Pergamon Copyright 0 1996 Elsewer Scm~ce Ltd Printed m Great Britain. All rights reserved G960-0779/96 $15.00 i- fl.Ml 0960-0779(95)00071-2 An Approach to the Ordering of One-Dimensional Quadratic Maps G. PASTOR, M. ROMERA and F. MONTOYA Institute de Fisica Aplicada, Consejo Superior de Investigaciones Cientfficas, Serrano 144, 28006 Madrid, Spain (Accepted 4 July 1995) Abstract-In this work the antenna of the Mandelbrot set is used to order hyperbolic components of one-dimensional quadratic maps by means of graphic and experimental tools. Successive partitions of the antenna have been made to classify hyperbolic components. All the separators used in partitions are Misiurewicz points and the pre-period and period of these points have been determined. 1. INTRODUCTION Since Ruelle and Takens suggested in 1970 that turbulent flow might be an example of dynamic chaos [l], dynamical systems have received more and more attention from physicists because of their wide range of applications as pattern formation in natural systems, turbulence and other topics [2-41. In the field of the one-dimensional discrete dynamical systems, parameter-dependent polynomical maps, f: x H P(x, c), have been widely studied. We shall focus our study on quadratic maps which are the more representative of the former maps, especially one-dimensional quadratic maps. The bifurcation diagram has been the normal tool used to study the chaotic or periodic behaviour of one-dimensional quadratic maps. We propose to use the real axis neighbour- hood (antenna) of the map complex form that offers better graphic advantages, but knowing that only projections on the real axis have a sense in one-dimensional quadratic maps [5, 61. As is well known, all one-dimensional quadratic maps are equivalent because they are topologically conjugate [7]. This means that any one-dimensional quadratic map can be used to study the others. We chose the map xntl = x’, + c, due to the historical importance of its complex form, the Mandelbrot set [g-lo]. As is well known, the Mandelbrot set can be defined by A = {c E @: f:(O) ,b 00 as k + ~0) (1) where f:(O) is the k-iteration of the c parameter-dependent complex quadratic function, h(z) = z* + c, z and c complex, for the initial value z. = 0. The Mandelbrot set is complex, so it has a real and an imaginary part. Therefore, &. = Re (A) + Im (A). But, although we manage complex figures in the real axis neigh- bourhood (antenna) for our own convenience, we insist that the one-dimensional quadratic map x,+~ =x’, + c is only the real part of the set, namely Re (A), which is the intersection of JU and the real axis. The real part of the Mandelbrot set that we are studying is defined for the parameter values -2 C c C l/4. In this part of the real axis there are several kinds of points according to the multiplier value h = 1 df,k(x)/dxl,=,rD, i.e., the slope value (the same for all the 56.5