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JOURNALS // Eurasian Mathematical Journal // Archive

Eurasian Math. J., 2024 Volume 15, Number 2, Pages 48–60 (Mi emj501)

This article is cited in 1 paper

Dynamics of relay systems with hysteresis and harmonic perturbation

A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva

Saint Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg, 199034, Russia

Abstract: We consider a system of ordinary differential equations with a relay hysteresis and a harmonic perturbation. We propose an approach that allows one to decompose an $n$-dimensional system into one- and two-dimensional subsystems. The approach is illustrated by a numerical example for the system of dimension $3$. As a result of the decomposition, a two-dimensional subsystem with non-trivial Jordan block in right-hand side is studied. For this subsystem we prove a theorem on the existence and uniqueness of an asymptotically stable solution with a period being multiple to period of the perturbation. Moreover, we show how to obtain this solution by tuning the parameters defining the relay. We also provide a supporting example in this regard.

Keywords and phrases: multidimensional system of ordinary differential equations, relay hysteresis, harmonic perturbation, decomposition, parametric matrix, subsystems, Jordan block, asymptotically stable periodic solution.

MSC: 34C25, 34C55, 93C15

Received: 23.07.2023
Accepted: 25.01.2024

Language: English

DOI: 10.32523/2077-9879-2024-15-2-48-60



© Steklov Math. Inst. of RAS, 2025