On reducibility of a class of nonlinear quasi-periodic systems with small perturbational parameters near equilibrium

Consider the reducibility of a class of nonlinear quasi-periodic systems with multiple eigenvalues under perturbational hypothesis in the neighborhood of equilibrium. That is, consider the following system x = （A ＋ εQ（ t） ）x ＋ eg（t） ＋ h（x, t）, where A is a constant matrix with multiple eigenvalues; h = O（x2） （x-4）） ; and h（x, t）, Q（t）, and g（t） are analytic quasi-periodic with respect to t with the same frequencies. Under suitable hypotheses of non-resonance conditions and non-degeneracy conditions, for most sufficiently small ε, the system can be reducible to a nonlinear quasi-periodic system with an equilibrium point by means of a quasi-periodic transformation.

Reducibility for a Class of Two-Dimensional almost Periodic System with Small Perturbation

This paper focuses on the reducibility of two-dimensional almost periodic system with small perturbation. We use the KAM iterative method to get the reducibility by an almost periodic transformation. The system has been reduced to a simple form. So we have dealt with the small perturbation problem of the almost periodic system.

Invariant Hyperbolic Tori for Gevrey-smooth Hamiltonian Systems Under Rfissmann's Non-degeneracy Condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.

Two-Dimensional Invariant Tori in the Neighborhood of an Elliptic Equilibrium of Hamiltonian Systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

We consider small perturbations of analytic non-twist area preserving mappings,and prove the existence of invariant curves with prescribed frequency by KAM iteration.Generally speaking,the frequency of invariant curve may undergo some drift,if the twist condition is not satisfied.But in this paper,we deal with a degenerate situation where the unperturbed rotation angle function r→w+r^(2n+1)is odd order degenerate at r=0,and prove the existence of invariant curve without any drift in its frequency.Furthermore,we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

Lyapunov Center Theorem of Infinite Dimensional Hamiltonian Systems

In this paper we reformulate a Lyapunov center theorem of infinite dimensional Hamiltonian systems arising from PDEs.The proof is based on a modified KAM iteration for periodic case.

Quasi-periodic Solutions for the Derivative Nonlinear Schrdinger Equation with Finitely Differentiable Nonlinearities

The authors are concerned with a class of derivative nonlinear Schr¨odinger equation iu_t + u_(xx) + i?f(u, ū, ωt)u_x=0,(t, x) ∈ R × [0, π],subject to Dirichlet boundary condition, where the nonlinearity f(z1, z2, ?) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.