Analytic methods for obstruction to integrability in discrete dynamical systems

Abstract

AbstractA unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized Borel summability.This continuation allows for Painlevé property methods to be extended to difference equations.It is shown that the Painlevé property (PP) induces, under relatively general assumptions, a dichotomy within first‐order difference equations: all equations with PP can be solved in closed form; on the contrary, absence of PP implies, under some further assumptions, that the local conserved quantities are strictly local in the sense that they develop singularity barriers on the boundary of some compact set.The technique produces analytic formulas to describe fractal sets originating in polynomial iterations. © 2004 Wiley Periodicals, Inc.