As I said before... I love the chaotic systems and complex nonlinear dynamical systems. Observing the whole world and trying to find the patterns ! it is really amazing to ask yourself and even your friends like daif, osama, Eng.Hazem, and even our humanoid Snap WHAT and HOW?
I will write some topics around the chaotic behavior study, I hope it will enriching simple views to know more and more about chaos. This topic target the clarification of chaos measure, actually there are two measuring paradigm:
1- Geometric Measuring:
- Fractal Dimension,
- Correlation dimension.
2- Dynamic Measuring:
- Lyapunov characteristic exponent,
- Kolmogorov Sinai Entropy rate.
It is pretty nice to show these 4 effective methods to measuring chaos. I advice my reader to skim the links contents and keep in mind the phase plane as efficient way to represent/study the non linearity especially chaotic systems.
- Fractal Dimension:
_ an Attractor said to be strange if it has a non integer dimension, for chaotic attractor nearby states diverge from each other exponentially. Notice the dimensions term means the minimum number of variables which describe the system states. Lorentz attractor the famous, the most appropriate to illustrate that. Edward Lorentz counted as the father of chaos defiantly "butterfly effect".
My Lorentz attractor powered by Mathematica 8.0
Strict speaking Box-Counting method is ultimate method describing the fractal dimensionality for systems.Simply partition the space into boxes of equal side length R then count then minimum number of boxes N(R) which strictly contain all points of the geometric object, that can done by this relation:
Where:
In a more expanded I can add a good piece of information about fractals, definitely related to cellular automaton representation for fractals which can seen by using rule90
My CA Rule 90 for 30 iterations powered by Mathematica 8.0
My CA Rule 90 for 300 iterations powered by Mathematica 8.0
My CA Rule 90 for 3000 iterations powered by Mathematica 8.0
Rest will be published soon isA
