Gradient-like Flows Research Articles

Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for its Distinction

Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for its Distinction

On the reduction of the topological classification of gradient-like flows problem to the classification of polar flows I. A. Saraev

In this paper we consider a class G(Mn) of gradient-like flows on connected closed manifolds of dimension n≥4 such that for any flow ft∈G(Mn) stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow from the class G(Mn) can be splitted into connected summ of the sphere Sn , gft≥0 copies of direct products Sn−1×S1 , and a simply connected manifold which is not homeomorphic to the sphere. The number gft is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension (n−1) (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension i∈{2,…,n−2} (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class G(Mn) without non-trivial saddles. In this paper we prove that for any flow ft∈G(Mn) the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow ft and transversally intersect its trajectories. The restriction of the flow ft to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows ft1,…,ftl defined on the components of a connected sum. Moreover, for any j∈1,…,l , the set of saddle equilibria of the flow ftj consists either only of trivial saddles or only of of non-trivial ones and then the flow ftj is polar. We introduce the notion of consistent topological equivalence for flows ft1,…ftj and show that flows ft,f′t∈G(Mn) are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.

On a Classification of Periodic Maps on the 2-Torus

In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

A combinatorial invariant of gradient-like flows on a connected sum of $\mathbb{S}^{n-1}\times\mathbb{S}^1$

We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections defined on the connected sum of a finite number of manifolds homeomorphic to $\mathbb{S}^{n-1}\times \mathbb{S}^1$, $n\geq 3$. For $n>3$, this result extends substantially the class of manifolds such that structurally stable systems on these manifolds admit a topological classification. Bibliography: 36 titles.

Topological Conjugacy of Gradient-Like Flows on Surfaces and Efficient Algorithms for Its Distinguition

Gradient-like flows on surfaces have simple dynamics, which inspired many mathematicians to search for invariants of their topological equivalence. Under assumptions of different generality on the class of gradient-like flows under consideration, such classical invariants as the Leontovich- Mayer scheme, the Peixoto graph, the equipped Peixoto graph, the two-color Wang graph, the threecolor Oshemkov-Sharko graph, the Fleitas circular scheme, etc. were obtained. Thus, the problem of classifying gradient-like flows on surfaces from the point of view of topological equivalence has been solved in an exhaustive way. In recent works by Kruglov, Malyshev, and Pochinka, it was proved that for gradient-like flows the topological equivalence classes coincide with the topological conjugacy classes. The obtained result allows us to use any invariants of their equivalence for topological conjugacy of gradient-like flows. The present study is a review of the results on topological conjugacy of gradient-like flows on surfaces and efficient algorithms for its distinguishing, that is, algorithms whose running time is limited by some polynomial on the length of the input information.

On perturbations of algebraic periodic automorphisms of a two-dimensional torus

According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface M2 there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when M2 is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a twodimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.

Morse Index of Saddle Equilibria of Gradient-Like Flows on Connected Sums of $$\mathbb S^{n-1}\times \mathbb S^1$$

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

Eigenpairs for the Analysis of Complete Lyapunov Functions

A complete Lyapunov function describes the qualitative behaviour of a dynamical system: the areas where the orbital derivative vanishes and where it is strictly negative characterise the chain recurrent set and the gradient‐like flow, respectively. Moreover, its local maxima and minima show the stability properties of the connected components of the chain recurrent set. In this study, we use collocation with radial basis functions to numerically compute approximations to complete Lyapunov functions and then localise and analyse the stability properties of the connected components of the chain recurrent set using its gradient and Hessian. In particular, we improve the estimation of the chain recurrent set, and we determine the dimension and the stability properties of its connected components.

Topological classification of flows without heteroclinic intersections on a connected sum of manifolds $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$

In this paper, we announce a result on the possibility of obtaining sufficient conditions for topological conjugacy of gradient-like flows without heteroclinic intersections, given on a connected sum of products $S^{n-1}\times S^1$ in combinatorial terms.

Topological classification of flows without heteroclinic intersections on a connected sum of manifolds $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$

In this paper, we announce a result on the possibility of obtaining sufficient conditions for topological conjugacy of gradient-like flows without heteroclinic intersections, given on a connected sum of products $S^{n-1}\times S^1$ in combinatorial terms.

