Methods of Nonlinear Analysis

Hermann Riecke
Sketchy lecture notes
  1. Introduction
    1. Examples:
      natural patterns, Briggs-Rauscher reaction, Taylor vortex flow, Rayleigh-Benard convection, directional solidification , combustion , rabbit heart
    2. Linear vs. Nonlinear

    3. Separation of Time Scales

  2. Linear Theory

  3. Simple Bifurcations in Normal Form

    1. Steady Bifurcation

    2. Hopf Bifurcation

  4. 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

  5. Mode Interaction

    1. Hopf-Hopf Interaction

    2. Takens-Bogdanov Double-Zero

  6. Symmetries

  7. Continuous Symmetries and Slow Dynamics

    1. Phase Dynamics Phase Diffusion in Taylor Vortex Flow

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

    3. Perturbed Solitons



Assignments:


HW 1 Solutions

HW 2 Solutions: problem 1 maple worksheets: problem 2 problem 3 problem 4

HW 3
References: Derivation of the equations in convection
Solitary traveling waves in Crystal Growth and in Taylor vortex flow

HW 4 Sketch of relevant part of the solutions
Final


Software:
dstool at Cornell. See also debian distribution and Geometry Center at Minnesota

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.

P.G. Drazin, Nonlinear Systems,, Cambridge University Press, 1992
call number 515.355 D769n

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.
call number 519.05 A652 v.42

R. Hoyle Pattern formation : an introduction to methods
call number 501.175 H868p

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer 1990.
call number 003.85 W655i

P. Manneville, Instabilities, chaos and turbulence : an introduction to nonlinear dynamics and complex systems
call number 515.39 M282i

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.

For downloading the two Rev. Mod. Phys. papers: Review of Modern Physics

Most of these books are on reserve the SEL.