Graphic tools to analyse one-dimensional quadratic maps

Abstract

The window structure of a 1-D quadratic map is normally graphically analysed starting from the bifurcation diagram. More information can be obtained by working in the real part neighbourhood of the corresponding complex quadratic map when the situation of “midgets’ and “hyperbolic components” are observed. A set of new rules to graphically determine the period of a “hyperbolic component” and a “midget” in the chaotic region of a 1-D quadratic map is presented. The real part neighbourhood of a complex quadratic map, the Mandelbrot set, has been used to obtain the rules. The escape lines method has been chosen among the several methods to draw the Mandelbrot set. Escape lines are associated to a number, like an equipotential line is associated to a potential. A correspondence between the period of an hyperbolic component and the number of escape lines attracted by its filaments can be given, which enables to determine the period.