Chaos Analysis


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Chaos Analysis - Mile stones in the chaos studies - Attractors - Fractal Geometry - Measuring chaos

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Chaos Analysis

  1. 1. Chaos Analysis Presented by: Divya Sindhu Lekha M. Tech (Technology Management) 2008-2010 [email_address]
  2. 2. Contents <ul><li>Introduction </li></ul><ul><li>Chaos </li></ul><ul><li>Mile stones </li></ul><ul><li>Attractors </li></ul><ul><li>Fractal Geometry </li></ul><ul><li>Measuring Chaos </li></ul><ul><li>Lyapunov Exponent </li></ul><ul><li>Entropy </li></ul><ul><li>Dimensions </li></ul><ul><li>Directions </li></ul>
  3. 3. Introduction <ul><li>“ Only Chaos Existed in the beginning” </li></ul><ul><li>“ Creation came out of chaos, is surrounded by chaos and will end in chaos” </li></ul>
  4. 4. Chaos - Definition <ul><li>“ Chaos is apparently noisy, aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions .” </li></ul>
  5. 5. Mile Stones <ul><li>1890 – Henri Poincare – non-periodic orbits while studying three body problem </li></ul><ul><li>1927 - Van der Pol - observed chaos in radio circuit. </li></ul><ul><li>1960 – Edward Lorenz - “Butterfly effect” </li></ul><ul><li>1975 – Li, Yorke coined the term “chaos”; Mandelbrot – “The Fractal Geometry of Nature” </li></ul><ul><li>1976 – Robert May – “Logistic Map” </li></ul>
  6. 9. Attractors <ul><li>An attractor is a set of points to which a dynamical system evolves after a long enough time. </li></ul><ul><li>Can be a cycle, point or torus attractors. </li></ul>
  7. 10. Point Attractor Cycle Attractor Torus Attractor
  8. 11. Strange Attractor <ul><li>An attractor is strange if it has non-integer dimension. </li></ul><ul><li>Attractor of chaotic dynamics. </li></ul><ul><li>Act strangely, once the system is on the attractor , the nearby states diverge from each other exponentially fast. </li></ul><ul><li>Term coined by David Ruelle and Floris Takens. </li></ul>
  9. 12. Strange Attractor (E.g.) <ul><li>Lorenz Attractor </li></ul><ul><li>Neither steady state nor periodic. </li></ul><ul><li>The output always stayed on a curve, a double spiral. </li></ul>
  10. 13. Fractal Geometry <ul><li>Geometry of fractal dimensions. </li></ul><ul><li>Has self similarity. </li></ul><ul><li>Can be explained by a simple iterative formula. </li></ul><ul><li>Bifurcation diagram, Lorenz Attractor </li></ul>
  11. 14. Some Fractals…
  12. 20. Measures of Chaos
  13. 21. Need to quantify the chaos <ul><li>To distinguish chaotic behavior from noisy behavior. </li></ul><ul><li>To determine the variables required to model the dynamics of the system. </li></ul><ul><li>To sort systems into universality classes. </li></ul><ul><li>To understand the changes in the dynamical behavior of the system. </li></ul>
  14. 22. Types of measures <ul><li>2 types </li></ul><ul><li>Dynamic (time dependence) measures </li></ul><ul><li>- Lyapunov Exponent </li></ul><ul><li>- Kolmogorov Entropy </li></ul><ul><li>Geometric measures </li></ul><ul><li>- Fractal Dimension </li></ul><ul><li>- Correlation Dimension </li></ul>
  15. 23. Lyapunov Exponent( λ ) <ul><li>Measure of divergence of near by trajectories. </li></ul><ul><li>For a chaotic system, the divergence is exponential in time. </li></ul>λ = ∑ λ (x i ) /N
