The document is a presentation by J.C. Sprott from the University of Wisconsin - Madison given to the Society for chaos theory in psychology and the life sciences on August 1, 1997 about strange attractors from art to science. It discusses modeling chaotic data, properties and examples of strange attractors, attractor dimension, the simplest chaotic flow, chaotic surrogate models, and the aesthetics of strange attractors.

1. Strange Attractors
From Art to Science
J. C. Sprott
Department of Physics
University of Wisconsin -
Madison
Presented to the
Society for chaos theory in
psychology and the life sciences
On August 1, 1997

2. Outline
 Modeling of chaotic data
 Probability of chaos
 Examples of strange attractors
 Properties of strange attractors
 Attractor dimension
 Simplest chaotic flow
 Chaotic surrogate models
 Aesthetics

3. Typical Experimental Data
Time0 500
x
5
-5

4. Determinism
xn+1 = f (xn, xn-1, xn-2, …)
where f is some model
equation with
adjustable parameters

5. Example (2-D Quadratic
Iterated Map)
xn+1 = a1 + a2xn + a3xn
2 +
a4xnyn + a5yn + a6yn
2
yn+1 = a7 + a8xn + a9xn
2 +
a10xnyn + a11yn + a12yn
2

6. Solutions Are Seldom Chaotic
Chaotic Data
(Lorenz equations)
Solution of
model equations
Chaotic Data
(Lorenz equations)
Solution of model equations
Time0 200
x
20
-20

7. How common is chaos?
Logistic Map
xn+1 = Axn(1 - xn)
-2 4A
LyapunovExponent
1
-1

8. A 2-D example (Hénon map)
2
b
-2
a-4 1
xn+1 = 1 + axn
2 + bxn-1

9. Mandelbrot set
a
b
xn+1 = xn
2 - yn
2 + a
yn+1 = 2xnyn + b

10. General 2-D quadratic map
100 %
10%
1%
0.1%
Bounded solutions
Chaotic solutions
0.1 1.0 10amax

11. Probability of chaotic solutions
Iterated maps
Continuous flows (ODEs)
100%
10%
1%
0.1%
1 10Dimension

12. % Chaotic in neural networks

13. Examples of strange
attractors
 A collection of favorites
 New attractors generated in real
time
 Simplest chaotic flow
 Stretching and folding

14. Strange attractors
 Limit set as t  
 Set of measure zero
 Basin of attraction
 Fractal structure
 non-integer dimension
 self-similarity
 infinite detail
 Chaotic dynamics
 sensitivity to initial conditions
 topological transitivity
 dense periodic orbits
 Aesthetic appeal

15. Correlation dimension
5
0.5
1 10System Dimension
CorrelationDimension

16. Simplest chaotic flow
dx/dt = y
dy/dt = z
dz/dt = -x + y2 - Az
2.0168 < A < 2.0577
02
 xxxAx 

17. Chaotic surrogate models
xn+1 = .671 - .416xn - 1.014xn
2 + 1.738xnxn-1 +.836xn-1 -.814xn-1
2
Data
Model
Auto-correlation function (1/f noise)

18. Aesthetic evaluation

19. References
 http://sprott.physics.wisc.edu/
lectures/satalk/
 Strange Attractors: Creating
Patterns in Chaos (M&T Books,
1993)
 Chaos Demonstrations software
 Chaos Data Analyzer software
 sprott@juno.physics.wisc.edu