Introduction to Chaos by: Saeed Heidary 29 Feb 2013

1. Introduction to Chaosby: Saeed Heidary29 Feb 2013

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

3. 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-linearterm,the system may be chaotic . • Nonlinearity is a necessary condition but not enough.

4. 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

5. Butterfly Effect 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.

6. Lyapunov Exponent Tow near points in phase space diverge exponentially

7. Lyapunov exponent • Stochastic (random ) systems: • Chaotic systems : • Regular systems :

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

9. Chaos and Randomness plot xn+1 versus xn(phase space) HenonMap Deterministic xn+1 = 1.4 - x2n + 0.3yn yn+1 = xn White Noise Non - deterministic

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

11. Fractals production • Applying simple rule against simple shape and iterate it

12. Fractal production

13. Sierpinsky carpet

14. Broccoli fractal!

15. Box counting dimension

16. Integer dimension • Point 0 • Line 1 • Surface 2 • Volume 3

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

18. Fractals in nature

19. Fractals in nature

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

21. 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?)

22. 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

23. 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:

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

25. THANK YOU