**MSC2010** 37D05

### Energy function for $\Omega$-stable flows without limit cycles on surfaces

#### A. E. Kolobyanina^{1}, V. E. Kruglov^{2}

Annotation | The paper is devoted to the study of the class of $\Omega$-stable flows without limit cycles on surfaces, i.e. flows on surfaces with non-wandering set consisting of a finite number of hyperbolic fixed points. This class is a generalization of the class of gradient-like flows, differing by forbiddance of saddle points connected by separatrices. The results of the work are the proof of the existence of a Morse energy function for any flow from the considered class and the construction of such a function for an arbitrary flow of the class. Since the results are a generalization of the corresponding results of K. Meyer for Morse-Smale flows and, in particular, for gradient-like flows, the methods for constructing the energy function for the case of this article are a further development of the methods used by K. Meyer, taking in sense the specifics of $\Omega$-stable flows having a more complex structure than gradient-like flows due to the presence of the so-called "chains" of saddle points connected by their separatrices. |
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Keywords | energy function, $\Omega$-stable flow, Morse function, a flow without limit cycles, a flow on a surface |

^{1}**Anna E. Kolobyanina**, Student, Faculty of Informatics, Mathematics and Computer Science, Research Assistant of Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics in Nizhny Novgorod (25/12, Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID: http://orcid.org/0000-0001-5312-4478, akolobyanina@mail.ru^{2}**Vladislav E. Kruglov**, Doctoral Student, Department of Fundamental Mathematics, Research Assistant of Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics in Nizhny Novgorod (25/12, Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), ORCID: http://orcid.org/0000-0003-4661-0288, kruglovslava21@mail.ru

**Citation**: A. E. Kolobyanina, V. E. Kruglov, "[Energy function for $\Omega$-stable flows without limit cycles on surfaces]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,21:4 (2019) 460–468 (In Russian)

**DOI** 10.15507/2079-6900.21.201904.460-468