Dynamics of High-Dimensional Systems

1
Dynamics of High-Dimensional Systems

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented at the
• Santa Fe Institute
• On July 27, 2004

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
5
x
-5
500
Time
0
4
How common is chaos?
1
Logistic Map xn1 Axn(1 - xn)
Lyapunov Exponent
-1
-2
4
A
5
A 2-D Example (Hénon Map)
2
b
xn1 1 axn2 bxn-1
-2
a
-4
1
6
General 2-D Iterated Quadratic Map

• xn1 a1 a2xn a3xn2 a4xnyn a5yn a6yn2
• yn1 a7 a8xn a9xn2 a10xnyn a11yn
  a12yn2

7
General 2-D Quadratic Maps
100
Bounded solutions
10
Chaotic solutions
1
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
100
Iterated maps
10
Continuous flows (ODEs)
1
0.1
Dimension
1
10
10
Correlation Dimension
5
Correlation Dimension
0.5
1
10
System Dimension
11
Lyapunov Exponent
10
1
Lyapunov Exponent
0.1
0.01
1
10
System Dimension
12
Neural Net Architecture
tanh
13
Chaotic in Neural Networks
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

• Let xi be population of the ith species
  (rabbits, trees, people, stocks, )
• dxi / dt rixi (1 - S aijxj )
• Parameters of the model
• Vector of growth rates ri
• Matrix of interactions aij
• Number of species N

N
j1
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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
a32 1 a34 a35 a36 a41 a42 a43 1 a45 a46 a51
a52 a53 a54 1 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
0
a
0
5
20
Typical Time History
15 species
xi
Time
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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)

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Minimal High-D Chaotic L-V Model
dxi /dt xi(1 xi-2 xi xi1)
1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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Space
Time
24
Route to Chaos in Minimal LV Model
25
Other Simple High-D Models
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(No Transcript)
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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

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References

• http//sprott.physics.wisc.edu/
  lectures/sfi2004.ppt (this talk)
• sprott_at_physics.wisc.edu (contact me)