Discrete-Time Nonautonomous Dynamical Systems

Abstract

These notes present and discuss various aspects of the recent theory for time-dependent difference equations giving rise to nonautonomous dynamical systems on general metric spaces:First, basic concepts of autonomous difference equations and discrete-time (semi-) dynamical systems are reviewed for later contrast in the nonautonomous case. Then time-dependent difference equations or discrete-time nonautonomous dynamical systems are formulated as processes and as skew products. Their attractors including invariants sets, entire solutions, as well as the concepts of pullback attraction and pullback absorbing sets are introduced for both formulations. In particular, the limitations of pullback attractors for processes is highlighted. Beyond that Lyapunov functions for pullback attractors are discussed.Two bifurcation concepts for nonautonomous difference equations will be introduced, namely attractor and solution bifurcations.Finally, random difference equations and discrete-time random dynamical systems are investigated using random attractors and invariant measures.