Methods of Nonlinear Analysis

0. Introduction

1. Linear vs. Nonlinear

2. Separation of Time Scales

3. Examples:
   Taylor vortex flow, Rayleigh-Benard convection, directional solidification , combustion, rabbit heart, granular Faraday

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

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

III. Slow Dynamics through Symmetries

1. Phase Dynamics

   1. Steady Patterns
      Phase Diffusion in Taylor Vortex Flow

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

Assignments:

Matlab: phaseplane program pplane5.m, ppn5out.m by J.C. Polking
Note: to get the orbits in pplane5 over longer times one can choose ode23 and pick a calculation window that is 10 times larger than the display window.

XPP-Aut is also often mentioned. I have not used it yet.

Demo:
sh.m shcode.m shini.m

Recommended Books and Papers:

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. call number 515.353 G629s

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. call number 003.85 W655i

P. Manneville, Dissipative Structures and Weak Turbulence, Academic Press, 1990. call number 531.11 M282d

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

These resources are on reserve the SEL.