This article is cited in 13 papers

Existence of $T/k$-Periodic Solutions of a Nonlinear Nonautonomous System Whose Matrix Has a Multiple Eigenvalue

V. V. Yevstafyeva

Saint Petersburg State University

Abstract: A system of $n$th-order ordinary differential equations with relay nonlinearity and periodic perturbation function on the right-hand side is studied. The matrix of the system has real nonzero eigenvalues, among which there is at least one positive and one multiple eigenvalue. A nonsingular transformation that reduces the matrix of the system to Jordan form is used. Continuous periodic solutions with two switching points in the phase space of the system are considered. It is assumed that the period of the perturbation function is a multiple of the periods of these solutions. Necessary conditions for the existence of such solutions are established. An existence theorem for a solution of period equal to the period of the perturbation function is proved. A numerical example confirming the obtained results is presented.

Keywords: system of ordinary differential equations, relay nonlinearity with hysteresis, periodic perturbation function, multiple eigenvalue, canonical transformation, Jordan matrix, periodic solution, switching points, switching points.

UDC: 517.925

PACS: N/A

Received: 14.04.2019
Revised: 24.11.2020

DOI: 10.4213/mzm12411

 English version: Mathematical Notes, 2021, 109:4, 551–562

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