chaotic dynamics on large networks j. c. sprott department of physics university of wisconsin -...
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Chaotic Dynamics on Large Networks
J. C. SprottDepartment 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? 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
jN
jjxax
1tanhout
Ninputs
tanh x
x
2 4
1
3
A General Model (artificial neural network)
N neurons
N
ijj
jijiii xaxbx1
tanh
“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 Trans-actions
Wealth Money
Political System
Voters Information Party affiliation
The Press
Other examples: War, religion, epidemics, organizations, …
Political System
tanh x
x
Republican
Democrat
Informationfrom others
Political “state”
N
jjj xabxx
1
tanh
a1
a2
a3 aj = ±1/√N, 0
Voter
Types of Dynamics
1. Static
2. Periodic
3. ChaoticArguably the most “healthy”Especially if only weakly so
“Dead”
“Stuck in a rut”
Equilibrium
Limit Cycle (or Torus)
Strange Attractor
Route to Chaos at Large N (=317)
jj
ijii xabxdtdx
317
1tanh/
“Quasi-periodic route to chaos”
400 Random networksFully connected
Typical Signals for Typical Network
Average Signal from all NeuronsAll +1
All −1
N =b =
3171/4
Simulated Elections100% Democrat
100% Republican
N =b =
3171/4
Strange AttractorsN =b =
101/4
Competition vs. Cooperation
jj
ijii xabxdtdx
317
1tanh/
500 Random networksFully connected
b = 1/4
Competition
Cooperation
Bidirectionality
jj
ijii xabxdtdx
317
1tanh/
250 Random networksFully connected
b = 1/4
Opposition
Reciprocity
Connectivity
jj
ijii xabxdtdx
317
1tanh/
250 Random networksN = 317, b = 1/4
Dilute Fully connected
1%
Network Size
jj
ijii xabxdtdxN
1
tanh/
750 Random networksFully connected
b = 1/4
N = 317
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)
StrangeAttractor
2-torus
Circulant Networksdxi /dt = −bxi + Σ ajxi+j
Fully Connected Circulant Network
jij
jii xabxdtdxN
1
1tanh/
N = 317
Diluted Circulant Network
)tanh(/ 25412642 iiiii xxxbxdtdx
N = 317
Near-Neighbor Circulant Network)tanh(/ 654321 iiiiiiii xxxxxxbxdtdx
N = 317
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
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)
[email protected] (contact me)