Abstract
In this paper, we study the following second-order periodic system: where has a singularity. Under some assumptions on the and 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, Levi [ ] considered the following equation:x + V (x, t) =, ( . )where V (x, t) satisfies some growth conditions and V (x, t) = V (x, t + )
The author reduced the system to a normal form and applied Moser twist theorem to prove the existence of quasi-periodic solution and the boundedness of all solutions
This result relies on the fact that the nonlinearity V (x, t) can guarantee the twist condition of KAM theorem
Summary
Remark It follows from the definitions of T+(h), T–(h) and Lemma that lim T–(h) = , 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). ] For I sufficient large and –αh ≤ x < , the following estimates hold: In. = (t, H, θ), dθ ∂t dθ ∂H is a Hamiltonian system with Hamiltonian function I and the action, angle and time variables are H, t and θ
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