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Chaos Theory. Seyed Ehsan Safavieh Computer Science Department Amirkabir University of Technology Apr-08. Chaotic Systems. Black Box. The game of Pinball. Black Box. Chaotic Specifications. Nonlinear, Deterministic, Iterative function Self similarity Strange Attractors
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Chaos Theory Seyed Ehsan Safavieh Computer Science Department Amirkabir University of Technology Apr-08
Chaotic Systems Black Box
The game of Pinball Black Box
Chaotic Specifications • Nonlinear, Deterministic, Iterative function • Self similarity • Strange Attractors • Sensitive to initial condition
Double Pendulum • Very sensitive to initial condition • A-periodic motion due to high damping effects Solved numerically using 4th order Runge-Kutta method in MATLAB
3-Body Problem • In the early 1900s Henri Poincaré, while studying the three-body problem, found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. • Also Khayyam and Laplace had stated the strangeness of this problem before.
3-Body Problem • The n-body problem has 10 independent algebraic integrals • 3 for the center of mass • 3 for the linear momentum • 3 for the angular momentum • 1 for the energy. • Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory.
Earth’s Climate System • There is a black box system with a deterministic nonlinear function that specifies the Earth’s climate changes
Edward Norton Lorenz(May 23 1917 – April 16 2008) • Lorenz built a mathematical model of the way air moves around in the atmosphere. (1960) • He realize that whether patterns did not always change as predicted. • Minute variations in the initial values of variables in his twelve variable computer weather model would result in grossly divergent weather patterns.
Lorenz’s System • the weather that the machine began to predict was completely different from the weather calculated before.
Butterfly Effect • The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number • Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. • Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?
Jackson Pollack • It is determined that some of his works display the properties of mathematical fractals • They even go on to speculate that on some level, Pollock may have been aware of the nature of chaotic motion, and was attempting to form what he perceived as a perfect representation of mathematical chaos - more than ten years before Chaos Theory itself was discovered. • “When I am in my painting, I'm not aware of what I'm doing …. It is only when I lose contact with the painting that the result is a mess. Otherwise there is pure harmony”
Cellular Automata • Cellular automata are Feedback Machines mathematical finite state machines which change the state of their cells step by step. • To run a cellular automaton we need two entities of information: an initial state of its cells (i.e., an initial layer) and a set of rules or laws.
Cellular Automata • Cellular automata have many applications nowadays: • Random number generators • Coding of images • Image compression • Video compression • …
Chaos Game • Choose a random starting point (o) within the triangle ABC. • Shoot dice. • Connect the point • with edge A, when you got 1 or 2 spots, • with edge B, when you got 3 or 4 spots, • with edge C, when you got 5 or 6 spots, • and determine the center of the distance. This is the new point to continue. • Goto 4
Random? Logistic Tent Bernouli Hp Rand
Maps • Logistic Map • Tent Map
Maps • Bernoulli Map • Hewlett-Packard calculators use this function to simulate random number generation.
How to distinguish random via chaotic • Lyapunov Exponents • More positive lyapunov exponents : More Chaotic • Fractal dimension • (Box counting dimension) • High fractal dimension : Randomness • Low fractal dimension : Chaotic • Correlation Dimension • , …
Practical difficulties with observed timeseries • We observe just one or a few variables of the system • Noise: if very high, it masks the chaotic signal • Finite length and missing data • The common tools for detecting chaos (Lyapunov exp, correlation dimension) are uneffective
Dynamical Systems • Any system whose state changes with time is a dynamical system. ξ(t) • If the current state ξ(0) uniquely determines the future states ξ(t), t>0, the system is deterministic dynamical system • If such a unique mapping between current and future states does not exist, the system is stochastic dynamical system
Chaotic Systems • A deterministic dynamical system which is very sensitive to its initial conditions, is a chaotic system.
Trajectory • The evolution of the state of the system through state space will form a curve named trajectory which is parameterized by t.
Attractors • There are various possibilities for the behavior of ξ(t) • Consequently after a sufficient long time the system will evolve on a reduced set of states, which is called attractor
Attractors • Fixed Point • Limit cycle • Limit tori quasiperiodic series: A discretely sampled sum of Nt periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines
Strange Attractor • A typical time series corresponding to strange attractor does not have any periodicities an dthe power spectrum is essentially broad banded. So such time series may appear to be stochastic at first sight
Delay Embedding theorem • delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes, but it does not preserve the geometric shape of structures in phase space.
References • Topics in nonlinear time series analysis, A. Galka, 2000 • Laws of Chaos, A. Boyarsky, P. Gora, 1997 • Chaos and Fractals New Frontiers of Science Second Edition, 2004 • Beautiful chaos, C. E. Slethaug, 2000 • Classical an quantum chaos, 2003 • Chaos under control, 2000 • wikipedia
