Dynamics of the Caputo fractional derivative

Abstract

In this article we analyse the dynamical behaviour of the Caputo complex fractional derivative. We prove that the Caputo complex fractional derivative operator is Devaney chaotic in the Mittag-Leffler Caputo space. We will also show that a tuple of different iterates of a Caputo derivative multiple is disjoint hypercyclic. Finally, we provide sufficient conditions that ensure chaos for one of the most common numerical approximation of the Caputo derivative, that is, its L1 discretization.