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ECEN 4423 - Chaotic Dynamics


Catalog Data ECEN 4423 (3). Chaotic Dynamics. Explores chaotic dynamics theoretically through computer simulation. Covers the standard computational and analytical tools used in nonlinear dynamics and concludes with an overview of leading-edge chaos research. Topics include time and phase-space dynamics, surfaces of section, bifurcation diagrams, fractal dimension, and Lyapunov exponents.
(Meets with CSCI 4446 and ECEN 5423.)
Credits and Design 3 credit hours. Elective course.
Prerequisite(s) CSCI 1200, Introduction to Programming, CSCI 1300, Computer Science 1: Programming, or ECEN 1030, C Programming for ECE
PHYS 1110, Physics 1
Recommended prerequisite(s) PHYS 1120, Physics 2
CSCI 3656, Numerical Computation
MATH 3130, Introduction to Linear Algebra.
Textbook
  
Course Objectives
Topics Covered
  1. General introduction to chaos
  2. Dynamics of iterated maps
    1. discrete- versus continuous-time chaos: definitions and examples
    2. return maps: the logistic map
    3. correlation plot vs. time domain
    4. stability, attractors, bifurcations
    5. accumulation points, reverse bifurcations
    6. Feigenbaum
    7. Lyapunov exponent and metric entropy
    8. symbolic dynamics
    9. stretching, folding, and sensitive dependence
    10. Sarkovskii and Yorke
  3. Fractals and fractal dimension
    1. definitions and examples
    2. box-counting dimension
    3. embedding
    4. the link between fractals and chaos
  4. Continuous-time dynamics
    1. ODE and linear algebra review
    2. atractors and idssipation: Liouville and Hamilton
    3. numerical integration
    4. numerical error
    5. integrator-induced dynamics: numerical bifurcations and chaos
  5. Applications
    1. Lorenz
    2. Rossler
    3. driven pendulum
  6. Poncare sections
    1. hyperplanes
    2. visulaization and examples
    3. algorithms
  7. Attractor characterization
    1. definitions
    2. the variational equation
    3. location and basin algorithms
    4. embedding: definitions and algorithms
    5. numeric Lyapunov algorithms
    6. experimental Lyapunov algorithms
    7. attractor dimension algorithms
    8. fractal basin boundaries
  8. Hamiltonian chaos
    1. introduction to classical mechanics
    2. chaos and KAM torii
    3. applications: Wisdom, Yip
  9. Prediction, modeling, and noise
    1. introduction
    2. algorithms
    3. applications
    4. noise
  10. Stability
    1. definitions
    2. eigenvectors and un/stable manifolds
    3. homoclinic/heteroclinic orbits
    4. algorithms
    5. Farmer's noise-reduction scheme
    6. Smale-Birkhoff: horseshoes and manifolds
    7. controlling chaos
  11. Modern topics

Last revised: 08-02-11, PM, ARP.