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MSC2020 37D05

Morse-Bott energy function for surface $\Omega$-stable flows

A. E. Kolobyanina1, V. E. Kruglov2

AnnotationIn this paper, we consider the class of $\Omega$-stable flows on surfaces, i.e. flows on surfaces with the non-wandering set consisting of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles. The class of $\Omega$ -stable flows is a generalization of the class of Morse-Smale flows, admitting the presence of saddle connections that do not form cycles. The authors have constructed the Morse-Bott energy function for any such flow. The results obtained are an ideological continuation of the classical works of S. Smale, who proved the existence of the Morse energy function for gradient-like flows, and K. Meyer, who established the existence of the Morse-Bott energy function for Morse-Smale flows. The specificity of $\Omega$-stable flows takes them beyond the framework of structural stability, but the decrease along the trajectories of such flows is still tracked by the regular Lyapunov function.
Keywords$\Omega$-stable flow, energy function, limit cycle, Morse-Bott function, surface

1Anna E. Kolobyanina, student of Faculty of Informatics, Mathematics and Computer Science, National Research University Higher School of Economics, Russian Federation (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0001-5312-4478, akolobyanina@mail.ru

2Vladislav E. Kruglov, graduate Student, Lecturer, Department of Fundamental Mathematics, Research Fellow, International Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, Russian Federation (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0003-4661-0288, kruglovslava21@mail.ru

Citation: A. E. Kolobyanina, V. E. Kruglov, "[Morse-Bott energy function for surface $\Omega$-stable flows]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,22:4 (2020) 434–441 (In Russian)

DOI 10.15507/2079-6900.22.202004.434-441