In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.
In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and conecting homomorphisms are introduced for the nonautonomous setting.
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…
This paper gives the definition of the Conley index for a piecewise continuous map, which is only well defined on compatible isolating neighborhoods with Wazewski property slightly weaker than continuous situation.
A theorem is established where the computation of the discrete Conley index for zero dimensional basic sets is given with respect to the dynamical information contained in the associated structure matrices. A classification of the reduced homology Conley index of a zero dimensional basic set in terms of its Jordan real form is presented.
Motivated by the problem of reconstructing dynamics from samples we revisit the Conley index theory for discrete multivalued dynamical systems. We introduce a new, less restrictive definition of the isolating neighbourhood. It turns out that then the main tool for the construction of the index, i.e. the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs.
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.
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.
We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions, and Conley-Morse graphs of the two dynamical systems are in one-to-one correspondence.
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.
In this paper, a modified van der Pol oscillator equation is considered which appears in several heart action models. We study its global dynamics and verify many interesting bifurcations such as a Hopf bifurcation, a heteroclinic saddle connection, and a homoclinic saddle connection. Some of these bifurcations are detected by using Conley index methods. We demonstrate how the study of connection matrices and transition matrices shows how to select interesting parameter values for simulations.
Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indices.
The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincaré compactification: this transformation allows to project a $n$-dimensional vector space $X$ on the $n$-dimensional unit hemisphere of $X\times \mathbb{R}$ and infinity on its $(n-1)$-dimensional equator called the sphere at infinity. Under normalizability condition, vector fields on $X$ transform into vector fields on the Poincaré hemisphere whose associated flows let the equator invar…
Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step towards this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, {\em SIAM J. Applied…
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…
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…
Conley index theory is a very powerful tool in the study of dynamical systems, differential equations and bifurcation theory. In this paper, we make an attempt to generalize the Conley index to discrete random dynamical systems. And we mainly follow the Conley index for maps given by Franks and Richeson in [6]. Furthermore, we simply discuss the relations of isolated invariant sets between time-continuous random dynamical systems and the corresponding time-$h$ maps. For applications we give several examples to illustrate our results.
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…
In this paper, we study Conley theory in Hilbert spaces and make some refinement of the construction of the stable Conley index developed by Gȩba, Izydorek, and Pruszko. For instance, we allow subspaces other than invariant subspaces in the the construction. As a main result, we show that the resulting stable Conley index for a given isolating neighborhood of a flow does not depend on choices in the construction. We also hope that some results can be applied to the construction of Floer homotopy type.
In this paper we give an alternative, purely Conley index based proof of the Arnold conjecture in $\mathbb C\mathbb P^n$ asserting that a Hamiltonian diffeomorphism of $\mathbb C\mathbb P^n$ endowed with the Fubini-Study metric has at least $n+1$ fixed points.
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…
Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley's results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even further. We also hope that our results provide a better bridge between the different fields. Furthermore we show how to interpret Conley's conditions in terms of averaging. The result are illustrated by …
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…
We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic oscillations for forced pendulum type equations are considered as applications of the main result.
This paper is concerned with index pairs in the sense of Conley index theory for flows relative to pseudo-gradient vector fields for $C^1$-functions satisfying Palais-Smale condition. We prove a deformation theorem for such index pairs to obtain a Lusternik-Schnirelmann type result in Conley index theory.
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…
This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.
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 …
This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete homological Conley index, which is periodic, that enforces a combinatorial behavior of higher indices. As a consequence, we prove that isolated (as an invariant set) fixed points of orientation-reversing homeomorphisms of $\mathds{R}^3$ have fixed point index $\le 1$ and, as a cor…
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…
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.
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 …
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…
We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties which characterize this index using normal forms for symplectic matrices obtained from geometrical considerations. We explore the relations to Robbin-Salamon index for paths of Lagrangians. We give an axiomatic characterization of the generalization of the Conley-Zehnder index fo…
We give a new proof of the strong Arnold conjecture for $1$-periodic solutions of Hamiltonian systems on tori, that was first shown by C. Conley and E. Zehnder in 1983. Our proof uses other methods and is shorter than the previous one. We first show that the $E$-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals. Then an existence result for the $E$-cohomological Conley index, which applies to the setting of the Arnold conjecture, paves the …
It was discovered by Gordon in 1977 that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this paper is to give a homotopy theoretical proof of these results through a self-contained, explicit and simple computation of the Conley-Zehnder index. The techniques developed in this paper can be used to investigate the higher dimensional case of Keplerian ellipses, where the classical variational proof no longer applies.
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…
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…
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…
In this article, we show the existence of an isolating block, a special neighborhood of an isolated invariant set, for multivalued semiflows acting on metric spaces (not locally compact). Isolating blocks play an important role in Conley's index theory for single-valued semiflows and are used to define the concepts of homology index. Although Conley's index was generalized in the context of multivalued (semi)flows, the approaches skip the traditional construction made by Conley, and later, Rybakowski. Our aim is to present a theory of isolating blocks for multivalued semiflows in which we unde…
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology.
In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the existence of infinitely many homoclinic orbits for a critical point with the Poincare index greater than one.
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…
The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,ω)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ ω(A) > 0 \quad \text{and} \quad -n \leq c_1(A) < 0, $$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $μ$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twist…
In this paper, we apply Conley index theory in a covering space of an invariant set $S$, possibly not isolated, in order to describe the dynamics in $S$. More specifically, we consider the action of the covering translation group in order to define a topological separation of $S$ which distinguish the connections between the Morse sets within a Morse decomposition of $S$. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non isolated invariant sets, as well as, for isolated invariant sets. Moreover,…
We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.
We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman- Lazer and strong resonance conditions. Obtained index formulas …
Let $ϕ^t$ be a continuous flow on a metric space $X$ and $I$ be an isolated invariant set with an index pair $(N,L)$ and a Morse decomposition $\{M_i\}^n_{i=1}$. For every category $ν$ on $N/L$, we prove that $ν(N/L)\leq ν([L])+\sum_{i=1}^n ν(M_i)$. As a result if $ϕ^t|_I$ is gradient-like and $X$ is semi-locally contractible, then $ϕ^t$ has at least $ν_H(h(I))-1$ rest points in $I$ where $h(I)$ is the Conley index of $I$ and $ν_H$ is the Homotopy Lusternik-Schnirelmann category.
