From limit cycles to periodic orbits through ultradiscretisation

We examine the transition between discrete and ultradiscrete (cellular-automaton-like) systems, the dynamics of which exhibit limit cycles. Motivated by results obtained previously for three-dimensional systems, we consider here a more manageable two-dimensional model. We show that one can follow the changes in dynamics of the system when a parameter that tunes the discrete–ultradiscrete transition is varied. In particular we explain the phenomenon of the splitting of a discrete limit cycle to a profusion of periodic orbits at the ultradiscrete limit.