Smale Flows Research Articles

Minimal number of periodic orbits for non-singular Morse–Smale flows in odd dimension

Minimal number of periodic orbits for non-singular Morse–Smale flows in odd dimension

Cancellations of periodic orbits for non-singular Morse–Smale flows

AbstractIn this work, we explore the dynamical implications of a spectral sequence analysis of a filtered chain complex associated to a non-singular Morse–Smale (NMS) flow $\varphi $ on a closed orientable $3$ -manifold $M^3$ with no heteroclinic trajectories connecting saddle periodic orbits. We introduce the novel concepts of cancellations and reductions of pairs of periodic orbits based on Franks’ morsification and Smale’s cancellation theorems. The main goal is to establish an algebraic-dynamical correspondence between the unfolding of this spectral sequence associated to $\varphi $ and a family of flows obtained by cancelling and reducing pairs of periodic orbits of $\varphi $ on $M^3$ . This correspondence is achieved through a spectral sequence sweeping algorithm (SSSA), which determines the order in which these cancellations and reductions of periodic orbits occur, producing a family of NMS flows that reaches a core flow when the spectral sequence converges.

Topology of Ambient 3-Manifolds of Non-Singular Flows with Twisted Saddle Orbit

In the present paper, nonsingular Morse – Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.

Structural stability for scalar reaction-diffusion equations

In this paper, we prove the structural stability for a family of scalar reaction-diffusion equations. Our arguments consist of using invariant manifold theorem to reduce the problem to a finite dimension and then, we use the structural stability of Morse–Smale flows in a finite dimension to obtain the corresponding result in infinite dimension. As a consequence, we obtain the optimal rate of convergence of the attractors and estimate the Gromov–Hausdorff distance of the attractors using continuous $\varepsilon$-isometries.

Nonsingular Morse–Smale Flows with Three Periodic Orbits on Orientable $$3$$-Manifolds

Nonsingular Morse–Smale Flows with Three Periodic Orbits on Orientable $$3$$-Manifolds

Non-singular Morse–Smale flows on n-manifolds with attractor–repeller dynamics

In the present paper the exhaustive classification up to topological equivalence of non-singular Morse–Smale flows on n-manifolds M n with exactly two periodic orbits is presented. Denote by G 2(M n ) the set of such flows. Let a flow f t : M n → M n belongs to the set G 2(M n ). Hyperbolicity of periodic orbits of f t implies that among them one is an attracting and the other is a repelling orbit. Due to the Poincaré–Hopf theorem, the Euler characteristic of the ambient manifold M n is zero. Only the torus and the Klein bottle can be ambient manifolds for f t in case of n = 2. The authors established that there are exactly two classes of topological equivalence of flows in G 2(M 2) if M 2 is the torus and three classes if M 2 is the Klein bottle. For all odd-dimensional manifolds the Euler characteristic is zero. However, it is known that an orientable three-manifold admits a flow from G 2(M 3) if and only if M 3 is a lens space L p,q . In this paper it is proved that every set G 2(L p,q ) contains exactly two classes of topological equivalence of flows, except the case when L p,q is homeomorphic to the three-sphere or the projective space , where such a class is unique. Also, it is shown that the only non-orientable n-manifold (for n > 2), which admits flows from G 2(M n ) is the twisted I-bundle over the (n − 1)-sphere . Moreover, there are exactly two classes of topological equivalence of flows in . Among orientable n-manifolds only the product of the (n − 1)-sphere and the circle can be ambient manifolds for flows from G 2(M n ) and splits into two topological equivalence classes.

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

Nontame Morse–Smale flows and odd Chern–Weil theory

AbstractUsing a certain well-posed ODE problem introduced by Shilnikov in the sixties, Minervini proved the currential “fundamental Morse equation” of Harvey–Lawson but without the restrictive tameness condition for Morse gradient flows. Here, we construct local resolutions for the flow of a section of a fiber bundle endowed with a vertical vector field which is of Morse gradient type in every fiber in order to remove the tameness hypothesis from the currential homotopy formula proved by the first author. We apply this to produce currential deformations ofodd degree closed formsnaturally associated to any hermitian vector bundle endowed with a unitary endomorphism and metric compatible connection. A transgression formula involving smooth forms on a classifying space for oddK-theory is also given.

Simple Smale flows and their templates on S3

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of 3-dimensional flows. We prove that for an embedded template in S3 with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set of knots and links. As an application, the isotopic classification of simple Smale flows on S3 is discussed.

Spectral analysis of Morse-Smale flows, II: Resonances and resonant states

The goal of the present work is to compute explicitely the correlation spectrum of a Morse-Smale flow in terms of the Lyapunov exponents of the Morse--Smale flow, the topology of the flow around periodic orbits and the monodromy of some given flat connection. The corresponding eigenvalues exhibit vertical bands when the flow has periodic orbits. As a corollary, we obtain sharp Weyl asymptotics for the dynamical resonances.

Optimal Morse–Smale Flows with Singularities on the Boundary of a Surface

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.

Topology of Pollicott-Ruelle resonant states

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott–Ruelleresonant states associated with Anosov and Morse–Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse–Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

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 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.

Realizing full n-shifts in simple Smale flows

Smale flows on 3-manifolds can have invariant saddle sets that are suspensions of shifts of finite type. We look at Smale flows with chain recurrent sets consisting of an attracting closed orbit a, a repelling closed orbit r and a saddle set that is a suspension of a full n-shift and draw some conclusions about the knotting and linking of a∪r. For example, we show for all values of n it is possible for a and r to be unknots. For any even value of n it is possible for a∪r to be the Hopf link, a trefoil and meridian, or a figure-8 knot and meridian.

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.

Every 3-manifold admits a structurally stable nonsingular flow with three basic sets

This paper is devoted to proving that every closed orientable 3 3 -manifold admits a simple Smale flow X t X_t . Here a simple Smale flow is a structurally stable nonsingular flow whose chain recurrent set is composed of a periodic orbit attractor, a periodic orbit repeller and a transitive saddle invariant set, i.e., a saddle basic set.

Smale flows on

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on$\mathbb{S}^{2}\times \mathbb{S}^{1}$. This classification gives necessary and sufficient conditions that must be satisfied by an (abstract) Lyapunov graph in order for it to be associated to a Smale flow on$\mathbb{S}^{2}\times \mathbb{S}^{1}$.

Simple Smale flows with a four band template

We study simple Smale flows on S3 and other 3-manifolds modeled by the Lorenz template and another template with four bands but that still has cross section a full 2-shift.