Simple Models of Complex Chaotic Systems
J. C. SprottDepartment of Physics
University of Wisconsin - Madison
Presented at the
AAPT Topical Conference on
Computational Physics in Upper
Level Courses
At Davidson College (NC)
On July 28, 2007
Background
Grew out of an multi-disciplinary chaos course that I taught 3 times
Demands computation
Strongly motivates students
Used now for physics undergraduate research projects (~20 over the past 10 years)
Minimal Chaotic Systems 1-D map (quadratic map)
Dissipative map (Hénon)
Autonomous ODE (jerk equation)
Driven ODE (Ueda oscillator)
Delay differential equation (DDE)
Partial diff eqn (Kuramoto-Sivashinsky)
21 1 nn xx
12
1 1 nnn bxaxx
02 xxxax
txx sin3
3 tt xxx
042 uuauuu xxxt
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 and adapt
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 ∞
Route to Chaos at Large N (=101)
jj
ijii xabxdtdx
101
1tanh/
“Quasi-periodic route to chaos”
Strange Attractors
Sparse Circulant Network (N=101)
jij
jii xabxdtdx 9
1tanh/
Labyrinth Chaos
x1 x3
x2
dx1/dt = sin x2
dx2/dt = sin x3
dx3/dt = sin x1
Hyperlabyrinth Chaos (N=101)
1sin/ iii xbxdtdx
Minimal High-D Chaotic L-V Modeldxi /dt = xi(1 – xi – 2 – xi – xi+1)
Lotka-Volterra Model (N=101)
)1(/ 12 iiiii xbxxxdtdx
Delay Differential Equation
txdtdx sin/
Partial Differential Equation
02/ 42 buuuuuu xxxt
Summary of High-N Dynamics Chaos is common for highly-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
Shameless PlugChaos and Time-Series Analysis
J. C. SprottOxford University Press (2003)
ISBN 0-19-850839-5
An introductory text for advanced undergraduateand beginning graduatestudents in all fields of science and engineering
References
http://sprott.physics.wisc.edu
/ lectures/models.ppt (this talk)
http://sprott.physics.wisc.edu/chao
stsa/
(my chaos textbook)
[email protected] (contact
me)
