Chaos Analysis

Presented by: Divya Sindhu Lekha M. Tech (Technology Management) 2008-2010 divi.lekha@gmail.com

Contents•Introduction•Chaos•Mile stones•Attractors•Fractal Geometry

•Measuring ChaosLyapunov ExponentEntropyDimensions

•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”

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…

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 types1. Dynamic (time

dependence) measures- Lyapunov Exponent- Kolmogorov Entropy

2. 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(xi) /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 = ∑ Pi ln(1/ Pi )0<=S<=ln r ; r – no.

of events

Entropy•Kolmogorov – Sinai Entropy

rate (Kn) – Rate of change of entropy as system evolves.

Kn = 1/τ(Sn+1 - Sn)

EntropyAvg. Kn = lim N → ∞1/Nτ ∑(Sn+1 -

Sn)

= lim N → ∞ 1/Nτ[SN – S0]

Kn = lim τ → 0lim L → 0 lim N → ∞ 1/Nτ[SN – S0]

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 Dimension2. 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 , Db.

N(R) = lim R → 0kR- Db ; k - constant

Fractal Dimension – Box counting

Db = -lim R → 0 log N(R)/log R

For a point in 2-D space, Db = 0

For a line segment of length L, Db

= 1

For a surface length L, Db = 1

Correlation Dimension (Dc )•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 = Ni(R)

•Relative number of points , Pi(R) = Ni(R)/N-1

Correlation Dimension (Dc )•Correlation, C(R) = 1/N ∑

Pi(R)

•C(R) = zero, No chaos.•C(R) = one, Absolute chaos.

DirectionsChaos 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.

DirectionsQuantum 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…