Chaotic Dynamics on
Large Networks
J. C. Sprott
Department of Physics
University of Wisconsin - Madison
Presented at the
Chaotic Modeling and Simulation
International Conference
in Chania, Crete, Greece
on June 3, 2008
What is a complex system?
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Complex ≠ complicated
Not real and imaginary parts
Not very well defined
Contains many interacting parts
Interactions are nonlinear
Contains feedback loops (+ and -)
Cause and effect intermingled
Driven out of equilibrium
Evolves in time (not static)
Usually chaotic (perhaps weakly)
Can self-organize, adapt, learn
A Physicist’s Neuron
N
N
xout  tanh  a j x j
j1
inputs
tanh x
x
A General Model
(artificial neural network)
1
N neurons
3
2
4
N
xi  bi xi  tanh  aij x j
j 1
j i
“Universal approximator,” N  ∞
Solutions are bounded
Examples of Networks
System
Agents
Interaction
State
Source
Brain
Neurons
Synapses
Firing rate
Metabolism
Food Web
Species
Feeding
Population
Sunlight
Financial
Market
Traders
Transactions
Wealth
Money
Political
System
Voters
Information
Party
affiliation
The Press
Other examples: War, religion, epidemics, organizations, …
Political System
Information
from others
Political “state”
a1
N
Voter
a2
a3
x  bx  tanh  a j x j
j 1
aj = ±1/√N, 0
tanh x
Democrat
x
Republican
Types of Dynamics
1.
2.
3.
Static
“Dead”
Equilibrium
Periodic Limit Cycle (or Torus)
“Stuck in a rut”
Chaotic Strange Attractor
Arguably the most “healthy”
Especially if only weakly so
Route to Chaos at Large N (=317)
317
dxi / dt  bxi  tanh  aij x j
j1
400 Random networks
Fully connected
“Quasi-periodic route to chaos”
Typical Signals for Typical Network
Average Signal from all Neurons
All +1
All −1
N = 317
b = 1/4
Simulated Elections
100% Democrat
100% Republican
N = 317
b = 1/4
Strange Attractors
N = 10
b = 1/4
Competition vs. Cooperation
317
dxi / dt  bxi  tanh  aij x j
j1
500 Random networks
Fully connected
b = 1/4
Competition
Cooperation
Bidirectionality
317
dxi / dt  bxi  tanh  aij x j
j1
250 Random networks
Fully connected
b = 1/4
Reciprocity
Opposition
Connectivity
317
dxi / dt  bxi  tanh  aij x j
j1
Dilute
250 Random networks
N = 317, b = 1/4
Fully connected
1%
Network Size
N
dxi / dt  bxi  tanh  aij x j
j1
750 Random networks
Fully connected
b = 1/4
N = 317
What is the Smallest Chaotic Net?
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dx1/dt = – bx1 + tanh(x4 – x2)
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dx2/dt = – bx2 + tanh(x1 + x4)
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dx3/dt = – bx3 + tanh(x1 + x2 – x4)
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dx4/dt = – bx4 + tanh(x3 – x2)
Strange
Attractor
2-torus
Circulant Networks
dxi /dt = −bxi + Σ ajxi+j
Fully Connected Circulant Network
N 1
dxi / dt  bxi  tanh  a j xi j
j1
N = 317
Diluted Circulant Network
dxi / dt  bxi  tanh( xi42  xi126  xi254)
N = 317
Near-Neighbor Circulant Network
dxi / dt  bxi  tanh( xi1  xi2  xi3  xi4  xi5  xi6)
N = 317
Summary of High-N Dynamics
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Chaos is generic for sufficiently-connected networks
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Sparse, circulant networks can also be chaotic (but
the parameters must be carefully tuned)
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Quasiperiodic route to chaos is usual
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Symmetry-breaking, self-organization, pattern
formation, and spatio-temporal chaos occur
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Maximum attractor dimension is of order N/2
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Attractor is sensitive to parameter perturbations, but
dynamics are not
References
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A paper on this topic is scheduled to
appear soon in the journal Chaos
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http://sprott.physics.wisc.edu/
lectures/networks.ppt (this talk)
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http://sprott.physics.wisc.edu/chaostsa/
(my chaos textbook)
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sprott@physics.wisc.edu (contact me)