Table Of ContentQuantifying Chaos:
Practical Estimation of the Correlation Dimension
Thesis by
James Theiler
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1988
(Submitted 11 September 1987)
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© 1988
James Theiler
All Rights Reserved
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and I have learned from experience
with mathematics and love
that neither is pure.
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ACKNOWLEDGEMENTS
wish to acknowledge Professor Noel R. Corngold for his clear
perspective, wise counsel, infinite patience, and uncountable criticisms (mostly
constructive), as my thesis advisor. In the same breath, acknowledge Mark
Muldoon for many fruitful discourses and digressions.
Various conversations with Jim Havstad, Peter Grassberger, Henry
Greenside, and J. Doyne Farmer have proved useful and interesting. I was first
introduced to nonlinear differential equations by J. Patrick Miller.
Finally, I would like to thank my parents, without whom dot dot dot, and
all my family and friends and relatives, too numerous to mention, and too
generous to forget.
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Contents
Copyright ....................................................................................................... ii
Pretext . .. . . . . . . . . . . . . . .. . . . . .... . .. ... ... . . . . . .. .. . . . . .. .. . .. ... . . . . . . . .. . . .. . . . . . .... .. ... .. .. .. ... . .. . . . . .. ... iii
Acknowledgements .......................................................................................... iv
Contents ......................................................................................................... v
Tables and Figures ......................................................................................... vi
Outline ......................................................................................................... viii
Abstract ....................................................................................................... xii
CHAPTER ONE: Introduction ............................................ .............. 1
CHAPTER TWO: Overview of dimension algorithms ........................... 35
CHAPTER THREE: Geometrical properties of the correlation integral ... , 53
CHAPTER FOUR: Statistical analysis of chaotic motion ...................... 61
CHAPTER FIVE: Noise and discretization.......................................... 80
CHAPTER SIX: Edges, sharp edges and singularities ........................ 90
CHAPTER SEVEN: Lacunarity in fractal sets....................................... 106
CHAPTER EIGHT: Nonchaotic attractors: quasiperiodicity ................... 138
CHAPTER NINE: Autocorrelation ....................................................... 155
CHAPTER TEN: Statistical error ...................................................... 172
CHAPTER ELEVEN: Limitations on dimension imposed by finite N ............ 184
CHAPTER TWELVE: Computation ............................................................ 192
CHAPTER THIRTEEN: Linear analysis of time series .................................. 213
CHAPTER FOURTEEN: Analysis of physical data ........................................ 241
Bibliography ................................................................................................ 249
Postscript .. . . .... .. ... .... .. .. .. . . .. .. .. . .. .. ... ... .. .. .. .. ............. .... ... . . .. . . . . . .. . . .. ... .. .. .. .. . . . . 255
vi
Tables and Figures
Figure 1.1 Trajectory of the Lorenz equations ............................................ 32
Figure 1.2 Trajectory of the Henon map ..................................................... 32
Figure 1.3 Simple attractors ........................................................................ 33
Figure 1.4 Diverging trajectories, shrinking volume ..................................... 34
Figure 2.1 log n(t) versus log ~; the slope is the box-counting dimension ...... 51
Figure 2.2 Scaling of "mass" with radius for d - 0, 1, 2 ............................... 52
Figure 3.1 "Circles" in various Lp norms ..................................................... 60
Figure 5.1 The "fuzzy" line: m =2, d =1 ..................................................... 87
Figure 5.2 The correlation integral for noisy data ....................................... 88
Figure 5.3 The correlation integral for discretized data ............................... 89
Figure 6.1 The set S={Cxl'x )&[0,1]X[0,1] such that lx -x l<r} ................ 102
2 1 2
Figure 6.2 "Butterfly" density on interval [ -1,1] ...................................... 103
Figure 6.3 Correlation integral for the uniformly dense circle in L2 ........... 104
Figure 6.4 Convergence to correlation dimension v for singularity index a.. lOS
Figure 7.1 Discrete self- similarity of a fractal: the Sierpinski gasket. ........ 130
Figure 7.2 The weighted Cantor measure ................................................... 131
Figure 7.3 The Devil's Staircase ............................................................... 132
Figure 7.4 Periodic undulations in the fraction F(r) ................................... 133
Figure 7.5 Self- similarity of Henon attractor at fixed point ....................... 134
Figure 7.6 log Cx ,y (r) versus log r for Henon attractor at fixed point. ... 135
0 0
Figure 7.7 Self-similarity of CXC on rotated axes ..................................... 136
Figure 7.8 Correlation integral for standard Cantor set. ............................. 137
Table 8.1 Rational approximations to the golden mean ............................... 144
