Impact oscillators of Hill's type with indefinite weight:Periodic and chaotic dynamics

Chao  Wang; Dingbian Qian; Qihuai  Liu

doi:10.3934/dcds.2016.36.2305

Article Contents

2016, Volume 36, Issue 4: 2305-2328. Doi: 10.3934/dcds.2016.36.2305

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Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics

1.
School of Mathematic Science, Yancheng Teacher's University, Yancheng 224001
2.
School of Mathematical Sciences, Soochow University, Suzhou 215006
3.
School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin, 541003

Received: October 31, 2014

Revised: June 30, 2015

Published: May 2017

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Abstract

In this paper, we are concerned with superlinear impact oscillators of Hill's type with indefinite weight $$ \left\{\begin{array}{lll} x''+f(x)x'+q(t)g(x)=0, ~\text{for}~ x(t)>0;\\ x(t)\geq0;\\ x'(t_0+)=-x'(t_0-),~\text{if}~x(t_0)=0,\end{array}\right.$$ where the indefinite weight $q(t)$, defined in $(a,b)$ with $-\infty\leq a< b \leq+\infty,$ has infinitely many zeros in $(a,b),$ $g$ is superlinear and $f$ is bounded. We prove the existence of globally defined bouncing solutions with prescribed number of impacts in the intervals of negativity and positivity of $q$. Furthermore, we show that when $q$ is periodic, the equation under consideration exhibits an interesting phenomenon of chaotic-like dynamics. Finally, in case that $q$ is even and periodic, we prove the existence and multiplicity of the even and periodic bouncing solutions for the Hill's type equation in case of $f\equiv0.$

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Mathematics Subject Classification: Primary: 34C15, 34C28; Secondary: 54H20.

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