Existence of quasiperiodic solutions and Littlewood’s boundedness problem of super-linear impact oscillators

So far most application of Kolmogorov–Arnold–Moser (KAM) theory has been restricted to smooth dynamical systems. In this paper, it is shown by a series of transformations that how KAM theory can be used to analyze the dynamical behavior of Duffing-type equations with impact. The analysis is carried out for the exampleequation(0.1)x¨+x2n+1=p(t),forx(t)>0,x(t)⩾0,x˙(t0+)=-x˙(t0-),ifx(t0)=0with p ∈ C5 being periodic. We prove that all solutions are bounded, and that there are infinitely many periodic and quasiperiodic solutions in this case.