On the Application of KAM Theory to Discontinuous Dynamical Systems

AbstractSo far the application of Kolmogorov–Arnold–Moser (KAM) theory has been restricted to smooth dynamical systems. Since there are many situations which can be modeled only by differential equations containing discontinuous terms such as state-dependent jumps (e.g., in control theory or nonlinear oscillators), it is shown by a series of transformations how KAM theory can be used to analyze the dynamical behaviour of such discontinuous systems as well. The analysis is carried out for the examplex+x+asgn(x)=p(t)withp∈C6being periodic. It is known that all solutions are unbounded for smalla>0. We prove that all solutions are bounded fora>0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case