Figure 8.1 Correlation integral C(N,r) for N equally spaced points .............. 150
Figure 8.2 C(N,r) for N randomly distributed points ................................... 151
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Figure 8.3 C(N,r) for N quasiperiodically generated points .......................... 152
Figure 8.4 r m•n as a function of winding number 1/J ...................................... 153
Figure 8.5 C(N,r) for N =100 quasiperiodically generated points: various 1/J • 154
Figure 9.1 Standard C(N,r) for autocorrelated random data ......................... 167
Figure 9.2 Modified C(N,W,r) for autocorrelated random data, various W .... 168
Figure 9.3 Modified C(N,W=10,r) for autocorrelated random data ................ 169
Figure 9.4 Standard C(N,r) for Mackey-Glass dynamical data ...................... 170
Figure 9.5 Modified C(N,W =10,r) for Mackey-Glass dynamical data ............ 171
Figure 11.1 Finite N edge effect ................................................................. 191
Figure 12.1 Points of attractor are distributed into boxes of size r0 ••••••••••• 210
Figure 12.2 Correlation integral for the Henon attractor ........................... :. 211
Figure 12.3 Prism-assisted correlation . . .. . . .. . . . . . .. .. ... . .. .. . . ............ .. .. .. .. ..... .. . .. 212
Figure 13.1 Linear unpredictability of the Henon map.................................. 238
Figure 13.2 Measures of nonlinearity versus logistic parameter ~ ................ 239
Figure 13.3 Decorrelated time series: En+l verses En for Henon attractor ..... 240
Figure 14.1 Typical "shot" time series from Caltech tokamak ...................... 246
Figure 14.2 Correlation integral for Caltech tokamak data ........................... 247
Figure 14.3 Correlation integral for Texas tokamak (TEXT) data ................. 248
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Outline
I. INTRODUCTION AND OVERVIEW
1. Introduction
1.1 Chaos
1.1.1 Simple systems with complicated behavior
1.1.2 Sensitive dependence on initial conditions
1.2 Dynamical systems
1.2.1 Evolution operators
1.2.1.1 Ordinary differential equations
1.2.1.2 Maps
1.2.2 Trajectories through state space
1.2.3 Conservative and dissipative dynamical systems
1.3 Attractors
1.3.1 Simple attractors
1.3.2 Strange and chaotic attractors
1.3.3 Formal definition of an attractor
1.3.4 Natural invariant measure
1.4 Time series
1.4.1 Time delay coordinates
1.4.1.1 Optimal reconstruction of state space
1.4.2 Deterministic time series
1.4.3 Stochastic time series
1.5 Quantifying chaos: numerical diagnostics
1.5.1 Kolmogorov entropy and Lyapunov exponents
1.5.2 Dimension
1.5.2.1 Geometrical effects
1.5.2.2 Dynamical effects
1.5.2.3 Statistical effects
1.5.3 Application
1.6 Notes and References
2. Overview of dimension algorithms
2.1 Hausdorff dimension
2.2 Box-counting algorithm
2.3 Generalized dimensions
2.4 Pairwise distance algorithms
2.4.1 Pointwise dimension
2.4.2 Correia tion integral
2.4.2.1 Takens' maximum likelihood estimate
2.4.3 Nearest neighbors algorithm
2.4.4 Recurrence time algorithm
2.4.5 Periodic orbits algorithm
2.5 Notes and References
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II. CORRELATION DIMENSION
A. Initial remarks
3. Geometrical Properties of the Correlation Integral
3.1 Correlation dimension is norm independent
3.2 Dimensions of direct products add
3.3 A geometric definition of the correlation integral
3.4 Notes and References
4. Statistical analysis of chaotic motion
4.1 Honest ensemble: E
0
4.2 Approximate ensembles: Ep E
2
4.3 Probability distribution
4.4 Distinguishing E and E
1 2
4.4.1 The "wraparound" metric
4.4.2 Ensemble E
1
4.4.2.1 E : Corre1ation integral
1
4.4.2.2 E : Shortest distance
1
4.4.3 Ensemble E
2
4.4.3.1 E : Correlation integral
2
4.4.3.2 E : Shortest distance
2
4.5 Notes and References
B. Geometrical effects
5. Noise and discretization
5.1 Noise
5.2 Discretization
5.2.1 Lattice model
5.2.1.1 Loo norm
5.2.1.2 L norm
1
5.2.1.3 Implications for C(r)
5.3 Notes and References
6. Edges, sharp edges and singularities
6.1 Wraparound model
6.2 One-dimensional models
6.2.1 Uniform distribution
6.2.2 "Butterfly" distribution
6.2.3 Gaussian distribution
6.2.4 General one-dimensional distribution
6.3 A two-dimensional model
6.4 A higher dimensional model
6.5 Singularities
6.5.1 Logistic map
6.5.2 Power law singularities
6.6 Notes and References
7. Lacunarity in fractal sets
7.1 Cantor sets
-X-
7 .1.1 Standard Cantor set
7.1.2 Weighted Cantor sets
7.2 Generating sample points
7.2.1 Iterated Function System (IFS)
7.2.2 Independent sample points
7.3 Pointwise dimension at the origin
7.3.1 The Fraction function
7.3.2 Standard Cantor set
7.3.3 Weighted Cantor set
7.3.4 Pointwise dimension at other points
7.3.5 Pointwise dimension on an attractor
7.4 The correlation integral
7 .4.1 Standard Cantor set
7.4.2 Weighted Cantor set
7.4:3 Strict and asymptotic self- similarity
7 .4.4 Correlation integral on an attractor
7.5 Conclusion
7.6 Notes and References
C. Dynamical effects
8. Nonchaotic attractors: quasiperiodicity
8.1 Wraparound metric
8.2 Two control experiments
8.3 Twist map
8.3.1 Rational ¢
8.3.2 Irrational ¢: the golden mean
8.3.3 Generic ¢
8.4 Notes and References
9. Autocorrelation
9.1 The modified correlation algorithm
9.2 Au tocorrela ted stochastic data
9.2.1 Uncorrelated limit
9.2.2 Effect of autocorrelation
9.2.3 Recommendations for W
9.3 Numerical results
9.3.1 Stochastic data
9.3.2 Deterministic dynamical data
9.4 Conclusion
9.5 Notes and References
D. Statistical effects
10. Statistical error
10.1 Least squares fit
10.2 Takens' maximum likelihood estimate
10.3 Average pointwise dimension
10.3.1 Nonuniformity
10.4 Notes and References
