Oberseminar "Dynamische Systeme und Kontrolltheorie" - Prof. Dr. Peter Kloeden

Caputo fractional differential equations (FDE) of order α ∈ (0, 1) in Rd do not generate a semi-dynamical system in Rd . Sell observed that the Volterra integral equation associated with an autonomous Caputo FDE of order α ∈ (0, 1) in Rd generates a semi-group on the space C(R, Rd ) of continuous functions f : R+ → Rd with the topology uniform convergence on compact subsets. Doan & Kloeden showed that this gives a semi-dynamical system for the Caputo FDE when restricted to initial functions f (t) ≡ idx0 for x0 ∈ Rd . Here it is shown that this semi-dynamical system has a global Caputo attractor in C, which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition. The analysis is complicated by the fact that the usual methods of classical calculus do not hold for fractional calculus.