1. Dynamics of High-Dimensional Systems
11/10/2018
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at the
Santa Fe Institute
On July 27, 2004
Entire presentation available on WWW

2. Collaborators David Albers, SFI & U. Wisc - Physics
Dee Dechert, U. Houston - Economics
John Vano, U. Wisc - Math
Joe Wildenberg, U. Wisc - Undergrad
Jeff Noel, U. Wisc - Undergrad
Mike Anderson, U. Wisc - Undergrad
Sean Cornelius, U. Wisc - Undergrad
Matt Sieth, U. Wisc - Undergrad

3. Typical Experimental Data
11/10/2018
Typical Experimental Data 5 x
Not usually shown in textbooks
Could be:
Plasma fluctuations
Stock market data
Meteorological data
EEG or EKG
Ecological data
etc...
Until recently, no hope of detailed understanding
Could be an example of deterministic chaos -5
Time
500

4. How common is chaos? Logistic Map xn+1 = Axn(1 − xn) 1
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Logistic Map
xn+1 = Axn(1 − xn)
Lyapunov Exponent
Simplest 1-D chaotic system
Chaotic over 13% of the range of A
Solutions are unbounded outside the range plotted (hence unphysical) -1 -2 A 4

5. A 2-D Example (Hénon Map)
11/10/2018 2 b
Simplest 2-D chaotic system
Two control parameters
Reduces to logistic map for b = 0
Chaotic "beach" on NW side of "island" occupies about 6% of area
Does probability of chaos decrease with dimension?
xn+1 = 1 + axn2 + bxn-1 −2 a −4 1

6. General 2-D Iterated Quadratic Map
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General 2-D Iterated Quadratic Map
xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2
yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)

7. General 2-D Quadratic Maps
11/10/2018
100 %
Bounded solutions
10%
Chaotic solutions 1%
12 coefficients ==> 12-D parameter space
Coefficients chosen randomly in -amax < a < amax (hypercubes)
0.1%
amax
0.1
1.0 10

8. High-Dimensional Quadratic Maps and Flows
Extend to higher-degree polynomials...

9. Probability of Chaotic Solutions
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100%
Iterated maps
10%
Continuous flows (ODEs) 1%
Quadratic maps and flows
Chaotic flows must be at least 3-D
Do the lines cross at high D?
Is this result general or just for polynomials?
0.1%
Dimension 1 10

10. Correlation Dimension
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Correlation Dimension 5
Correlation Dimension
As described by Grassberger and Procaccia (1983)
Error bars show standard deviation
Scaling law approximately SQR(d)
Useful experimental guidance
0.5 1 10
System Dimension

11. Lyapunov Exponent 10 1 0.1 0.01 1 10 Lyapunov Exponent
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Lyapunov Exponent 10 1
Lyapunov Exponent
0.1
Little difference in maps and flows
At high-D chaos is more likely but weaker
May relate to complex systems evolving “on the edge of chaos”
0.01 1 10
System Dimension

12. Neural Net Architecture
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Neural Net Architecture
Many architectures are possible
Neural nets are universal approximators
Output is bounded by squashing function
Just another nonlinear map
Can produce interesting dynamics
tanh

13. % Chaotic in Neural Networks
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% Chaotic in Neural Networks
Large collection of feedforward networks
Single (hidden) layer (8 neurons, tanh squashing function)
Randomly chosen weights (connection strengths with rms value s)
d inputs (dimension / time lags) D

14. Attractor Dimension
N = 32
DKY = 0.46 D D

15. Routes to Chaos at Low D

16. Routes to Chaos at High D

17. Multispecies Lotka-Volterra Model
11/10/2018
Let xi be population of the ith species (rabbits, trees, people, stocks, …)
dxi / dt = rixi (1 − Σ aijxj )
Parameters of the model:
Vector of growth rates ri
Matrix of interactions aij
Number of species N N
j=1

18. Parameters of the Model
Growth
rates
Interaction matrix 1 r2 r3 r4 r5 r6
1 a12 a13 a14 a15 a16
a21 1 a23 a24 a25 a26 a31 a a34 a35 a36 a41 a42 a a45 a46 a51 a52 a53 a a56 a61 a62 a63 a64 a65 1

19. Choose ri and aij randomly from an exponential distribution:1
P(a) = e−a
P(a)
a = − LOG(RND)
mean = 1 a 5

20. Typical Time History
15 species xi
Time

21. Probability of Chaos
One case in 105 is chaotic for N = 4 with all species surviving
Probability of coexisting chaos decreases with increasing N
Evolution scheme:
Decrease selected aij terms to prevent extinction
Increase all aij terms to achieve chaos
Evolve solutions at “edge of chaos” (small positive Lyapunov exponent)

22. Minimal High-D Chaotic L-V Model
dxi /dt = xi(1 – xi-2 – xi – xi+1) 1

23. Space
Time

24. Route to Chaos in Minimal LV Model

25. Other Simple High-D Models

27. Summary of High-D Dynamics
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Summary of High-D Dynamics
Chaos is the rule
Attractor dimension is ~ D/2
Lyapunov exponent tends to be small (“edge of chaos”)
Quasiperiodic route is usual
Systems are insensitive to parameter perturbations

28. 11/10/2018
References
lectures/sfi2004.ppt (this talk)
(contact me)