Singularity Analysis and Integrability for Discrete Dynamical Systems

Evidence that suggests a necessary consequence of integrability for discrete dynamical systems is presented. The evidence is analogous to that provided by the Painlevé test of integrability for continuous dynamical systems. Autonomous discretizations of several integrable continuous systems are studied. It is shown that those discrete systems exhibiting no chaotic behaviour, in the complex space of the iterates, possess solutions that have a property analogous to the Painlevé property of their corresponding continuous dynamical system.