Interdisciplinary Nonlinear Dynamics (438)

Hermann Riecke


Lecture Notes

Overview

  1. Introduction
    Taylor vortex flow, Rayleigh-Benard Convection, Flames, Chemical Oscillations Waves in Rabbit Heart
    Nonlinear vs. linear, phase space
  2. One-dimensional Flows
    1. Flows on the Line
      Population Growth
      Fixed points, stability, potentials, uniqueness of solution
    2. Bifurcations
      Saddle-node bifurcation, transcritical bifurcation, pitch-fork bifurcation, catastrophes
    3. Flows on the Circle
      Saddle-node bifurcation, excitability, synchronization
      (for a recent paper on the old clock experiment by Huygens and its modern version see Bennett et al., Proc. Royal Society (A) (2001))
  3. Two-dimensional Flows
    1. Classification of linear systems
    2. Phase plane
      Phase portrait, attractors, basin of attraction, structural stability
    3. Limit Cycles
      1. Weakly Nonlinear Oscillator. Regular Perturbation Theory
      2. Weakly Nonlinear Oscillator. Multiple Scales
      3. Hopf Bifurcation
    4. Steady Bifurcations
      Implicit function theorem, invariant manifolds (stable, unstable, center)
      1. Center-Manifold Reduction
        Example
  4. Pattern Formation
    Rayleigh-Benard convection, Swift-Hohenberg model
    1. Ginzburg-Landau Equation
      Solvability condition, role of symmetries
    2. Phase Dynamics
      Eckhaus instability, phase slips
  5. Chaos
    1. Differential Equations: Lorenz System
      Java demos of M.C. Cross (CalTech), Pictures for logistic map
      Chaos in a chemical system
    2. Strange attractors, Lyapunov exponents
    3. Maps
      1. Logistic Map
        Cobweb diagram, period-doubling cascade and chaos, Lyapunov exponents
        Experiments in Rayleigh-Benard convection
      2. Universality and Renormalization
    4. Cantor Sets and Fractal Dimension
    5. Diagnostic Tools
      Attractor reconstruction, Poincare section, power spectrum


    The course uses the textbook Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering by S. Strogatz.

    Sketch of Course Notes Postscript PDF (poor quality on screen, prints o.k.))

    Please note, the pdf-version has poor font quality and may not be completely up to date.
    Please note that the notes are still evolving while the class is progressing. They may still have mistakes and typos: please point them out to me if you find any.



    Assignments:

    HW 1 PS HTML PDF
    HW 2 PS HTML PDF
    HW 3 PS PDF
    HW 4 PS PDF
    HW 5 PS PDF
    HW 6 PS PDF
    HW 7 PS PDF
    Final PS PDF
    Aitta, Ahlers, Cannell paper

    If your browser does not show the html-file appropriately you may tune it by following these instructions


    Discussion section: Wednesday 5-6 in L251
    The homework will be done in groups which I have assigned. Please let me know if you are not on that list. The homework will not be graded. Instead students will present and explain the homework to the other students in the class in the discussion section at the blackboard.
    Ted Rieger t-rieger1@northwestern.edu will lead the discussion sections.

    The grade for the class will be based on the presentations (25 %) and a take-home final (75 %).


    MATLAB.There are a number of tutorials and more detailed manuals of MATLAB on the web. A good starting page is the tutorial from UMD which also points to more detailed and extensive web resources (.e.g. the tutorial by S. Davis (Rice)). Please note that the current version of Matlab is 6 and it differs in some relevant respects from the older versions.

    Template for logistic growth program: logistic.m euler.m

    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.

    Program for solving the Swift-Hohenberg model:
    sh.m init.m default.m ic.m go.m eckhaus.m

    Demo


    There will be no class on Tuesday October 8 and Thursday October 10. The class will be made up in two extra sessions.
    Office Hours: Mo 5-6 and Tu, Th 5-6 in L550

    Seminars


    The following seminars may be of interest to students in this class:

    Interdisciplinary Seminar in Nonlinear Science on Fridays at 2pm in M416.


    Graduate Student Seminar

    You can subscribe to the listserver announcing the nonlinear seminars by sending a message to esam.northwestern.edu with the subject subscribe nonlinear Both seminars are part of the IGERT Program : Dynamics of Complex Systems in Engineering and Science