On the application of KAM theory to discontinuous dynamical systems
This paper deals with the following discontinuous systems \[ x''+ x + a \text{ sgn}(x) =p(t), \] where \(p\in C^6(S^1)\). It is proved via Moser's twist theorem that the above-mentioned equation possesses infinitely many periodic and quasiperiodic solutions and all its solutions are bounded if \(a>0\) is sufficiently large. There have been many papers concerning the application of KAM theory to continuous systems [\textit{M. Levi}, Commun. Math. Phys. 143, No. 1, 43-83 (1991; Zbl 0744.34043); \textit{R. Dieckerhoff} and \textit{E. Zehnder}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 1, 79-95 (1987; Zbl 0656.34027); \textit{R. Ortega}, J. Lond. Math. Soc., II. Ser. 53, No. 2, 325-342 (1996; Zbl 0860.34017); etc.], but this paper is the first about discontinuous systems.
scientific article; zbMATH DE number 1067761
scientific article