Strange Attractors
From Art to Science
J. C. Sprott
Department of Physics
University of Wisconsin Madison
Presented to the
University of Wisconsin Madison Physics Colloquium
On November 14, 1997
Outline









Modeling of chaotic data
Probability of chaos
Examples of strange attractors
Properties of strange attractors
Attractor dimension
Lyapunov exponent
Simplest chaotic flow
Chaotic surrogate models
Aesthetics
Acknowledgments



Collaborators
 G. Rowlands (physics) U. Warwick
 C. A. Pickover (biology) IBM Watson
 W. D. Dechert (economics) U. Houston
 D. J. Aks (psychology) UW-Whitewater
Former Students
 C. Watts - Auburn Univ
 D. E. Newman - ORNL
 B. Meloon - Cornell Univ
Current Students
 K. A. Mirus
 D. J. Albers
Typical Experimental Data
5
x
-5
0
Time
500
Determinism
xn+1 = f (xn, xn-1, xn-2, …)
where f is some model
equation with
adjustable parameters
Example (2-D Quadratic
Iterated Map)
xn+1 = a1 + a2xn + a3xn2 +
a4xnyn + a5yn + a6yn2
2
a9xn
yn+1 = a7 + a8xn +
+
a10xnyn + a11yn + a12yn2
Solutions Are Seldom Chaotic
20
Chaotic Data
Chaoticequations)
Data
(Lorenz
(Lorenz equations)
x
Solution of
model equations
Solution of model equations
-20
0
Time
200
1
How common is chaos?
Lyapunov Exponent
Logistic Map
-1
xn+1 = Axn(1 - xn)
-2
A
4
A 2-D Example (Hénon Map)
2
b
2
xn+1 = 1 + axn + bxn-1
-2
-4
a
1
The Hénon Attractor
xn+1 = 1 - 1.4xn2 + 0.3xn-1
Mandelbrot Set
xn+1 = xn2 - yn2 + a
yn+1 = 2xnyn + b
a
zn+1 = zn2 + c
b
Mandelbrot Images
General 2-D Quadratic Map
100 %
Bounded solutions
10%
Chaotic solutions
1%
0.1%
0.1
1.0
amax
10
Probability of Chaotic Solutions
100%
Iterated maps
10%
Continuous flows (ODEs)
1%
0.1%
1
Dimension
10
Neural Net Architecture
tanh
% Chaotic in Neural Networks
Types of Attractors
Fixed Point
Spiral
Limit Cycle
Radial
Torus
Strange Attractor
Strange Attractors




Limit set as t  
Set of measure zero
Basin of attraction
Fractal structure




Chaotic dynamics




non-integer dimension
self-similarity
infinite detail
sensitivity to initial conditions
topological transitivity
dense periodic orbits
Aesthetic appeal
Stretching and Folding
Correlation Dimension
Correlation Dimension
5
0.5
1
System Dimension
10
Lyapunov Exponent
Lyapunov Exponent
10
1
0.1
0.01
1
System Dimension
10
Simplest Chaotic Flow
dx/dt = y
dy/dt = z
2
dz/dt = -x + y - Az
2.0168 < A < 2.0577
x  Ax  x  x  0
2
Simplest Chaotic Flow
Attractor
Simplest Conservative Chaotic Flow
...
.
x + x - x = - 0.01
2
Chaotic Surrogate Models
xn+1 = .671 - .416xn - 1.014xn2 + 1.738xnxn-1 +.836xn-1 -.814xn-12
Data
Model
Auto-correlation function (1/f noise)
Aesthetic Evaluation
Summary

Chaos is the exception at low D

Chaos is the rule at high D

Attractor dimension ~ D1/2



Lyapunov exponent decreases
with increasing D
New simple chaotic flows have
been discovered
Strange attractors are pretty
References





http://sprott.physics.wisc.edu/
lectures/sacolloq/
Strange Attractors: Creating
Patterns in Chaos (M&T
Books, 1993)
Chaos Demonstrations
software
Chaos Data Analyzer software
[email protected]