A differential equation for undamped forced non-linear oscillations. I

Abstract

In this paper and its sequels I consider the differential equationin which e(t) is an even periodic function of t and dots denote differentiation with respect to t. We can regard (1·1) as a special case of the equation(the ‘differential equation of non-linear oscillations’), which describes the motion on the x–axis of a particle of unit mass subject to a restoring force g(x), a variable damping force f(x, ẋ)ẋ and an external force p(t); the dynamical description used in the title appeals to this interpretation. The most interesting problems on (1·2) and its specializations concern the behaviour of solutions as t → ∞ for example, we may seek conditions on f, g and p which allow us to assert that some or all solutions are bounded or, having assumed that p(t) is periodic, seek conditions on f and g which allow us to assert the existence of periodic solutions.