Exploding the Dark Heart of Chaos: Part I

Abstract

This paper, with its associated graphical software and movies, is an investigation of the universality of the cardioid at the centre of the cyclone of chaotic discrete dynamics, the quadratic ‘heart’ forming the main body of the classic Mandelbrot set. Using techniques investigating and exploring the continuity, bifurcations and explosions in its related Julia sets, we demonstrate its universality across a wide spread of analytic functions of a complex variable, extending from the classical quadratic, through higher polynomials and rational functions, to transcendental functions and their compositions. The approach leads to some interesting and provocative results, including decoding dendritic island periodicities, and multiple critical point analysis, leading to layered Mandelbrot set ‘parameter planes’, and intriguing issues of critical point sensitivity in the irregular structures in the Mariana trenches of the more complex functions. Part I of this article includes: 1. Real ‘Boom and Bust’ Origins of the Dark Heart; 2. Illuminating the Writhing Dark Heart of Complexity; 3. The Dark Heart’s Magic Numerology; and 4. A Heart is only Half a Hamburger: Degenerate Critical Points.