On classification of Morse-Smale flows on projective-like manifolds

In this paper, the problem of topological classification of gradient-like flows without heteroclinic intersections, given on a four-dimensional projective-like manifold, is solved. We show that a complete topological invariant for such flows is a bi-color graph that describes the mutual arrangement of closures of three-dimensional invariant manifolds of saddle equilibrium states. The problem of construction of a canonical representative in each topological equivalence class is solved.

On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

Synchronization Conditions of a Mixed Kuramoto Ensemble in Attractive and Repulsive Couplings

We study the emergent dynamics of a mixed Kuramoto ensemble in the presence of both attractive and repulsive coupling strengths. To be precise, we consider coupled Kuramoto-type systems consisting of two ensembles in which the oscillators in the same group interact attractively with a positive intra-group coupling strength, whereas the oscillators in the different group communicate repulsively with a negative inter-group coupling strength. For the modeling perspective of the Kuramoto model (KM for brevity), two types of systems are treated in this paper: an attractive–repulsive inertial KM–KM and an attractive–repulsive KM–KM. For these two models, we provide sufficient frameworks leading to complete bi-polar synchronization in which asymptotic configuration tends to a bi-polar state. Our estimates mainly rely on the gradient-like flow formulation and the energy estimate.

Minimization with differential inequality constraints applied to complete Lyapunov functions

Motivated by the desire to compute complete Lyapunov functions for nonlinear dynamical systems, we develop a general theory of discretizing a certain type of continuous minimization problems with differential inequality constraints. The resulting discretized problems are quadratic optimization problems, for which there exist efficient solution algorithms, and we show that their unique solutions converge strongly in appropriate Sobolev spaces to the unique solution of the original continuous problem. We develop the theory and present examples of our approach, where we compute complete Lyapunov functions for nonlinear dynamical systems. A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is divided into the chain-recurrent set, where the complete Lyapunov function is constant along solutions, and the part characterizing the gradient-like flow, where the complete Lyapunov function is strictly decreasing along solutions. We propose a new method to compute a complete Lyapunov function as the solution of a quadratic minimization problem, for which no information about the chain-recurrent set is required. The solutions to the discretized problems, which can be solved using quadratic programming, converge to the complete Lyapunov function.

Morse-Bott energy function for surface Ω-stable flows

In this paper, we consider the class of Ω-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of Ω -stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of saddle connections that do not form cycles. The authors have constructed the Morse-Bott energy function for any such flow. The results obtained are an ideological continuation of the classical works of S. Smale, who proved the existence of the Morse energy function for gradient-like flows, and K. Meyer, who established the existence of the Morse-Bott energy function for Morse-Smale flows. The specificity of Ω-stable flows takes them beyond the framework of structural stability, but the decrease along the trajectories of such flows is still tracked by the regular Lyapunov function.

On Topological Classification of Gradient-like Flows on an $$n$$-sphere in the Sense of Topological Conjugacy

In this paper, we study gradient-like flows without heteroclinic intersections on an $$n$$ -sphere up to topological conjugacy. We prove that such a flow is completely defined by a bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we show that such a tree is a complete invariant for these flows with respect to the topological equivalence also. This result implies that for these flows with the same (up to a change of coordinates) partitions into trajectories, the partitions for elements, composing isotopies connecting time-one shifts of these flows with the identity map, also coincide. This phenomenon strongly contrasts with the situation for flows with periodic orbits and connections, where one class of equivalence contains continuum classes of conjugacy. In addition, we realize every connected bicolor tree by a gradient-like flow without heteroclinic intersections on the $$n$$ -sphere. In addition, we present a linear-time algorithm on the number of vertices for distinguishing these trees.

Criterion for the Topological Conjugacy of Multi-Dimensional Gradient-Like Flows with No Heteroclinic Intersections on a Sphere

We study gradient-like flows with no heteroclinic intersections on an n-dimensional (n ≥ 3) sphere from the point of view of topological conjugacy. We prove that the topological conjugacy class of such a flow is completely determined by the bicolor tree corresponding to the frame of separatrices of codimension 1. We show that for such flows the notions of topological equivalence and topological conjugacy coincide (which is not the case if there are limit cycles and connections.

On functional moduli of surface flows

Currently, an complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9, 7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in [3]. In the classical paper [8], it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.

Topological Classification of Gradient-Like Flows with Surface Dynamics on 3-Manifolds

Topological Classification of Gradient-Like Flows with Surface Dynamics on 3-Manifolds