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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 CoursesAt Davidson College (NC)
On July 28, 2007
Collaborators
� David Albers, Univ California - Davis
� Konstantinos Chlouverakis, Univ Athens (Greece)
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 ωsin3 =+&&
3ττ −− −= 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
j jxax ∑==1
tanhoutN
inputs
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 Chaosx1 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
τ−= txdtdx sin/
Partial Differential Equation02/ 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/davidson.ppt (this talk)
� http://sprott.physics.wisc.edu/chao
stsa/ (my chaos textbook)
� [email protected] (contact
me)