dynamics of high-dimensional systems
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Dynamics of High-Dimensional Systems
J. C. SprottDepartment of Physics
University of Wisconsin - Madison
Presented at the
Santa Fe Institute
On July 27, 2004
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
Typical Experimental Data
Time0 500
x
5
-5
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 Iterated Quadratic Map
xn+1 = a1 + a2xn + a3xn2 +
a4xnyn + a5yn + a6yn2
yn+1 = a7 + a8xn + a9xn2 +
a10xnyn + a11yn + a12yn2
General 2-D Quadratic Maps100 %
10%
1%
0.1%
Bounded solutions
Chaotic solutions
0.1 1.0 10amax
High-Dimensional Quadratic Maps and Flows
)1(1
)1(1
)( 0 tkxD
k jkajatD
j jxatix
D
k kxjkajaD
j jxadt
idx
110
Extend to higher-degree polynomials...
Probability of Chaotic Solutions
Iterated maps
Continuous flows (ODEs)
100%
10%
1%
0.1%1 10Dimension
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
Neural Net Architecture
tanh
11
1tanh
1
jnn XD
jijW
N
iiX
% Chaotic in Neural Networks
D
Attractor Dimension
DKY = 0.46 D
D
N = 32
Routes to Chaos at Low D
Routes to Chaos at High D
Multispecies Lotka-Volterra Model
j=1
N
• 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
Parameters of the Model
1r2
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
Growthrates Interaction matrix
Choose ri and aij randomly from an exponential distribution:
P(a)
a00 5
1 P(a) = e−a
a = − LOG(RND)
mean = 1
Typical Time History
Time
xi
15 species
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)
Minimal High-D Chaotic L-V Model
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
dxi /dt = xi(1 – xi-2 – xi – xi+1)
Time
Spac
e
Route to Chaos in Minimal LV Model
jxN
jijaix
dtidx
1
tanh
Other Simple High-D Models
)1(1
tanh)(
tjxN
j ijatix
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
References
http://sprott.physics.wisc.edu/ l
ectures/sfi2004.ppt
(this talk)
sprott@physics.wisc.edu
(contact me)
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