Chaotic Dynamics on Large Networks

1
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

2
Collaborators

• David Albers
• Sean Cornelius

3
What is a complex system?

• 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

4
A Physicists Neuron
N
inputs
tanh x
x
5
A General Model (artificial neural network)
N neurons
Universal approximator, N ? 8 Solutions are
bounded
6
Examples of Networks
Other examples War, religion, epidemics,
organizations,
7
Political System
Information from others
Political state
a1
Voter
a2
a3
aj 1/vN, 0
tanh x
Democrat
x
Republican
8
Types of Dynamics

• Static
• Periodic
• Chaotic

Equilibrium
Dead
Limit Cycle (or Torus)
Stuck in a rut
Strange Attractor
Arguably the most healthy Especially if only
weakly so
9
Route to Chaos at Large N (317)
400 Random networks Fully connected
Quasi-periodic route to chaos
10
Typical Signals for Typical Network
11
Average Signal from all Neurons
All 1
N b
317 1/4
All -1
12
Simulated Elections
100 Democrat
N b
317 1/4
100 Republican
13
Real Electroencephlagrams
14
Strange Attractors
N b
10 1/4
15
Competition vs. Cooperation
500 Random networks Fully connected b 1/4
Competition
Cooperation
16
Bidirectionality
250 Random networks Fully connected b 1/4
Reciprocity
Opposition
17
Connectivity
250 Random networks N 317, b 1/4
Dilute
Fully connected
1
18
Network Size
750 Random networks Fully connected b 1/4
N 317
19
What is the Smallest Chaotic Net?

• dx1/dt bx1 tanh(x4 x2)
• dx2/dt bx2 tanh(x1 x4)
• dx3/dt bx3 tanh(x1 x2 x4)
• dx4/dt bx4 tanh(x3 x2)

Strange Attractor
2-torus
20
Circulant Networks
dxi /dt -bxi S ajxij
21
Fully Connected Circulant Network
N 317
22
(No Transcript)
23
Diluted Circulant Network
N 317
24
(No Transcript)
25
Near-Neighbor Circulant Network
N 317
26
(No Transcript)
27
Summary of High-N Dynamics

• Chaos is generic for sufficiently-connected
  networks
• Sparse, circulant networks can also be chaotic
  (but the parameters must be carefully tuned)
• Quasiperiodic route to chaos is usual
• Symmetry-breaking, self-organization, pattern
  formation, and spatio-temporal chaos occur
• Maximum attractor dimension is of order N/2
• Attractor is sensitive to parameter
  perturbations, but dynamics are not

28
References

• A paper on this topic is scheduled to appear soon
  in the journal Chaos
• http//sprott.physics.wisc.edu/
  lectures/networks.ppt (this talk)
• http//sprott.physics.wisc.edu/chaostsa/ (my
  chaos textbook)
• sprott_at_physics.wisc.edu (contact me)