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

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  • 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 $f\equiv0.$
    Mathematics Subject Classification: Primary: 34C15, 34C28; Secondary: 54H20.


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