Strange Attractors From Art to Science
J. C. SprottDepartment of Physics
University of Wisconsin - Madison
Presented at the
Santa Fe Institute
On June 20, 2000
Outline Modeling of chaotic data Probability of chaos Examples of strange attractors Properties of strange attractors Attractor dimension scaling Lyapunov exponent scaling Aesthetics Simplest chaotic flows New chaotic electrical circuits
Typical Experimental Data
Time0 500
x
5
-5
General 2-D Iterated Quadratic 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
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
Aesthetic Evaluation
Sprott (1997)
dx/dt = y
dy/dt = z
dz/dt = -az + y2 - x
5 terms, 1 quadratic
nonlinearity, 1 parameter
“Simplest Dissipative Chaotic Flow”
xxxax 2
Linz and Sprott (1999)
dx/dt = y
dy/dt = z
dz/dt = -az - y + |x| - 1
6 terms, 1 abs nonlinearity, 2 parameters (but one =1)
1 xxxax
First Circuit
1 xxxax
Bifurcation Diagram for First Circuit
Second Circuit
Third Circuit
)sgn(xxxxax
Chaos Circuit
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
New chaotic circuits have been developed