that $0$ is always a trivial stationary solution of the equation. We show that the global dynamic bifurcation branch $Γ$ of a bifurcation point $(0,λ_0)$ either meets another bifurcation point $(0,λ_1)$, or is unbounded, completely extending the well-known Rabinowitz Global Bifurcation Theorem on operator equ…
Linking Theorems of Local Semiflows on Complete Metric Spaces
本文证明了完全度量空间上局部半流动力系统的一些联结定理和山路型结果。我们的结果提供了一种替代方法来检测存在的紧凑不变集，而不使用康利指标理论。它们也可以应用于椭圆型方程的变分问题，而不需要验证经典的P. S.条件作为例子，我们研究了有界区域上非自治抛物型方程$ u_t-Δu-μu=f(u)+g(x,t) $的共振问题。在某些Landesman-Laser型条件下，利用半流的一个适当的连接定理证明了常返解的存在性。另一个例子是关于$R^n$的椭圆型方程$-Δu+a(x)u=f(x,u)$。我们证明了正解的存在性，通过应用山路引理的半流的抛物流的问题。
In this paper we investigate Deep Learning Models using topological dynamical systems, index theory, and computational homology. These mathematical machinery was invented initially by Henri Poincare around 1900 and developed over time to understand shapes and dynamical systems whose structure and behavior is too complicated to solve for analytically but can be understood via global relationships. In particular, we show how individual neurons in a neural network can correspond to simplexes in a simplicial complex manifold approximation to the decision surface learned by the NN, and how these si…
The Fundamental Theorem of Dynamical Systems: all at once and all in the same place
本文概述了动力系统基本定理的多种形式，发展了同时处理连续时间与离散时间拓扑动力学框架的统一方法，并首次为连续时间流和半流提供了完全 Lyapunov 函数。
On the Lyndon Dynamical Systems
给定一个全有限序字母表$ A $，赋予$ A $上的词集交替的字典序，我们定义了一类新的Lyndon词.我们研究了Lyndon系统的基本性质。将负基β位移与Lyndon系统联系起来，得到了负基β位移的一些基本性质。我们发现，独立于W。Steiner的方法，条件，其中一个字是$(-β)$ -扩展的$-\fracβ{β+ 1} $为一些$ β> 1$。
Discretizing stochastic dynamical systems using Lyapunov equations
随机动力系统是状态估计、系统辨识和控制的基础。系统模型通常是在连续时间内提供的，而应用理论的主要部分是为离散时间系统开发的。因此，连续时间模型的离散化是基本的。我们提出了一种新的算法，使用相结合的李雅普诺夫方程和解析解，使高效的软件实现。所提出的方法规避了文献中的标准算法所表现出的数值问题。理论和仿真结果。
We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for or…
Entropy of nonautonomous dynamical systems
熵的不同概念在动力系统的经典理论中起着基础性的作用。与许多其他用于分析自治动力学的概念不同，测度论和拓扑熵都可以很自然地扩展到过程公式中给出的离散时间非自治动力系统。本文概述了作者在这方面的工作。还提出了一个例子，以前没有出现在文献中。
We consider stationary stochastic dynamical systems evolving on a compact metric space, by perturbing a deterministic dynamics with a random noise, added according to an arbitrary probabilistic distribution. We prove the maximal and pointwise ergodic theorems for the transfer operators associated to such systems. The results are extensions to noisy systems of some of the fundamental ergodic theorems for deterministic systems.
Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks
研究了线性动态互联网络在外部随机干扰下的性能退化问题。本文的重点是一阶和二阶线性共识网络。我们采用的期望值的稳定状态分散的整个网络的状态作为一个性能指标，并开发了一个图论的方法，网络的基础图形的结构规格的性能指标。我们明确量化了线性共识网络中最佳性能水平的几个固有的基本限制，并表明这些性能限制仅仅是由底层图的特定结构所施加的。此外，我们发现新的稀疏性和性能指标的概念之间的联系。特别是，我们描述了几个基本的权衡，揭示了线性共识网络的性能指标和各种稀疏性措施之间的相互作用。最后，我们将我们的结果应用到两个真实世界的动态网络，并提供能量解释所提出的性能指标。结果表明，在同步电力网络的总功率损耗和总动能的自动驾驶车辆的网络形成这些网络是可行的性能指标，这些措施的基本限制量化的最佳实现水平的能源效率在这些动态网络。
Fundamental Markov systems
我们继续发展的理论马尔可夫系统发起的\cite{Wer1}。本文引入了与随机动力系统相联系的基本Markov系统，证明了随机动力系统的平稳初始分布的唯一性和唯一性的证明可简化为与之相联系的基本Markov系统的证明，后者的稳定性判据更为清晰。
A tale of two vortices: how numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems
本文开发基于转移算子奇异值分解的算法，用于定位非自治系统中的相干结构并检测其合并与分裂等变化，在周期、准周期驱动模型及南极极涡分裂数据上验证。
A quantitative dynamical Zhang fundamental inequality and Bogomolov-type problems
本文通过证明张基本不等式的定量版本，扩展了Néron-Tate高度的间隙原理，并在算术和几何情形中提供了动力学波戈莫洛夫猜想的有效实例。
An Abstract Category of Dynamical Systems
本文通过泛性质刻画时间，并据此提出动力系统范畴的抽象定义，为其他类型系统的具体实例化提供指导。
Unifying Dynamical and Structural Stability of Equilibriums
我们展示了一个基本的动态和结构稳定性的平衡，从真正的动力系统所产生的措施之间的关系。我们表明，动态稳定性，量化通过系统的局部响应外部扰动，符合最小的内部扰动能够破坏平衡。首先，通过重新制定的控制理论的结果，我们解释，谐波外部扰动反映了雅可比矩阵在平衡的频谱灵敏度，相对于其系数的常数变化。然而，为了保持这种等价性，必须允许雅可比系数的虚变化。因此，对于真实的动力系统，与动力稳定性的联系就消失了。我们表明，这个问题是可以避免的，从而恢复动力学和结构稳定性之间的基本联系，考虑随机噪声的外部和内部扰动。更准确地说，我们证明了系统的局部响应白噪声扰动直接反映了内部白噪声的强度，它可以容纳之前的渐近均方稳定的平衡丢失。
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