Singular perturbations of quadratics maps

We give a complete description of the dynamics of the mapping f(z) = z^2 + e*1/z for positive real values of e. We then consider two generalizations: the case of complex e and the mapping z -> z^n + e/z^m , where e is positive and real. In both of those 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.

PS

Yakov Shapiro has written a java applet that illustrates the results in our paper on singularly perturbed quadratic maps. The applet can be found here.