Boundedness of Solutions for a Class of Second-Order Periodic Systems

In this paper we study the following second-order periodic system:x′′+V′(x)+p(x,t)=0,whereV(x)has a singularity. Under some assumptions on theV(x)andp(x,t)by Ortega’ small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems

We study the following two-order differential equation,(Φp(x'))'+f(x,t)Φp(x')+g(x,t)=0,whereΦp(s)=|s|(p-2)s,p>0.f(x,t)andg(x,t)are real analytic functions inxandt,2aπp-periodic inx, and quasi-periodic intwith frequencies(ω1,…,ωm). Under some odd-even property off(x,t)andg(x,t), we obtain the existence of invariant curves for the above equations by a variant of small twist theorem. Then all solutions for the above equations are bounded in the sense ofsupt∈R|x′(t)|<+∞.