Abstract
In this paper we study the following second-order periodic system: x �� + V � (x )+ p(x)f (t )=0 , where V(x) has a singularity. Under some assumptions on the V(x), p(x )a ndf (t), by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.
Highlights
Introduction and main resultIn the early s, Littlewood [ ] asked whether or not the solutions of the Duffing-type equations, x + g(x, t) =, ( . )are bounded for all time, i.e., whether there are resonances that might cause the amplitude of the oscillations to increase without bound.The first positive result of boundedness of solutions in the superlinear case (i.e., g(x,t) x →∞ as |x| → ∞) was due to Morris [ ]
In this paper we study the following second-order periodic system: x + V (x) + p(x)f (t) = 0, where V(x) has a singularity
Are bounded for all time, i.e., whether there are resonances that might cause the amplitude of the oscillations to increase without bound
Summary
The proof of the second inequality is similar to that of the first one, so we only give a brief proof. . Remark It follows from the definitions of T+(h), T–(h) and Lemma that lim h→+∞. Remark Note that h = h (I ) is the inverse function of I. We carry out the standard reduction to t√he action-angle variables. For this purpose, we define the generating function S(x, I) = C (h – V (s)) ds, where C is the part of the closed curve h connecting the point on the y-axis and point (x, y). Lemma For I sufficient large and –αh ≤ x < , the following estimates hold: In. Dθ ∂H is a Hamiltonian system with the Hamiltonian function I and the action, angle and time variables are H, t and θ
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