Existence of \(T/k\)-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvalue
The author considers the \(n\)-dimensional system of differential equations of the following form \[ \dot{Y} = AY + BF(\sigma) + Kf(t). \] It is assumed that the matrix \(A\) has real nonzero eigenvalues, among which there are at least one positive and one multiple eigenvalues, the vectors \(B\) and \(K\) are nonzero, and the function \(F(\sigma),\)\,\(\sigma = (C,Y)\) describes a nonideal two-position relay with two output values. The perturbation function \(f(t)\) is \(T\)-periodic with \(T = 2\pi/\omega,\,\omega >0\) of the form \[ f(t) = f_0 +f_1\sin(\omega t + \varphi_1) + f_2\sin (2\omega t + \varphi_2). \] Necessary conditions for the existence of a \(T/k\)-periodic solution of the system with \(k \in \mathbb{N}\) having two switching points in the phase space are studied. An existence theorem for a \(T\)-periodic solution is proved. A numerical example is presented.
system of ordinary differential equations
scientific article; zbMATH DE number 7369730
scientific article