Skew-product attractors of non-autonomous Caputo fractional differential equations

Abstract

A non-autonomous Caputo fractional differential equation (FDE) of order α∈(0,1) in Rd with a driving system on a compact base space P is shown to generate a skew-product semi-flow on Cα×P, where Cα is the space of continuous functions f:R+→Rd with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here Cα. The properties and structure of this attractor in Cα×P are also discussed.