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

Vladislav Kruglov; Dmitry Malyshev; Olga Pochinka

doi:10.3934/dcds.2018188

Article Contents

2018, Volume 38, Issue 9: 4305-4327. Doi: 10.3934/dcds.2018188

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

HSE; Bolshaya Pecherskaya 25/12, Nizhniy Novgorod, 603155, Russia

^* Corresponding author: Olga Pochinka
^* Corresponding author: Olga Pochinka

Received: May 2017

Revised: April 2018

Published: September 2018

Authors are grateful to participants of the seminar "Topological Methods in Dynamics" for fruitful discussions. The classification results (Sections 1–6 without Subsections 5.2, 5.3) were obtained with the support of the Russian Science Foundation (project 17-11-01041). The realisation results (Subsection 5.2, Section 7) were obtained as an output of the research project "Topology and Chaos in Dynamics of Systems, Foliations and Deformation of Lie Algebras (2018)" implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE). The algorithmic results (Subsection 5.3, Section 8) were obtained with the support of Russian Foundation for Basic Research 16-31-60008-mol-a-dk and with LATNA laboratory, National Research University Higher School of Economics

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Abstract

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to $Ω$-stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.

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Mathematics Subject Classification: 37D05.

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Figure 1. The case when $U_\mathfrak c$ is homeomorphic to a Möbius band

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Figure 2. $\phi^t$ and $\Upsilon_{\phi^t}$

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Figure 3. The cases of the consistent (leftward) and the inconsistent (rightward) orientation of boundary's connecting component of some $\mathcal E$-region

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Figure 4. A polygonal region

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Figure 5. An example of the flow $f^t$ together with the polygonal regions

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Figure 6. An example of $f^t$ and its four-colour graph

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Figure 7. Two flows from $G$ and their equipped graphs

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Figure 8. Two examples of flows from $G$ differing only by orientation of the limit cycle between $\mathcal M$ and $\mathcal A$ and their equipped graphs

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Figure 9. Two examples of flow from $G$ without $\mathcal A$- and $\mathcal M$-regions differing only by orientation of the limit cycle and their equipped graphs

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Figure 10. $f^t$, $\Gamma_{\mathcal M}$ and $\Gamma^*_{{\mathcal M}}$

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