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Boundedness of solutions for a class of impact oscillators with time-dependent polynomial potentials. (English) Zbl 1311.34070

By making use of action-angle variables and Moser’s small twist theorem, the authors prove the boundedness of the solutions of a nonlinear impact oscillator with periodically time-dependent polynomial potential given by \[ \ddot{x}+x^{2n+1}+\sum_{i=1}^{2n} p_i(t)x^i=0, \] with the impact condition \[ x(t_0)=0\implies \dot{x}(t_0^+)=-\dot{x}(t_0^-), \] and the regularity assumption \(p_i\in C^5\).

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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