Speaker: Dr. Jonathan Meddaugh from Baylor University (invited by Will Brian)
Title: Shadowing in Dynamical Systems: When are approximate orbits meaningful?
Abstract: In a discrete dynamical system f: X → X, the orbit of a point x ϵ X is the sequence x, f(x), f²(x), . . . . In even modestly complex systems, small errors compound rapidly in computations of orbits, leading to true orbits and computed orbits that are wildly different. Surprisingly, however, there is a large class of systems for which these approximate orbits can yield useful qualitative information about the dynamics of the system—systems with shadowing. Informally, a dynamical system f: X → X has the shadowing property provided that for any specified level of tolerance, there is a level of precision that guarantees that any approximate orbit which is computed with this level of precision is itself approximated (shadowed) within the specified tolerance by a true orbit for the system. In this talk, we will discuss the shadowing property with a focus on recent results concerning characterizations of this property and its relative prevalence in certain classes of dynamical systems.