On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions

Abstract

We establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems (set-valued dynamical systems). We also investigate the dynamical behavior of nonautonomous periodic differential inclusion x ′ ( t ) ∈ f ( t , x ( t ) ) on R m with only upper semi-continuous right-hand side by applying the abstract results. Firstly, we show that if the system has a compact uniformly attracting set, then it has a periodic nonautonomous uniform forward attractor A . Secondly, we prove that A is robust with respect to both internal and external perturbations. Finally, we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system.