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We consider the existence of invariant curves of real analytic reversible mappings which are quasi-periodic in the angle variables. By the normal form theorem, we prove that under some assumptions, the original mapping is changed into its linear part via an analytic convergent transformation, so that invariants curves are obtained. In the iterative process, by solving the modified homological equations, we ensure that the transformed mapping is still reversible. As an application, we investigate the invariant curves of a class of nonlinear resonant oscillators, with the Birkhoff constants of the corresponding Poincaré mapping all zeros or not. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Littlewood's problem for the second order differential equation x ¨ + x 2 n + 1 + q (t) x l = p (t) , l < n , where the functions p (t) and q (t) do not need to be periodic. Using the theory of non-periodic twist mappings which is developed by Kunze and Ortega in [3–7] , we obtained longtime closeness results on the bounded and unbounded solutions of the equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the boundedness of all solutions for the periodic equation , where ω satisfies the Diophantine condition, is bounded, and is bounded for . [Copyright &y& Elsevier]