Previous: Quick Start  2 Attractors  Next: Rendering modes 
This chapter is an exhaustive description
of the attractors found in Chaoscope. The equations are shown
here solely for the mathematical
inclined (even though some equations may have their validity
disputed) as there is no algebra skills involved in rendering attractors.
The pictures displayed represent a typical "find" for each equation. 

2.1 Chaotic Flow 

project file 
Chaotic Flow, a family of attractors which includes Lorenz and Lorenz84, is a generalization of the equations described by Pr. Julien Sprott in his paper "Some Simple Chaotic Flows". This is the only equation in Chaoscope where the position of the variables (x, y and z) is itself a parameter, M_{i} Op. This means that during a Search, not only the parameters are randomized, so is the equation. 

22 parameters :  M0, M1... M11, M0 Op., M1 Op.... M10 Op and dT  
equation :  
2.2 Icon 

project file 
This equation was used by Michael Field and Martin Golubitsky
to demonstrate how Chaos could yield symmetry. 

6 parameters :  Degree, Alpha, Beta, Lambda, Gamma and Omega  
equation :  
2.3 IFS 

project file 
Invented by Michael Barnsley, it is the only
attractor with fractal properties available in Chaoscope along
with Julia, and probably the most famous of them all. To obtain
these convoluted attractors, sets of affine transformations (translation,
scale, rotation, etc.) are applied to the orbit in a random order.
Each matrix in the equation corresponds to a set of transformation.
You can set the number of matrix (from 2 to 8) to use when you
create a project or later while a project is already open. Attractors
saved under the previous file format (.cla) will have their parameters
shown as matrix coefficients instead of affine transformations.


up to 128 parameters :  Fifteen affine transformation variables plus a probability factor
for each matrix ( M rot., M sc., M sh., M tr., N rot., N sc., N sh., N tr., etc.) 

equation :  
2.4 Julia 

project file 
Alan Norton was the first to render Quaternions in the early eighties. He was followed later by John C. Hart, whose code was used for Chaoscope. The inverse of the equation yields a slightly different result than the regular z = z^{2}+ c: the set is perfectly symmetrical. Also, because iterating the equation like a normal attractor proved not very effective, the depth first tracing method has been implemented. The Level parameter defines the depth of the square roots tree. A high level (i.e. > 16) will produce more detail and won't slow down the rendering. 

4 parameters :  Level, Creal, Cimag and Phi  
equation :  
2.5 Lorenz 

project file 
A beautifully simple equation created by Edward
Lorenz to demonstrate the chaotic behavior of dynamic systems.
This attractor is also historically important because Lorenz
discovered, while working on weather patterns simulation, one
of the fundamental laws of the Chaos Theory : "the sensitive
dependence on initial conditions" he himself dubbed "the
butterfly effect".
Although moving the initial orbit won't affect the shape of a strange attractor, the position of the orbit on the attractor after several iterations will vary considerably from one initial position to the other. Interesting attractors can be found with a relatively high value for dT, i.e. dT > 0.3. 

4 parameters :  A, B, C and dT  
equation :  
2.6 Lorenz84 

project file 
First introduced in Lorenz's paper entitled "Irregularity:
a fundamental property of the atmosphere", this
equation is a lowdimensional model for long term atmospheric
circulation. Rather than a graphical representation of atmospheric
currents, the orbit coordinate are the three variables of
the model.


5 parameters :  A, B, F, G and dT  
equation :  
2.7 Pickover 

project file 
Probably the best equation to start experimenting
with, since the orbit will never escape the attractor as it is "trapped" in
sinusoids. A good set of parameters is
{1, 1.8, 0.71, 1.51} from
which nice attractors can be found, after modifying each parameter
slightly. 

4 parameters :  A, B, C and D  
equation :  
2.8 Polynomial, Type A 

project file 
The Polynomial type A (which probably is polynomial
only by name !) uses the most simple equation, it is therefore
the fastest attractor to render. The equation is a special case
of Type B where P1, P3 and P5 = 0.


3 parameters :  P0, P1 and P2  
equation :  
2.9 Polynomial, Type B 

project file 
A slightly more complex equation for a challenging
specimen. This attractor equation is rather feeble and will see
the orbit escapes more than often. When not escaping, it won't
produce a great variety of shapes.


6 parameters :  P0, P1... P5  
equation :  
2.10 Polynomial, Type C 

project file 
Next step up the Polynomial evolution ladder,
it has three times more parameters than its predecessor. This
equation is a good compromise between speed and complexity.


18 parameters :  P0, P1... P17  
equation :  
2.11 Polynomial Function (Abs, Power and Sin) 

project file project file project file 
Polynomial Function is a group of three
equations which make use of algebraic or trigonometric functions
rather than the normal 2^{nd} order structure.
They were adapted to 3 dimensions from Julien Sprott's book on
Strange
Attractors. Abs will yield very typical angular forms, Power will add some flexibility to Abs straight lines, while Sin will
produce wavy attractors.


21 to 39 parameters :  P0, P1... P38  
equations :  Abs: Power: Sin: 

2.12 Polynomial, Sprott 

project file 
Described in Pr. Julien Sprott's book under
its scientific name, "three dimensional quadratic map",
it is the most complex equation found in Chaoscope. 

up to 168 parameters :  P0, P1... P167  
equation :  
2.13 Unravel 

project file 
The only equation created for Chaoscope. The
idea was to have an attractor from which the orbit couldn't escape,
regardless of its parameters, without using trigonometric functions
(like Pickover does) thus achieving a relatively good rendering
speed.
The first attempt was to use a polynomiallike equation, and limit the volume the orbit could evolve into to a unit cube. Each time the orbit was leaving the cube space it would reappear near the opposite face, inside the cube. The best result obtained was remotely looking like a Menger Sponge. Later, the cube was replaced by a sphere of radius R and the equation was simplified. The best looking attractors are found setting R to a negative value, which makes no sense considering the concept behind the equation, leaving the author puzzled. 

6 parameters :  A, E, U, L, N, V and R  
equation : 
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