SINGULAR PERTURBATIONS OF QUADRATIC MAPS

Abstract

We give a complete description of the dynamics of the mapping fε(z)=z2+(ε/z) for positive real values of ε. We then consider two generalizations: the case of complex ε and the mapping z→zn+(ε/zm), where ε is positive and real. In both cases we provide a full characterization of the map for a certain set of parameters, and give observations based on numerical evidence for all other parameter values. The dynamics of all maps that we consider bears striking resemblance to that of complex quadratic maps.