Morse-Smale Flows Research Articles

On embedding of invariant manifolds of the simplest Morse-Smale flows with heteroclinical intersections

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. More precisely, we consider a class G(S4) of Morse-Smale flows on the sphere S4 such that for any flow f∈G(S4) its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves that belong to intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of saddle equilibria’s invariant manifolds; that is the first step in the solution of topological classification problem. In particular, we prove that the closures of invariant manifolds of saddle equlibria that do not contain heteroclinical curves are locally flat 2-sphere and closed curve. These manifolds are attractor and repeller of the flow. In set of orbits that belong to the basin of attraction or repulsion we construct a section that is homeomoprhic to the direct product S2×S1. We study a topology of intersection of saddle equlibria’s invariant manifolds with this section.

SPECTRAL ANALYSIS OF MORSE–SMALE FLOWS I: CONSTRUCTION OF THE ANISOTROPIC SPACES

We prove the existence of a discrete correlation spectrum for Morse–Smale flows acting on smooth forms on a compact manifold. This is done by constructing spaces of currents with anisotropic Sobolev regularity on which the Lie derivative has a discrete spectrum.

On the simplest Morse-Smale flows with heteroclinical intersections on the sphere $S^n$

On the simplest Morse-Smale flows with heteroclinical intersections on the sphere $S^n$

Morse-Smale flows and the model of the topology of magnetic fields in plasma

Morse-Smale flows and the model of the topology of magnetic fields in plasma

On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections

It is shown that if a closed smooth orientable manifold M n , n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.

Generalized topological transition matrix

This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this more general transition matrix satisfies the additional requirement that it covers flow-defined Conley-index isomorphisms, one proves algebraic and connection-existence properties. These general transition matrices with this covering property are referred to as generalized topological transition matrices and are used to consider connecting orbits of Morse-Smale flows without periodic orbits, as well as those in a continuation associated to a dynamical spectral sequence.

Cherry flow: physical measures and perturbation theory

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows.Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

Continuous Morse-Smale flows with three equilibrium positions

Continuous Morse-Smale flows on closed manifolds whose nonwandering set consists of three equilibrium positions are considered. Necessary and sufficient conditions for topological equivalence of such flows are obtained and the topological structure of the underlying manifolds is described. Bibliography: 36 titles.

Continuous Morse-Smale flows with three equilibrium positions

Рассматриваются непрерывные потоки Морса-Смейла, неблуждающее множество которых состоит из трех состояний равновесия, на замкнутых многообразиях. Получены необходимые и достаточные условия топологической эквивалентности таких потоков и описана топологическая структура несущих многообразий. Библиография: 36 названий.

Behavior $0$ nonsingular Morse Smale flows on $S^3$

In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits(abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions:  &nbsp 1. we use WLG to list behavior $0$ NMS flows on $S^3$;  &nbsp 2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;  &nbsp 3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;  &nbsp 4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.

A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces

In a paper of Oshemkov and Sharko, three-colour graphs were used to make the topological equivalence of Morse-Smale flows on surfaces obtained by Peixoto more precise. In the present paper, in the language of three-colour graphs equipped with automorphisms, we obtain a complete (including realization) topological classification of gradient-like cascades on surfaces. Bibliography: 25 titles.

Fat Handles and Phase Portraits of Non Singular Morse-Smale Flows on S3 with Unknotted Saddle Orbits

Abstract In this paper we build Non-singular Morse-Smale flows on S3 with unknotted and unlinked saddle orbits by identifying fat round handles along their boundaries. This way of building the flows enables to get their phase portraits. We also show that the presence of heteroclinic trajectories imposes an order in the round handle decomposition of these flows; this order is total for NMS flows composed of one repulsive, one attractive and n unknotted saddle orbits, for n ≥ 1.

Morse–Smale systems with few non-wandering points

Let MSflow(Mn,k) and MSdiff(Mn,k) be Morse–Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold Mn. For k=3, we show that the only values of n possible are n∈{2,4,8,16}, and M2 is the projective plane. For n⩾4, Mn is simply connected and orientable. We prove that the closure of any separatrix of ft∈MSflow(Mn,3) is a locally flat n2-sphere while there is ft∈MSflow(Mn,4) such that the closure of separatrix of ft is a wildly embedded codimension two sphere. This allows us to classify flows from MSflow(M4,3). For n⩾6, one proves that the closure of any separatrix of f∈MSdiff(Mn,3) is a locally flat n2-sphere while there is f∈MSdiff(M4,3) such that the closure of any separatrix is a wildly embedded 2-sphere.

Locally flat and wildly embedded separatrices in simplest morse-smale systems

Let MS flow (M n , k) and MS diff (M n , k) be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold M n (n???4). We prove that the closure of any separatrix of f t ? MS flow (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f t ? MS flow (M n , 4) the closure of separatrix of those is a wildly embedded codimension two sphere. For n???6, one proves that the closure of any separatrix of f ? MS diff (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f ? MS diff (M 4, 3) such that the closure of any separatrix is a wildly embedded 2-sphere.

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

In this paper we find topological conditions for the non existence of heteroclinic trajectories connecting saddle orbits in non singular Morse-Smale flows on S 3. We obtain the non singular Morse-Smale flows that can be decomposed as connected sum of flows and we show that these flows are those who have no heteroclinic trajectories connecting saddle orbits. Moreover, we characterize these flows in terms of links of periodic orbits.

Poincaré-Hopf inequalities for periodic orbits

The interplay between the dynamics of a nonsingular Morse-Smale flow on a smooth, closed, n-dimensional manifold, M, and the topology of M, was exhibited in Franks (Comment Math Helv 53(2):279–294, 1978), Smale (Bull Am Math Soc 66:43–49, 1960), by means of a collection of inequalities, which we refer to as Morse-Smale inequalities. These inequalities relate the number of closed orbits of each index to the Betti numbers of M. These well-known inequalities provide the necessary conditions for a given dynamical data in the form of a specified number of closed orbits of a given index to be realized as a nonsingular Morse-Smale flow on M. In this article we provide two inequalities, hereby referred to as Poincare-Hopf inequalities for periodic orbits, which imposes constraints on the dynamics of periodic orbits without reference to the Betti numbers of the manifold M. The main theorem establishes the necessity and sufficiency of the Poincare-Hopf inequalities in order for the Morse-Smale inequalities to hold.

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold Mn on which one can define a Morse-Smale gradient flow ft with two nodes and two saddles such that the closure of the separatrix of some saddle of ft is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.

Lorenz like Smale flows on three-manifolds

In this paper, we discuss how to realize Lorenz like Smale flows (LLSF) on 3-manifolds. It is an extension of M. Sullivan's work about Lorenz Smale flows on S 3 . We focus on two questions: (1) Classify the topological conjugate classes of LLSF which can be realized on S 3 ; (2) Which 3-manifolds admit LLSF? If some 3-manifold admits LLSF, how does it admit LLSF? This paper is in some sense parallel to the work of J. Morgan and M. Wada on Morse–Smale flows on 3-manifolds.

Transverse Foliations to Nonsingular Morse–Smale Flows on the 3-Sphere and Bott-Integrable Hamiltonian Systems

In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse–Smale flows on S3 have transverse foliations. We then observe that there is a connection to flows arising from certain Hamiltonian systems and from certain contact structures.

Complete topological invariants of Morse-Smale flows and handle decompositions of 3-manifolds

We construct a topological invariant for the canonical decomposition on prime and round handles associated with a Morse-Smale flow on a closed 3-manifold. We prove that the flows are topologically equivalent if and only if their invariants coincide.