M. K. Barinova, E. K. Shustova

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

Key Words: Direct product, diffeomorphism, chain recurrent set, energy function

For citation: M. K. Barinova, E. K. Shustova. Energy Function for Direct Products of Discrete Dynamical Systems. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:2(2023), 11–21. DOI: https://doi.org/10.15507/2079-6900.25.202302.11-21

Information about the authors:

Marina K. Barinova, Senior Research Fellow, National Research University «High School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-4406-583X, mkbarinova@yandex.ru

Evgenia K. Shustova, student, National Research University «High School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-4998-2186, ekshustova@gmail.com