Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs

Kruglov, V. E.; Malyshev, D. S.; Pochinka, O. V.

doi:10.3934/dcds.2018188

Cited by 9 publications

(7 citation statements)

References 10 publications

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“…Classification of such flows follows from classification of Morse-Smale flows of surfaces (see e.g. [2], [3], [4]). We provide an independent classification of class G 2 (M 2 ) in section 4.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

Pochinka,

Shubin

2021

Preprint

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In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of n-manifolds with two limit cycles. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincaré-Hopf theorem Euler characteristic of closed manifold M n which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of n = 2. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle. There are no constraints for odd-dimensional manifolds which follow from the fact that Euler characteristic is zero. However, it is known that orientable 3-manifold admits a flow of considered class if and only if it is a lens space. In this paper, it is proved that up to topological equivalence each of S 3 and RP 3 admit one such flow and other lens spaces two flows each. Also, it is shown that the only non-orientable n-manifold (for n > 2), which admits considered flows is the twisted I-bundle over (n − 1)sphere. Moreover, there are exactly two classes of topological equivalence of such flows. Among orientable n-manifolds only the product of (n − 1)sphere and the circle can be ambient manifold of a considered flow and the flows are split into two classes of topological equivalence.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

Pochinka,

Shubin

2021

Preprint

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“…From theoretical point of view, it is important to note that the flow of finite type contains Morse-Smale vector fields, which are dense in the set of surface flows. The classification of streamline topologies for Morse-Smale flows has been considered in [9,16,17] to which our theory would make contributions in terms of discrete combinatorics. In addition, we can apply the theory to the projection or the restriction of 3D vector fields on 2D sections.…”

Section: Discussionmentioning

confidence: 99%

“…The flow of finite type is thus a generalized class of vector fields containing structurally stable Hamiltonian flows as well as Morse-Smale flows, which are of mathematical significance class of vector fields [9,11,16,17,22]. In addition, from the application points of view, experimental/measurement data subject to noises and errors and a 2D projection of 3D incompressible flows admit a finite number of local orbit structures belonging to the flow of finite type.…”

Section: Flow Of Finite Type and The Topological Structure Theoremmentioning

confidence: 99%

Discrete representations of orbit structures of flows for topological data analysis

Sakajo¹,

Yokoyama²

2020

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This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequence of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of 2D incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse-Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows, called Topological Flow Data Analysis (TFDA), when applied to data brought by measurements, experiments and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.

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“…Oshemkov's construction of a three-coloured graph was generalized to the case of saddle-saddle connections, i.e. for flows that are not of Morse or Morse-Smale type [14]. An equipped multi-coloured graph is a complete invariant for Ω-stable flows on closed surfaces, each of which is a "Morse-Smale" flow without the non-existence condition of heteroclinic separatrices.…”

Section: Topological Invariants Of Flows On Surfacesmentioning

confidence: 99%

Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows

Kibkalo¹,

Yokoyama²

2021

Preprint

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It is known that C r Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any r ∈ Z>0. In particular, C r Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of C r gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows and instantaneous non-Morse gradient flows. To illustrate the generic transitions, we characterize and list all generic non-Morse gradient flows. To construct such characterizations, we characterize isolated singular points of gradient flows on surfaces. In fact, such a singular point is a non-trivial finitely sectored singular point without elliptic sectors. Moreover, considering Morse-Smale flows as "generic gradient flows with limit cycles", we characterize and list all generic non-Morse-Smale flows.

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Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs

Cited by 9 publications

References 10 publications

Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

Discrete representations of orbit structures of flows for topological data analysis

Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows

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