# American Institute of Mathematical Sciences

April  2016, 36(4): 2305-2328. doi: 10.3934/dcds.2016.36.2305

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

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

Received  November 2014 Revised  July 2015 Published  September 2015

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 $f\equiv0.$
Citation: Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305
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