Open AccessEnergy function for Ω-stable flows without limit cycles on surfaces
Author(s) -
Anna E. Kolobyanina,
V. E. Kruglov
Publication year - 2019
Publication title -
žurnal srednevolžskogo matematičeskogo obŝestva
Language(s) - English
Resource type - Journals
eISSN - 2587-7496
pISSN - 2079-6900
DOI - 10.15507/2079-6900.21.201904.460-468
Subject(s) - mathematics , generalization , flow (mathematics) , saddle point , function (biology) , class (philosophy) , saddle , balanced flow , limit (mathematics) , mathematical analysis , maxima and minima , morse theory , pure mathematics , energy (signal processing) , geometry , computer science , mathematical optimization , evolutionary biology , artificial intelligence , biology , statistics
The paper is devoted to the study of the class of Ω-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 Ω-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.