Oscillations of a forced asymmetric oscillator at resonance

Abstract

We consider the equation\nx " + mu x(+) - vx(-) = f(x) + g(x) + e(t)\nwhere x(+) = max{x, 0}; x(-) = max{-x, 0}, in a situation of resonance for the period 2 pi, i.e. when 1/root mu +1 root upsilon = 2/n for some integer n. We assume that e is 2 pi-periodic, that f has limits f(+/-infinity) at +/-infinity, and that the function g has a sublinear primitive. Denoting by phi a solution of the homogeneous equation x " + mu x(+) - vx(-) = 0, we show that the behaviour of the solutions of the full nonlinear equation depends crucially on whether the function\nPhi(theta) = n/pi [f(+infinity)/mu - f(-infinity)/upsilon] +1/2 pi integral(0)(2 pi) e(t)phi(t+theta) dt\nis of constant sign or not. In particular, existence results for 2 pi-periodic and for subharmonic solutions, based on the function Phi, are given.