Strange Attractors From Art to Science
J. C. SprottDepartment 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
Time0 500
x
5
-5
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
yn+1 = a7 + a8xn + a9xn2 +
a10xnyn + a11yn + a12yn2
Solutions Are Seldom ChaoticChaotic Data (Lorenz equations)
Solution of model equations
Chaotic Data(Lorenz equations)
Solution of model equations
Time0 200
x
20
-20
How common is chaos?
Logistic Map
xn+1 = Axn(1 - xn)
-2 4A
Lya
puno
v
Exp
onen
t1
-1
A 2-D Example (Hénon Map)2
b
-2a-4 1
xn+1 = 1 + axn2 + bxn-1
The Hénon Attractorxn+1 = 1 - 1.4xn
2 + 0.3xn-1
Mandelbrot Set
a
b
xn+1 = xn2 - yn
2 + a
yn+1 = 2xnyn + b
zn+1 = zn2 + c
Mandelbrot Images
General 2-D Quadratic Map100 %
10%
1%
0.1%
Bounded solutions
Chaotic solutions
0.1 1.0 10amax
Probability of Chaotic Solutions
Iterated maps
Continuous flows (ODEs)
100%
10%
1%
0.1%1 10Dimension
Neural Net Architecture
tanh
% Chaotic in Neural Networks
Types of AttractorsFixed Point Limit Cycle
Torus Strange Attractor
Spiral Radial
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
Stretching and Folding
Correlation Dimension5
0.51 10System Dimension
Cor
rela
tion
Dim
ensi
on
Lyapunov Exponent
1 10System Dimension
Lya
puno
v E
xpon
ent
10
1
0.1
0.01
Simplest Chaotic Flow
dx/dt = ydy/dt = zdz/dt = -x + y2 - Az
2.0168 < A < 2.0577
02 xxxAx
Simplest Chaotic Flow Attractor
Simplest Conservative Chaotic Flow
x + x - x2 = - 0.01... .
Chaotic Surrogate Modelsxn+1 = .671 - .416xn - 1.014xn
2 + 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 Pat
terns in Chaos (M&T Books, 1993)
Chaos Demonstrations software
Chaos Data Analyzer software [email protected]