  16. 24. Λ Value <ul><li>Zero - System’s trajectory is periodic. </li></ul><ul><li>Negative - System’s trajectory is stable periodic. </li></ul><ul><li>Positive - System’s trajectory is chaotic. </li></ul>
  17. 25. Entropy <ul><li>A measure of the time rate of creation of information as a chaotic orbit evolves. </li></ul><ul><li>Shannon Entropy (S) gives the amount of uncertainty concerning the outcome of a phenomenon </li></ul>S = ∑ P i ln(1 / P i ) 0<=S<=ln r ; r – no. of events
  18. 26. Entropy <ul><li>Kolmogorov – Sinai Entropy rate (K n ) – Rate of change of entropy as system evolves. </li></ul>K n = 1/ τ (S n+1 - S n )
  19. 27. Entropy Avg. K n = lim N -> ∞ 1/N τ ∑ (S n+1 - S n ) = lim N -> ∞ 1/N τ [S N – S 0 ] K n = lim τ -> 0 lim L -> 0 lim N -> ∞ 1/N τ [S N – S 0 ] By complete definition of K-S Entropy,
  20. 28. Geometric Measures <ul><li>Focuses on the geometric aspects of the attractors. </li></ul><ul><li>Dimensionality of an attractor gives the actual degrees of freedom for the system. </li></ul><ul><li>1. Fractal Dimension </li></ul><ul><li>2. Correlation Dimension </li></ul>
  21. 29. Fractal Dimension <ul><li>Dimensionality is the minimum number of variables needed to describe the state of the system. </li></ul><ul><li>Chaotic systems are of non integer dimension, i.e. fractal dimension. </li></ul><ul><li>Strange attractor. </li></ul><ul><li>Measured by box-counting method </li></ul>
  22. 30. Fractal Dimension – Box counting <ul><li>Boxes of side length “R” to cover the space occupied by the object. </li></ul><ul><li>Count the minimum number of boxes, N(R) needed to contain all the points of the geometric object. </li></ul><ul><li>Box counting dimension , D b. </li></ul>N(R) = lim R -> 0 kR - Db ; k - constant
  23. 31. Fractal Dimension – Box counting D b = -lim R -> 0 log N(R)/log R For a point in 2-D space, D b = 0 For a line segment of length L, D b = 1 For a surface length L, D b = 1
  24. 32. Correlation Dimension ( D c ) <ul><li>A simpler approach to determination of dimension using correlation sum. </li></ul><ul><li>Uses trajectory points directly. </li></ul><ul><li>Number of trajectory points lying within the distance, R of point i = N i (R) </li></ul><ul><li>Relative number of points , </li></ul><ul><li>P i (R) = N i (R)/N-1 </li></ul>
  25. 33. Correlation Dimension ( D c ) <ul><li>Correlation, C(R) = 1/N ∑ Pi(R) </li></ul><ul><li>C(R) = zero, No chaos. </li></ul><ul><li>C(R) = one, Absolute chaos. </li></ul>
  26. 34. Directions <ul><li>Chaos theory in many scientific disciplines: mathematics, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. </li></ul><ul><li>Chaos theory in ecology - show how population growth under density dependence can lead to chaotic dynamics. </li></ul><ul><li>Chaos theory in medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions. </li></ul>
  27. 35. Directions <ul><li>Quantum Chaos - interdisciplinary branch of physics which arose from the modeling of quantum/wave phenomenon with classical models which exhibited chaos. </li></ul><ul><li>Fractal research - Fractal Image Compression, Fractal Music </li></ul>
  28. 36. References <ul><li>Chaos and nonlinear dynamic- Robert.C.Hiborn </li></ul><ul><li>Chaos Theory: A Brief Introduction </li></ul><ul><li>A Sound Of Thunder </li></ul><ul><li>Chaos Theory </li></ul><ul><li>Chaotic Systems </li></ul><ul><li>Math and Real Life: a Brief Introduction to Fractional Dimensions </li></ul><ul><li>Wikipedia </li></ul>
  29. 37. Thank You…