The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x ′ = y , y ′ = − x − y z , and z ′ = y 2 − a , where a is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever a is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.

Response Solutions for a Singularly Perturbed System Involving Reflection of the Argument

In this paper, the existence and uniqueness of response solutions, which has the same frequency ω with the nonlinear terms, are investigated for a quasiperiodic singularly perturbed system involving reflection of the argument. Firstly, we prove that all quasiperiodic functions with the frequency ω form a Banach space. Then, we obtain the existence and uniqueness of quasiperiodic solutions by means of the fixed-point methods and the B-property of quasiperiodic functions.

A New Approach to the Existence of Quasiperiodic Solutions for Second-Order Asymmetric p-Laplacian Differential Equations

For p≥2 and ϕp(s):=sp-2s, we propose a new estimate approach to study the existence of Aubry-Mather sets and quasiperiodic solutions for the second-order asymmetric p-Laplacian differential equations ϕpx′′+λϕp(x+)-μϕp(x-)=ψ(t,x), where λ and μ are two positive constants satisfying λ-1/p+μ-1/p=2/ω with ω∈R+, ψ(t,x)∈C0,1(Sp×R) is a continuous function, 2πp-periodic in the first argument and continuously differentiable in the second one, x±=max⁡{±x,0}, πp=2π(p-1)1/p/psin⁡π/p, and Sp=R/2πpZ. Using the Aubry-Mather theorem given by Pei, we obtain the existence of Aubry-Mather sets and quasiperiodic solutions under some reasonable conditions. Particularly, the advantage of our approach is that it not only gives a simpler estimation procedure, but also weakens the smoothness assumption on the function ψ(t,x) in the existing literature.