Approximate determination of periodic solutions of a class of non-linear differential equations

Abstract

In this paper a special technique is presented for the approximate representation of periodic solutions for a class of second order non-linear differential equations of the form x + x + g(x) = a sin ωt. Here g(x) is Lipschitz continuous, bounded and of saturation type. Our method is to approximate g(x) arbitrarily closely by a multi-step relay and then solve the resulting multi-step problem. In an earlier paper [2], the authors have derived an explicit representation for a harmonic solution of the multi-step problem as a linear combination of phase-shifted (known) solutions of a simpler (ideal relay) equation. The phase shifts, which are the ‘switching times’ in the solution of the multi-step relay problem, are the roots of a system of transcendental equations. Thus, an accurate and efficient solution of the system of transcendental equations, is a crucial step in our procedure. A good part of the paper is devoted to this and related questions. To illustrate our technique, we also present some numerical results.