Nonuniqueness of families of periodic solutions in a four dimensional mapping

Abstract

If we follow a family of periodic solutions along a closed path in a parameter space of two dimensions we may not return to the original solution when the parameters return to the original values. We study such nonuniqueness phenomena in simple and double period familis. Nonuniqueness appears if a closed path in the parameter space goes around a critical point. In some cases we find Riemann sheets in the same way as in multiply valued functions. In other cases the connections of various families change in a complicated way around the critical point. All these phenomena are explained analytically. At the critical point there is a ‘collision of bifurcations’. The changes of the connections of various families at such collisions of bifurcations are studied in some detail.