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Энергетическая функция для Омега-устойчивых потоков без предельных циклов на поверхностях

Колобянина А. Е., Круглов В. Е.

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.