Chaos Analysis

Chaos Analysis Presented by: Divya Sindhu Lekha M. Tech (Technology Management) 2008-2010 [email_address]

Contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Introduction ,[object Object],[object Object]

Chaos - Definition ,[object Object]

Mile Stones ,[object Object],[object Object],[object Object],[object Object],[object Object]

Attractors ,[object Object],[object Object]

Point Attractor Cycle Attractor Torus Attractor

Strange Attractor ,[object Object],[object Object],[object Object],[object Object]

Strange Attractor (E.g.) ,[object Object],[object Object],[object Object]

Fractal Geometry ,[object Object],[object Object],[object Object],[object Object]

Need to quantify the chaos ,[object Object],[object Object],[object Object],[object Object]

Types of measures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Lyapunov Exponent( λ ) ,[object Object],[object Object],λ = ∑ λ (x i ) /N

Λ Value ,[object Object],[object Object],[object Object]

Entropy ,[object Object],[object Object],S = ∑ P i ln(1 / P i ) 0<=S<=ln r ; r – no. of events

Entropy ,[object Object],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 ,[object Object],[object Object],[object Object],[object Object]

Fractal Dimension ,[object Object],[object Object],[object Object],[object Object]

Fractal Dimension – Box counting ,[object Object],[object Object],[object Object],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 ) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Correlation Dimension ( D c ) ,[object Object],[object Object],[object Object]

Directions ,[object Object],[object Object],[object Object]

Directions ,[object Object],[object Object]

References ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Chaos Analysis

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Chaos Analysis