Robert John Morton
About my interest in:
Chaos Theory
Introduction
Difference Equation
Bifurcation Maps
Mandelbrot Set
Hénon's Attractor
Iterative Function
Lissajous Figures
Program: chaos.c
Clique na bandeira
para ler em Português.
Robert John Morton
About my interest in:
Introduction
Difference Equation
Bifurcation Maps
Mandelbrot Set
Hénon's Attractor
Iterative Function
Lissajous Figures
Program: chaos.c
Clique na bandeira
para ler em Português.
[Português]
This is an example of how phase space is methodically distorted in order to get a clearer picture of what is going on. It represents the behaviour of a complex dynamical system which is described in terms of 3 interdependent variables x, y and z. The two variables x and y form a dissipative function:
x = 1·4x² + y − 1 and y = 0·3x
The system is driven (or pumped) by a periodic variable z whose axis is perpendicular to the screen.
If with Microsoft Windows a Security box pops up saying that the application has been blocked because it's untrusted please click here. If you get warning messages with Linux, please click here.
Each point on the display shows where the orbit of the system's 3-Dimensional strange attractor passes through the x-y plane of the screen. The screen is thus a slice of the complex orbit at a particular position around it. This was used to illustrate the chaotic aspects of a star's orbit around and through the galaxy or cluster to which it belonged.
However, it is not quite that straight-forward. The pattern of a slice through such a complex orbit changes form according to the total energy of the system. The Hénon strange attractor is an attractor to which all these real-world attractors are attracted. It is formed from those other attractors when normal space is bent and folded to form a particular phase space in which this 'attractor of attractors' looks simple.
The important point is as follows: These mathematical transformations which appear to bend and fold real space into some distorted contortion does in no way actually bend or fold the phenomenon which occupies that space. It is merely the observer who is bending and folding his 'lens of perception'. The entire reason for all this bending and folding is to 'rotate' those aspects of the phenomenon the observer deems interesting into full view, while turning those aspects of the phenomenon he deems uninteresting 'edge on' so that they cease to cloud his view of the interesting stuff.
I first wrote this program in C from which the above applet was produced in Java.