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Dynamical behaviors for generalized pendulum type equations with p-Laplacian
Authors:Yanmin NIU  Xiong LI
Affiliation:1. School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China2. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Abstract:We consider a pendulum type equation with p-Laplacian (ϕp(x))+Gx(t,x)=p(t), where ϕp(u)=|u|p2u,p>1,G(t,x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser's twist theorem, we find infinitely many invariant tori whenever 01p(t)dt=0, which yields the bounded-ness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and Gx(t,x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.
Keywords:p-Laplacian  invariant tori  quasi-periodic solutions  boundedness  complex dynamics  
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