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On Algorithms that Effectively Distinguish Gradient-Like Dynamics on Surfaces

  • Vladislav E. KruglovEmail author
  • Dmitry S. Malyshev
  • Olga V. Pochinka
Research Contribution
  • 6 Downloads

Abstract

In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas’s invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time.

Keywords

Gradient-like dynamics Effective distinguishing algorithms Graph invariants Surface 

Notes

Acknowledgements

The algorithm for the Peixoto invariants was created with a support of the Russian Foundation for Basic Research Project 18-31-00022 mol\(\underline{\,\,\,}\)a. The polynomial-time algorithm for the Wang invariant was obtained as an output of the research project of the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019. The algorithms for the Fleitas’ graph invariant was created under a support of RF President Grant MD-879.2019.1 and Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics. The polynomial-time algorithm for the Oshemkov-Sharko invariant and conjugacy theorem were implemented in the framework of the Russian Science Foundation Project 17-11-01041.

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Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019

Authors and Affiliations

  1. 1.HSENizhny NovgorodRussia

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