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Chaos Analysis
 

Chaos Analysis

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

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

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    Chaos Analysis Chaos Analysis Presentation Transcript

    • Chaos Analysis Presented by: Divya Sindhu Lekha M. Tech (Technology Management) 2008-2010 [email_address]
    • Contents
      • Introduction
      • Chaos
      • Mile stones
      • Attractors
      • Fractal Geometry
      • Measuring Chaos
      • Lyapunov Exponent
      • Entropy
      • Dimensions
      • Directions
    • Introduction
      • “ Only Chaos Existed in the beginning”
      • “ Creation came out of chaos, is surrounded by chaos and will end in chaos”
    • Chaos - Definition
      • “ Chaos is apparently noisy, aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions .”
    • Mile Stones
      • 1890 – Henri Poincare – non-periodic orbits while studying three body problem
      • 1927 - Van der Pol - observed chaos in radio circuit.
      • 1960 – Edward Lorenz - “Butterfly effect”
      • 1975 – Li, Yorke coined the term “chaos”; Mandelbrot – “The Fractal Geometry of Nature”
      • 1976 – Robert May – “Logistic Map”
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    • Attractors
      • An attractor is a set of points to which a dynamical system evolves after a long enough time.
      • Can be a cycle, point or torus attractors.
    • Point Attractor Cycle Attractor Torus Attractor
    • Strange Attractor
      • An attractor is strange if it has non-integer dimension.
      • Attractor of chaotic dynamics.
      • Act strangely, once the system is on the attractor , the nearby states diverge from each other exponentially fast.
      • Term coined by David Ruelle and Floris Takens.
    • Strange Attractor (E.g.)
      • Lorenz Attractor
      • Neither steady state nor periodic.
      • The output always stayed on a curve, a double spiral.
    • Fractal Geometry
      • Geometry of fractal dimensions.
      • Has self similarity.
      • Can be explained by a simple iterative formula.
      • Bifurcation diagram, Lorenz Attractor
    • Some Fractals…
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    • Measures of Chaos
    • Need to quantify the chaos
      • To distinguish chaotic behavior from noisy behavior.
      • To determine the variables required to model the dynamics of the system.
      • To sort systems into universality classes.
      • To understand the changes in the dynamical behavior of the system.
    • Types of measures
      • 2 types
      • Dynamic (time dependence) measures
      • - Lyapunov Exponent
      • - Kolmogorov Entropy
      • Geometric measures
      • - Fractal Dimension
      • - Correlation Dimension
    • Lyapunov Exponent( λ )
      • Measure of divergence of near by trajectories.
      • For a chaotic system, the divergence is exponential in time.
      λ = ∑ λ (x i ) /N
    • Λ Value
      • Zero - System’s trajectory is periodic.
      • Negative - System’s trajectory is stable periodic.
      • Positive - System’s trajectory is chaotic.
    • Entropy
      • A measure of the time rate of creation of information as a chaotic orbit evolves.
      • Shannon Entropy (S) gives the amount of uncertainty concerning the outcome of a phenomenon
      S = ∑ P i ln(1 / P i ) 0<=S<=ln r ; r – no. of events
    • Entropy
      • Kolmogorov – Sinai Entropy rate (K n ) – Rate of change of entropy as system evolves.
      K n = 1/ τ (S n+1 - S n )
    • 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,
    • Geometric Measures
      • Focuses on the geometric aspects of the attractors.
      • Dimensionality of an attractor gives the actual degrees of freedom for the system.
      • 1. Fractal Dimension
      • 2. Correlation Dimension
    • Fractal Dimension
      • Dimensionality is the minimum number of variables needed to describe the state of the system.
      • Chaotic systems are of non integer dimension, i.e. fractal dimension.
      • Strange attractor.
      • Measured by box-counting method
    • Fractal Dimension – Box counting
      • Boxes of side length “R” to cover the space occupied by the object.
      • Count the minimum number of boxes, N(R) needed to contain all the points of the geometric object.
      • Box counting dimension , D b.
      N(R) = lim R -> 0 kR - Db ; k - constant
    • 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
    • Correlation Dimension ( D c )
      • A simpler approach to determination of dimension using correlation sum.
      • Uses trajectory points directly.
      • Number of trajectory points lying within the distance, R of point i = N i (R)
      • Relative number of points ,
      • P i (R) = N i (R)/N-1
    • Correlation Dimension ( D c )
      • Correlation, C(R) = 1/N ∑ Pi(R)
      • C(R) = zero, No chaos.
      • C(R) = one, Absolute chaos.
    • Directions
      • Chaos theory in many scientific disciplines: mathematics, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics.
      • Chaos theory in ecology - show how population growth under density dependence can lead to chaotic dynamics.
      • Chaos theory in medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.
    • Directions
      • Quantum Chaos - interdisciplinary branch of physics which arose from the modeling of quantum/wave phenomenon with classical models which exhibited chaos.
      • Fractal research - Fractal Image Compression, Fractal Music
    • References
      • Chaos and nonlinear dynamic- Robert.C.Hiborn
      • Chaos Theory: A Brief Introduction http://www.imho.com/grae/chaos/chaos.html
      • A Sound Of Thunder http://www.urbanhonking.com/universe/2006/09/a_sound_of_thunder.html
      • Chaos Theory http://www.genetologisch-onderzoek.nl/wp-content
      • Chaotic Systems http://dept.physics.upenn.edu/courses/gladney/mathphys/subsection3_2_5.html
      • Math and Real Life: a Brief Introduction to Fractional Dimensions http://www.imho.com/grae/chaos/fraction.html
      • Wikipedia
    • Thank You…