Nonsingular Morse-Smale Flows Research Articles

The indexed links of non-singular Morse-Smale flows on graph manifolds

The indexed links of non-singular Morse-Smale flows on graph manifolds

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.

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.

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

Топологической эквивалентности неособых потоков Морса-Смейла в предположениях различной общности посвящен целый ряд статей, например, [1]-[4]. Однако, в случае малого числа периодических орбит известные инварианты можно значительно упростить и, главное, довести задачу классификации до реализации, описав допустимость полученных инвариантов. В недавней работе [5] была получена исчерпывающая классификация потоков с двумя орбитами на произвольных замкнутых $n$-многообразиях. В настоящей статье полная топологическая классификация получена для потоков с тремя периодическими орбитами, заданных на ориентируемых $3$-многообразиях. Библиография: 10 названий.

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.

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.

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.

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.

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.

Round-handle decomposition of S 2×S 1

A round-handle decomposition is associated with a non-singular Morse–Smale flow on 3-manifolds prime to S 2× S 1. This decomposition has been built only for the 3-sphere S 3. In this paper we obtain the round-handle decomposition of non-singular Morse–Smale flows on S 2× S 1, in order to get all the different fattened round handles in this manifold. Some of them include non-separating boundary components that induce the topology of the links of periodic orbits.

L 2-invariants and morse-smale flows on manifolds

We study the homotopy invariants of free cochain and Hilbert complexes. These L 2-invariants are applied to the calculation of exact values of minimum numbers of closed orbits of some indices of nonsingular Morse-Smale flows on manifolds of large dimensions.

Bifurcations of links of periodic orbits in non-singular Morse–Smale systems with a rotational symmetry on [formula omitted

In this paper we consider a rotational symmetry on a non-singular Morse–Smale (NMS) system analyzing the restrictions this symmetry imposes on the links defined by the set of its periodic orbits and to the appearance of local generic codimension one bifurcations in the set of NMS flows on S 3 . The topological characterization is obtained by writing the involved links in terms of Wada operations. It is also obtained that symmetry implies that in general bifurcations have to be multiple. On the other hand, we also see that there exists a set of links that cannot be related to any other by sequences of this kind of bifurcation.

Gradient flows within plane fields

We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple (i.e., a graph-manifold). Furthermore, there are strong restrictions on the types of gradient flows realized within plane fields: such flows lie on the boundary of the space of nonsingular Morse-Smale flows. This relationship translates to knot-theoretic obstructions for the link of singularities in the flow. In the case of an integrable plane field, the restrictions are even finer, forcing taut foliations on surface bundles. The situation is completely different in the case of contact plane fields, however: it is easy to realize gradient fields within overtwisted contact structures (the nonintegrable analogue of a foliation with Reeb components).

KNOTS AND LINKS IN INTEGRABLE HAMILTONIAN SYSTEMS

The main purpose of this paper is to prove that Bott integrable Hamiltonian flows and non-singular Morse-Smale flows are closely related. As a consequence, we obtain a classification of the knots and links formed by periodic orbits of Bott integrable Hamiltonians on the 3-sphere and on the solid torus. We also show that most of Fomenko's theory on the topology of the energy levels of Bott integrable Hamiltonians can be derived from Morgan's results on 3-manifolds that admit non-singular Morse-Smale flows.

Bifurcations of links of periodic orbits in non-singular Morse - Smale systems on

The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.

Extending flows from isolated invariant sets

AbstractLet M be a closed n-dimensional manifold, and let U be a n-dimensional isolating block such that U is smoothly embedded in M. Let φ be a smooth semi-flow on U and let Λ contained in U, be isolated and invariant under φ Then there exists a semi-flow φ′ on M which extends φ such that φ′ is Morse-Smale outside of U, and no new recurrence is introduced in U. The theorem is true for any finite number of pairwise-disjoint Ui. Furthermore, if Λ is hyperbolic, topologically transitive and is the closure of periodic orbits, then φ′ is an Axiom A flow and is Ω-stable. In dimensions two and three, we have the stronger result that φ′ is structurally stable. Also, as a corollary, we give sufficient conditions for the flow φ′ to be nonsingular. One application of the corollary permits the formation of allowable knots and links in three-manifolds such that there exists a structurally stable nonsingular Morse-Smale flow φ′ which contains the specified knots and links in Ω(φ′) Moreover, the knots and links can be specified to be any combination of attractors, repellers or saddles.

On Closed Orbits of Morse-Smale Flows on 3-Manifolds

Any link in a 3-manifold is the closed orbits of a non-singular Morse-Smale flow after taking the split sum with the unlink and the connected sum with S2 × S1s.

Closed orbits of non-singular Morse-Smale flows on S3

Closed orbits of non-singular Morse-Smale flows on S3

Primitive links of non-singular Morse-Smale flows on the special Seifert fibered manifolds

In this paper we consider a non-singular Morse-Smale flow Φ t on an irreducible, simple, closed, orientable 3-manifold M. We define a primitive flow ψ t from Φ t , and call the link type of the closed orbits of ψ t a primitive link of Φ t . We show that the link types of the primitive links are finite and every non-singular Morse-Smale flow on M is obtained from a primitive flow by exchanging the flow in a regular neighborhood of attracting or repelling closed orbits.