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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 article, we prove Lyapunov behavior and dynamical localization for analytic quasi-periodic CMV matrices in the regime of positive Lyapunov exponents. The localization results of Wang and Damanik [28] are extended. [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]

In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for almost all frequencies and large coupling parameter, which establish the analogous results of one-dimensional Schrödinger operators proved by Bourgain et al. [7]. [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]

Abstract: In this paper we consider the problem of the boundedness of all solutions for the reversible system It is shown that all the solutions are bounded provided that the are of bounded variation in and the derivatives of and are Lipschitzian. It is also shown that there exist ''s being discontinuous everywhere such that all solutions of the equation are bounded. This implies that the continuity of ''s is not necessary for the boundedness of solutions of the equation. [Copyright &y& Elsevier]

In this paper we investigate the existence and uniqueness of periodic and almost periodic solution of the differential equation with reflection of argument. The relationship between modules of forced term and solution of the equation is considered. [Copyright &y& Elsevier]

The authors discuss the existence of pseudo almost periodic solutions of differential equations with piecewise constant argument by means of introducing new concept, pseudo almost periodic sequence. [ABSTRACT FROM AUTHOR]

The authors study the Lagrangian stability for the sublinear Duffing equations ẍ + e(t)∣x∣α−1x = p(t) with 0 < α < 1, where e and p are real analytic quasi-periodic functions with frequency ω. It is proved that if the mean value of e is positive and the frequency ω satisfies Diophantine condition, then every solution of the equation is bounded. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we generalize the comparison result of Bostan and Namah (2007)  to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses. [Copyright &y& Elsevier]