ECE 6950: Nonlinear Systems

 

 

Course Description
  ECE 6950 is a first course in nonlinear systems. In it you will learn to characterize nonlinear phenomena such as limit cycles and chaotic behavior, both analytically and numerically. We will also delve into the world of strange attractors and fractals. All this will be applied to a number of engineering, mechanical, biological and chemical problems. Specifically we will consider the family nonlinear control problems (such as the inverted pendulum) and chaotic communication systems (such as the Cummo and Chua circuits.)

 

Course Topics

 

Helpful Downloads
 

Introduction

  • History and perspective
  • Chaos, fractals and dynamical systems

Flows on the number line

  • Fixed points
  • Stability
  • Potentials
  • Existance theorems
  • Examples

Bifurcations

  • Zeeman machines
    • Catastrophies
  • Bifurcations
    • Saddle-node
    • Transcritical
    • Pitchfork
  • Examples

Flows on a circle

  • Oscillators
  • Uniform
  • Non-uniform

Flows in two-dimensions - linear Systems

  • Phase plane analysis
  • Classification

Two-dimensional phase plane analysis

  • Phase portraits
  • Existance and uniqueness theorems
  • Linearization
  • Conservative systems
  • Reversable systems
  • Examples

Limit cycles

  • Poincare-Bendixson Theorem
  • Weakly nonlinear oscillators
  • Perturbation theory

Bifurcations - applications

  • Hopf birfuctaions
  • Global cyclic bifurcations
  • Poincare maps

Flows in Three-dimensions - chaos

  • Lorenz equations
  • Chaotic waterwheel
  • Chaos in communications

One-dimensional maps

  • Logistic maps
  • Periodic windows
  • Logistic equation
  • Liapunov exponents

Fractals

  • Cantor sets
  • Strange attractors
  • Fractional dimensions
  • Lorentz attractor

Strange attractors

  • Correlation of chaos and fractals
  • Examples

Course Syllabus

Schedule

Lecture Notes

Assignments

Computer Programs

Sample Tests

Tests and Solutions

Zeeman Machine Experiment

 

 

 

Class of Summer, 2005 with Chaotic Waterwheels