Invariant graphs for forced systems

Abstract

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When such contraction is uniform, it is well known that there exists a globally attracting invariant set which is the graph of a function φ from the driving state space to the driven state space. If the contraction is sufficiently strong, then φ is smooth. We describe the theoretical framework for such results and discuss a number of applications to the filtering of time series, to synchronization and to quasiperiodically forced systems. We go on to give recent generalizations to the case of non-uniform contraction; that is contraction measured by Lyapunov exponent like quantities. We conclude with a number of new results for quasiperiodic forcing; a corollary of these is that a strange non-chaotic attractor for such a system cannot be the graph of a continuous function, nor roughly speaking can it have as open attracting neighbourhood; in other words its closure must contain some repelling orbits.