Dynamics of a Forced Oscillator having an Obstacle

Abstract

Consider the scalar differential equation

$$\ddot x + g(x) = f(t)$$

(1.1)

where f is 2π-periodic, say $f \in C(\mathbb{T}{\text{)}} {\text{with}} \mathbb{T}{\text{ = }}\mathbb{R}{\text{/2}}\pi \mathbb{Z}$ and g satisfies

$$\mathop {\lim }\limits_{x \to \pm \infty } g(x) = \pm \infty , \mathop {\lim \sup }\limits_{|x| \to \infty } \frac{{g(x)}}{x} < \infty .$$

(1.2)

The existence of 2π-periodic solutions has been analyzed by many authors using different variational and topological methods. For the linear case it (g(x) = w ² x) is well known that the existence of a periodic solution is equivalent to the boundedness of all solutions, and one can ask whether such an equivalence still holds in nonlinear cases. In this paper we report on several results which give partial answers to this question. First we shall assume that g satisfies the assumptions of Lazer and Leach in [13] and we shall show that the condition for existence of a periodic solution obtained in that paper guarantees, in many cases, the boundedness of all solutions. For this class of nonlinearities the situation resembles the linear theory. Later we shall consider the asymmetric nonlinearities that were first discussed by Fucik [11] and Dancer [6, 5]. The situation now is more delicate because unbounded and periodic solutions can coexist. After this brief review of published results we shall analyze in detail the problem of boundedness for a forced linear oscillator which bounces elastically against a wall.