Exploding the Dark Heart of Chaos

Exploding the Dark Heart of Chaos (by Chris King): 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.

Part II of this article includes: 5. Hearts Layered as Onions of Complexity: Generalizing to Higher Polynomials; 6. The Dark Heart of Division: Rational Functions and Newton’s Method; 7. Transcendent Chaos out of Quadratic Disorder; and 8: Falling into the Heart of Deep Transcendental Variation.

Part III of this article includes: 9. Getting to the Heart of the Riemann Zeta Function; Conclusion; Software and Demonstrations; Appendix 1: Combined Methods of Depicting Julia Sets and Parameter Planes; and Appendix 2: Ray Tracing Hypercomplex and Multi-dimensional Chaotic Iterations.