TY - JOUR

T1 - On One Type of Oscillatory Solutions of a Nonautonomous System with Relay Hysteresis

AU - Yevstafyeva, V.V.

PY - 2024/6/1

Y1 - 2024/6/1

N2 - Abstract: We consider an -dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.

AB - Abstract: We consider an -dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.

KW - continuous bounded perturbation function

KW - discontinuous nonlinearity of relay type with hysteresis

KW - essentially nonlinear system

KW - multidimensional system of ordinary differential equations

KW - nonautonomous system

KW - oscillatory solution

KW - return period

KW - switching points and hyperplanes

KW - system of transcendental equations

UR - https://www.mendeley.com/catalogue/456436a6-8e1d-3bac-8406-0f5e583ebd78/

U2 - 10.1134/s0001434624050080

DO - 10.1134/s0001434624050080

M3 - Article

VL - 115

SP - 734

EP - 747

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 5-6

ER -