Introduction to Chaos by: Saeed Heidary 29 Feb 2013

Outline: Chaos in Deterministic Dynamical systems Sensitivity to initial conditions Lyapunov exponent Fractal geometry Chaotic time series prediction

Chaos in Deterministic Dynamical systems There are not any random terms in the equation(s) which describe evolution of the deterministic system. If the these equations have non-linear term,the system may be chaotic. Nonlinearity is a necessary condition but not enough.

Characteristics of chaotic systems Sensitivity to initial conditions(butterfly effect) Sensitivity measured by lyapunov exponent. complex shape in phase space (Fractals ) Fractals are shape with fractional (non integer) dimension !. Allow short-term prediction but not long-term prediction

The butterfly effect describes the notion that the flapping of the wings of a butterfly can ‘cause’ a typhoon at the other side of the world.

Tow near points in phase space diverge exponentially

Lyapunov exponent Stochastic (random ) systems: Chaotic systems : Regular systems :

Chaos and Randomness Chaos is NOT randomness though it can look pretty random. Let us have a look at two time series:

Chaos and Randomness x n+1 = x 2 n y n y n+1 = x n White Noise Non - deterministic Henon Map Deterministic plot x n+1 versus x n (phase space)

fractals Geometrical objects generally with non-integer dimension Self-similarity (contains infinite copies of itself) Structure on all scales (detail persists when zoomed arbitrarily)

Fractals production Applying simple rule against simple shape and iterate it

Fractal production

Sierpinsky carpet

Broccoli fractal!

Box counting dimension

Integer dimension Point 0 Line 1 Surface 2 Volume 3

Exercise for non-integer dimension Calculate box counting dimension for cantor set and repeat it for sierpinsky carpet?

Fractals in nature

Complexity - disorder Nature is complicated but Simple models may suffice I emphasize: “Complexity doesn’t mean disorder.”

Prediction in chaotic time series Consider a time serie : The goal is to predict T is small and in the worth case is equal to inverse of lyapunov exponent of the system (why?)

Forecasting chaotic time series procedure (Local Linear Approximation) The first step is to embed the time series to obtain the reconstruction (classify) The next step is to measure the separation distance between the vector and the other reconstructed vectors And sort them from smalest to largest The (or ) are ordered with respect to

Local Linear Approximation (LLA) Method the next step is to map the nearest neighbors of forward forward in the reconstructed phase space for a time T These evolved points are The components of these vectores are as follows: Local linear approximation:

Local Linear Approximation (LLA) Method Again the unknown coefficients can be solved using a least – squares method Finally we have prediction

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