Geodesic
Energy function for $\Omega$-stable flows without limit cycles on surfaces
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 460-468

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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.
Keywords: energy function, $\Omega$-stable flow, Morse function, a flow without limit cycles, a flow on a surface. 
@article{SVMO_2019_21_4_a4,
     author = {A. E. Kolobyanina and V. E. Kruglov},
     title = {Energy function for $\Omega$-stable flows without limit cycles on surfaces},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {460--468},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a4/}
}
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A. E. Kolobyanina; V. E. Kruglov. Energy function for $\Omega$-stable flows without limit cycles on surfaces. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 460-468. http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a4/