1. Chaos Analysis

2. Divya Sindhu Lekha Assistant Professor (Information Technology)
College of Engineering and Management Punnapra

3. Contents Introduction Chaos Mile stones Attractors Fractal Geometry
Measuring Chaos
Lyapunov Exponent
Entropy
Dimensions
Directions

4. Introduction “Only Chaos Existed in the beginning”
“Creation came out of chaos, is surrounded by chaos and will end in chaos”

5. Chaos - Definition
“ Chaos is apparently noisy, aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.”

6. Mile Stones
1890 – Henri Poincare – non-periodic orbits while studying three body problem
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”

10. 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.

11. Point Attractor
Cycle Attractor
Torus Attractor

12. 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.

13. Strange Attractor (E.g.)
Lorenz Attractor
Neither steady state nor periodic.
The output always stayed on a curve, a double spiral.

14. Fractal Geometry Geometry of fractal dimensions. Has self similarity.
Can be explained by a simple iterative formula.
Bifurcation diagram, Lorenz Attractor

15. Some Fractals…

21. Measures of Chaos

22. 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.

23. Types of measures 2 types Dynamic (time dependence) measures
- Lyapunov Exponent
- Kolmogorov Entropy
Geometric measures
- Fractal Dimension
- Correlation Dimension

24. Lyapunov Exponent(λ) λ = ∑ λ(xi) /N
Measure of divergence of near by trajectories.
For a chaotic system, the divergence is exponential in time.
λ = ∑ λ(xi) /N

25. Λ Value Zero - System’s trajectory is periodic.
Negative - System’s trajectory is stable periodic.
Positive - System’s trajectory is chaotic.

26. 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

27. Entropy Kn = 1/τ(Sn+1 - Sn)
Kolmogorov – Sinai Entropy rate (Kn) – Rate of change of entropy as system evolves.
Kn = 1/τ(Sn+1 - Sn)

28. Entropy Avg. Kn = lim N → ∞1/Nτ ∑(Sn+1 - Sn) = lim N → ∞ 1/Nτ[SN – S0]
By complete definition of K-S Entropy,
Kn = lim τ → 0lim L → 0 lim N → ∞ 1/Nτ[SN – S0]

29. 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

30. 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

31. 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

32. 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

33. 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

34. Correlation Dimension (Dc )
Correlation, C(R) = 1/N ∑ Pi(R)
C(R) = zero, No chaos.
C(R) = one, Absolute chaos.

35. 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.

36. 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

37. References Chaos and nonlinear dynamic- Robert.C.Hiborn
Chaos Theory: A Brief Introduction
A Sound Of Thunder
Chaos Theory
Chaotic Systems
Math and Real Life: a Brief Introduction to Fractional Dimensions
Wikipedia

38. Thank You…