Syllabus D-12

Methods of Nonlinear Analysis

0. Introduction

  1. Linear vs. Nonlinear

    Taylor vortex flow, Rayleigh-Benard convection, directional solidification, combustion

  2. Separation of Time Scales



I. Bifurcations & Low-dimensional Systems

  1. Linear Theory

  2. Simple Bifurcations in Normal Form

  3. Reduction of Dynamics

  4. Normal Forms

  5. Mode Interaction

  6. Symmetries & Non-simple Eigenvalues

  7. Bifurcations in Maps



II. Bifurcations in Large Systems

  1. Newell-Whitehead Equation

  2. Complex Ginzburg-Landau Equation

  3. Coupled Complex Ginzburg-Landau Equations

  4. Long-Wave Equations



III. Slow Dynamics through Symmetries

  1. Phase Dynamics

    1. Steady Patterns

    2. Oscillatory Patterns

    3. Coupled Phase Equations

  2. Fronts

    1. Single Fronts

    2. Front Interaction

  3. Perturbed Solitons



Homework Assignments:

  1. HW 1
  2. HW 2
  3. HW 3
    Solution for Hopf bifurcation in Brusselator
    Maple worksheet for Hopf bifurcation in Brusselator
  4. HW 4
  5. HW 5
  6. HW 6
  7. HW 7
  8. HW 8

FINAL

For simulation of dynamical systems dstool is very useful, which was written by Kim and Guckenheimer (Cornell U.). I have installed it on
korf.esam.nwu.edu
and you can log in using the account
leo
Note you will need the one-time passwords handed out in class.

Recommended Books and Papers:

J.D. Crawford, Introduction to Bifurcation Theory, Rev. Mod. Phys. 63 (1991) 991.

P. Manneville, Dissipative Structures and Weak Turbulence, Academic Press, 1990.

M Golubitsky, I. Stewart and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer, 1988.
Relevant pages 1 Relevant pages 2 Relevant pages 3 Relevant pages 4

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer 1990.