ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)
Energy Function for Direct Products of Discrete Dynamical Systems
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
Submitted: 27.01.2023; Revised: 04.03.2023; Accepted: 25.05.2023
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, firstname.lastname@example.org
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, email@example.com
All authors have read and approved the final manuscript.
Conflict of interest: The authors declare no conflict of interest.