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, entrainment
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
      Demos of M.C. Cross (CalTech)
      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

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

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   Assignments: HW 1 HW 2 HW 3 HW 4 HW 5 HW 6 HW 7 HW 8 (postscript)
   
   Take-home final (due Thursday, December 7, 5 p.m.)
   

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   Discussion section: We 4-5 in M177
   The goal of the discussion section is to provide feed-back for the homework. Every student will be asked to present their homework at the blackboard and to explain how the homework to the other students in the class.
   The grade will be based on the presentations.
   Alex Roxin (a-roxin@nwu.edu) will lead the discussion sections.

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   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 5 and it differs in some relevant respects from the older versions 3 and 4.
   
   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

   Demo

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   There will be no class on Tuesday November 21. The class will be made up by extending the lecture time of the first 5 classes.

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   Office Hours: Mo Tu Th 5-6 in M458

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   Seminars

   
   The following seminars will be of interest to students in this class:
   Interdisciplinary Seminar in Nonlinear Science on Fridays at 2pm in M416.
   
   Graduate Student Seminar Tuesdays at 4:30 in M416.
   Both seminars are part of the IGERT Program Dynamics of Complex Systems in Engineering and Science

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