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

Abstract

In this paper, we are concerned with superlinear impact oscillatorsof 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 $f$ isbounded. We prove the existence of globally defined bouncingsolutions with prescribed number of impacts in the intervalsof negativity and positivity of $q$. Furthermore, we showthat when $q$ is periodic, the equation under consideration exhibits an interesting phenomenonof 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 forthe Hill's type equation in case of $f\equiv0.$