Scaling symmetries in nonlinear dynamics. A view from parameter space

Abstract

The orbit of the critical point of a discrete nonlinear dynamical system defines a family of polynomials in the parameter space of that system. We show here that for the important class of quadratic-like maps, these polynomials become indistinguishable under a suitable scaling transformation. The universal representation of these polynomials produced by such a scaling leads directly to accurate approximations for those parameter values where windows of order appear, the sizes of such windows, measures of window distortion, and the characterization of the internal structure of the windows in terms of generalized Feigenbaum numbers.