On the application of KAM theory to discontinuous dynamical systems

So 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 example (x) double over dot + x + a sgn(x) = p(t) with p is an element of C-6 being periodic. It is known that all solutions are unbounded for small a > 0. We prove that all solutions are bounded for a > 0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case. (C) 1997 Academic Press.