Saturday, June 16, 2007

Chaos, complexity, and the Cat map.

Chaos and Complexity theory studies nonlinear processes: Chaos explores how complexly interwoven patterns of behaviour can emerge out of relatively simply-to-describe nonlinear dynamics, while Complexity tries to understand how relatively simply-to-describe patterns can emerge out of complexly interwoven dynamics.

In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat. One of this map's features is that image being apparently randomized by the transformation but returning to its original state after a number of steps.

Chaos and Complexity emerge from non-linear mathematics. Theoretical and applied development of mathematical cybernetics and computer science made it possible for many mathematicians and physicists to step out of the framework of linearity, continuity and smoothness, and to approach problems belonging to the world of non-linearity, discontinuity and transformations.

With their pioneering works on local stability (instability) of dynamical systems in the last decade of the 19 century, the Russian mathematicians Andrey Lyapunov and Sophia Kovalevskaya are viewed as the founders of the single and most creative and prolific stand of thought in the analysis of dynamic discontinuities and non-linearities up to the present day, the Russian School. Significant successors to Lyapunov and Kovalevskaya include A. Andropov and L. Pontryagin (1937) who crucially advanced the theory of structural stability, A. Kolmagorov (1941) who developed foundation of the mathematical theory of turbulence (chaotic fluid dynamics) and V. Arnold (1968) with his most completely classification of mathematical singularities (catastrophes).

In 1962 Kolmagorov and Obukhov mathematically demonstrated possibility of an intermittency of chaotic fluid dynamics and emergence of patterns of order out of a turbulent flow.

The famous KAM theorem of Kolmagorov-Arnold-Moser (1978) threw light over the unresolved 3-body problem of Laplacean-Newtonian celestial mechanics - a problem firstly approached by the French mathematician Henri Poincaré (1890) who, facing its unsurmountable computational difficulty, saw the possibility of existence of a non-wandering (dynamically stable) solution of extreme complexity, and thus firstly predicted the existence of an attractor in chaotic dynamics. According to the KAM theorem, the trajectories studied in classical mechanics are neither completely regular nor completely irregular, but they depend very sensitively on the chosen initial states: tiny fluctuations can cause chaotic development.

The sensitive dependence of initial conditions as a true sign of chaotic dynamics was firstly studied mathematically by E. Lorenz (1963) in his three-equation model of atmospheric flow. A slight change in parameter value of the model had generated significantly different behaviour. Lorenz labelled this phenomenon as "butterfly effect". He found also that dynamic trajectories described by the equations moved very quickly along a branched, S-shaped, two-dimensional attractor with of a estrange form.

Self-organization behaviour can be exhibited by far-from-equilibrium chemical systems as it was shown by the Nobel-prize winner (1977) in chemistry Ilya Prigogine. According to the results of his studies, the inorganic chemical systems can exist in highly non-equilibrium conditions impregnated with a potential for emergence of self-organizing chemical structures. The more complex the aggregation of these structures, the stronger the tendency for macro-molecules to organize themselves.

There are a number of constraints.
1) Prediction and determinism are incompatible: we cannot predict long-term behaviour of complex systems, even if know their precise mathematical description.
2) Reducing does not simplify: interaction is important and interaction means inseparability.
3) Simple linear causality does not apply to Chaos and Complexity.
4) Complex dynamics give birth to forces of self-organisation.

The reason why we cannot say much about a complex dynamic system is because of its enormous sensitivity: even an infinitely small change in the starting conditions of a complex process can result in drastically different future developments

What does this all mean?.Simplistically if we cannot measure the initial values exactly,prediction is mathematically impossible.


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