Morse Decompositions for Delay-Difference Equations

Abstract

Scalar difference equations x k+1 = f (x k , x k−d ) with delay d ∈ N are well-motivated from applications e.g. in the life sciences or discretizations of delay-differential equations. We investigate their global dynamics by providing a (nontrivial) Morse decomposition of the global attractor. Under an appropriate feedback condition on the second variable of f , our basic tool is an integer-valued Lyapunov functional.