On almost periodic solutions for undamped systems with almost periodic forcing

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Using the method of averaging, we give sufficient conditions for the existence of an almost periodic solution of an undamped oscillatory system with almost periodic forcing, and show that these can be applied to a Duffing equation with almost periodic forcing provided the nonlinear term is sufficiently small, and the natural frequency of the linear part of the system is included in the set of Fourier exponents of the forcing function. It is well known that the real linear second order scalar differential equation + x = f (t), with f almost periodic (a.p. for short), cannot have almost periodic solutions, or even a solution bounded on the real line, whenever 1 is a Fourier exponent off It is the purpose of this note to obtain a result which will show that for such so-called resonance cases, there exist sufficiently small perturbations of this equation, involving only functions of x which have a.p. solutions. More precisely, iff has Fourier exponent 1 and one of the corresponding coefficients of the sine or cosine term in the Fourier series is zero, then there exist positive numbers v and E0 = E0(v) such that for 0 < E < E0 the equation (1) X +X-EVX+ E3X3f=(t) will have an a.p. solution. The magnitude of v is proportional to the absolute value of the Fourier coefficient corresponding to the Fourier exponent 1 in the series forf This a.p. solution is unstable in the sense of exhibiting the so-called saddle-point property, and the suprema of its absolute value and that of its derivative over all real t become unbounded as E --0. This property of (1) seems interesting for cases wheref has a sequence {2k}, k = 1, 2, * , of Fourier exponents, 2' < 1, and 2'-? 1 as k-xo. Received by the editors February 22, 1971. AMS 1970 subject classifications. Primary 34C25; Secondary 34C30.