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)

� sprott@physics.wisc.edu (contact

me)