Saturday, February 6, 2010

Navigating the Dark Heart of Chaos

I've complemented the Riemann zeta viewer with a suite of Mac XCode applications to visualize a full spectrum of complex functions, from the familiar 'dark heart' of the Mandelbrot set, through polynomials, and rational functions to a variety of transcendental functions.

The dark heart viewer enables exploration of functions with multiple critical points and shows where the higher dimensional variants of the parameter plane (Mandelbrot set) cardioid appear in subtle places in functions like the 'cubic' Cos(z) +c (compare Coz(z)+c below with the cubic z^3-z+c in the above icon) and the 'quartic' c(Cos(z)+Cos(2z))

The most up-to-date downloadable releases of the major XCode applications, which have been tested for both Tiger and Snow Leopard are as follows:

  1. Riemann Zeta Viewer: Application - Source - RZ Flight Manual
  2. Dark Heart Viewer: Application - Source - DH Flight Manual
  3. Wave Function Method Viewer: Application - Source - WF Flight Manual
  4. Modified Inverse Iteration Viewer: Application - Source
  5. Herman Ring 4D parameter Viewer: Application - Source
  6. Eight Critical Point csin(z)/z Viewer: Application - Source
  7. Collatz 3n+1 Complex Map Viewer: Application - Source

DHViewer and RZViewer are the twin primary applications covering most of the research examples.

RZViewer deals with all the weird analogues of the Riemann zeta function, while the twin application DHViewer examines a wide variety of rational and transcendental functions. This means that the widest variety of complex functions are explored including some of the most difficult ones to model.

The Wave Function method is a method I invented, which performs effective inverse iteration by forward mapping the domain and colouring by a wave function of the eventual iterated range. As far as I know it is the only method which enables inverse iteration of functions like zeta or compex functions whose inverses cannot be solved explicitly. For comparison see the modified inverse application which works only for the standard map f(z)=z2+c.

The Herman viewer examines the Herman ring map h(z)=cz2(z-r)/(rz-1), which has two complex parameters. Following each parmeter in sequence to give an effective 4D parameter exploration involving whole classes of Mandelbrot sets for the second parameter c.

The remaining XCode viewers and source code cover specific techniques with the standard quadratic function, have also een tested on Tiger and Snow Leopard and can be accessed through links in the papers.

Research Papers

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