Syllabus D-12

Methods of Nonlinear Analysis

Fall Quarter

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

    1. Steady Bifurcation

    2. Hopf Bifurcation

  3. Reduction of Dynamics

    1. Center Manifold

    2. Local Representation of Center Manifold

    3. Intervals in Parameter Space

    4. Multiple-Scale Approach

    5. Formal Perturbation Expansion: Solvability Condition
      Steady Patterns
      Hopf Bifurcation

  4. Poincare-Birkhoff Normal Forms

    1. General Case

    2. Steady Bifurcation

    3. Hopf Bifurcation

  5. Mode Interaction

    1. Hopf-Hopf Interaction

    2. Takens-Bogdanov Double-Zero

  6. Symmetries

    1. Elements of Group Theory

    2. Symmetry Breaking

    3. Steady Bifurcation with Dn-Symmetry on C

    4. O(2)-Hopf Bifurcation

  7. Degenerate Bifurcations. Unfolding

    1. Perturbing Saddle-Node Bifurcation

    2. Perturbing the Pitch-Fork Bifurcation

    3. Weakly Subcritical Pitch-Fork Bifurcation

    4. Unfolding in 2 Dimensions

  8. Periodically Driven Systems

    1. Periodically-Forced Hopf Bifurcation. Weak Forcing

    2. Strong Periodic Forcing of Waves: Maps

Winter Quarter

0. Introduction

  1. Bifurcations in Systems with Large Aspect Ratio

    Envelope equations

  2. Slow Dynamics through Symmetries

  3. Secondary Bifurcations

II. Bifurcations in Large Systems

  1. Newell-Whitehead Equation

    1. 1 Dimension
      Eckhaus Instability in Taylor Vortex Flow

    2. 2 Dimensions
      Eckhaus Instability: Defects
      Zig-Zag Instability in Reaction-Diffusion System

  2. Complex Ginzburg-Landau Equation

    1. Homogeneous Oscillations
    2. Traveling Waves
      Dispersion
      Convective vs. absolute instability

  3. Brief Discussion of the Phenomenology of the Complex Ginzburg-Landau Equation
    Spatio-temporal chaos

  4. Coupled Complex Ginzburg-Landau Equations

  5. Long-Wave Equations

    1. Swift-Hohenberg Equation

    2. Kuramoto-Sivashinsky Equation

III. Slow Dynamics through Symmetries

  1. Phase Dynamics

    1. Steady Patterns
      Phase Diffusion in Taylor Vortex Flow

    2. Hopf Bifurcation at Vanishing Wavenumber

    3. Hopf Bifurcation at Non-Zero Wavenumber

      1. Traveling Waves
      2. Standing Waves

  2. Fronts

    1. Single Fronts

    2. Front Interaction and Localized Structures

      Locking of Fronts, Binary-Mixture Convection, Electroconvection in nematics

  3. Perturbed Solitons

    1. Nonlinear Schrodinger Equation

    2. Dissipative Perturbations

      Binary-Mixture Convection

IV. Secondary Bifurcations

  1. Parity-Breaking Bifurcation

    directional solidification (video of localized parity-breaking waves in Taylor vortex flow)

  2. Secondary Hopf Bifurcation (there was no time for this any more)



Assignments D-12-1 (Fall 1999):

  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

  9. FINAL

Assignments D-12-2 (Winter 2000):

  1. HW 1

  2. HW 2
    Partial Solutions (courtesy V. Moroz): Problem 1, Problem 2, Problem 3

  3. HW 3

  4. HW 4

  5. HW 5

  6. HW 6

  7. HW 7

List of suggested projects.

Poster Session for the projects will be on Wednesday March 15, 12-2, in the Applied Math conference room.

There will be weekly discussion sections in which students present the results of their homework. Vadim Moroz will be leading this sections. They are Thursdays at 12 in M349.

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:

Fall:

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

M. Golubitsky, I. Stewart and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer, 1988.

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.

Winter:

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

D. Walgraef, Spatio-Temporal Pattern Formation

M.C. Cross and P.C. Hohenberg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 851.

These sources are on reserve.