Simple Models of Complex Chaotic Systems

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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

J. C. Sprott

Department of Physics

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
• 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?
• 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)
A General Model (artificial neural network)

N neurons

“Universal approximator,” N  ∞

Route to Chaos at Large N (=101)

“Quasi-periodic route to 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

dxi /dt = xi(1 – xi –2– xi – xi+1)

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

Chaos and Time-Series Analysis

J. C. SprottOxford University Press (2003)

ISBN 0-19-850839-5

An introductory text for