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Simple Models of Complex Chaotic Systems. J. C. Sprott Department 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.

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simple models of complex chaotic systems

Simple Models of Complex Chaotic Systems

J. C. Sprott

Department 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
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
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)
what is a complex system
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 general model artificial neural network
A General Model (artificial neural network)

N neurons

“Universal approximator,” N  ∞

route to chaos at large n 101
Route to Chaos at Large N (=101)

“Quasi-periodic route to chaos”

labyrinth chaos
Labyrinth Chaos

dx1/dt = sin x2

dx2/dt = sin x3

dx3/dt = sin x1

x1

x3

x2

minimal high d chaotic l v model
Minimal High-D Chaotic L-V Model

dxi /dt = xi(1 – xi –2– xi – xi+1)

summary of high n dynamics
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 plug
Shameless Plug

Chaos and Time-Series Analysis

J. C. SprottOxford University Press (2003)

ISBN 0-19-850839-5

An introductory text for

advanced undergraduate

and beginning graduate

students in all fields of

science and engineering

references
http://sprott.physics.wisc.edu/ lectures/models.ppt (this talk)

http://sprott.physics.wisc.edu/chaostsa/ (my chaos textbook)

sprott@physics.wisc.edu (contact me)

References