Dynamical behaviors for generalized pendulum type equations with <i>p</i>-Laplacian期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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 $(ϕ p (x ′)) ′ + G x ′ (t, x) = p (t)$ , where $ϕ p (u) = \| u \| p − 2 u, p > 1, G (t, x)$ 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 $∫ 01 p (t) d t = 0,$ 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 $G x ′ (t, x)$ 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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