MSC2010 37D05
Energy function for $\Omega$-stable flows without limit cycles on surfaces
A. E. Kolobyanina1, V. E. Kruglov2
| 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 |
1Anna 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
2Vladislav 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
