- Introduction

Taylor vortex flow, Rayleigh-Benard Convection, Flames, Chemical Oscillations Waves in Rabbit Heart

Nonlinear vs. linear, phase space - One-dimensional Flows
- Flows on the Line

Population Growth

Fixed points, stability, potentials, uniqueness of solution - Bifurcations

Saddle-node bifurcation, transcritical bifurcation, pitch-fork bifurcation, catastrophes - 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))

- Flows on the Line
- Two-dimensional Flows
- Classification of linear systems
- Phase plane

Phase portrait, attractors, basin of attraction, structural stability - Limit Cycles

- Weakly Nonlinear Oscillator. Regular Perturbation Theory
- Weakly Nonlinear Oscillator. Multiple Scales
- Hopf Bifurcation

- Steady Bifurcations

Implicit function theorem, invariant manifolds (stable, unstable, center)- Center-Manifold Reduction

Example

- Center-Manifold Reduction

- Pattern Formation

Rayleigh-Benard convection, Swift-Hohenberg model- Ginzburg-Landau Equation

Solvability condition, role of symmetries - Phase Dynamics

Eckhaus instability, phase slips

- Ginzburg-Landau Equation
- Chaos
- Differential Equations: Lorenz System

Java demos of M.C. Cross (CalTech), Pictures for logistic map

Chaos in a chemical system - Strange attractors, Lyapunov exponents
- Maps
- Logistic Map

Cobweb diagram, period-doubling cascade and chaos, Lyapunov exponents

Experiments in Rayleigh-Benard convection - Universality and Renormalization

- Logistic Map
- Cantor Sets and Fractal Dimension
- 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 paperIf 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.mPhaseplane 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

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. You can subscribe to the listserver announcing the nonlinear seminars by sending a message to

Graduate Student Seminar`esam.northwestern.edu`with the subject*subscribe nonlinear*Both seminars are part of the IGERT Program : Dynamics of Complex Systems in Engineering and Science

- Differential Equations: Lorenz System