Math 350, Spring 2019
Dynamics and Chaos
John E. McCarthy
Class MWF 9.00-10.00, Cupples I Rm 218
Midterm Friday March 1st
Final Friday May 3rd, 8.00-10.00 a.m.
Office 105 Cupples I
Office Hours M 4.00-5.00, We 10.00-11.00, Th. 3.00-3.45, and by appointment
Prerequisites Math 217 (Differential Equations) and 233 (Multivariable Calculus)
One of the great mathematical discoveries of the 20th century is that nearly all naturally occuring systems are chaotic -
small changes in the initial conditions will lead to large changes in the long-term behavior.
This means that predictions can only be made for a characteristic time period.
This can be days for the weather, hundreds of millions of years for planetary motion, or seconds for some chemical reactions.
But there is a theoretical limit to how well predictions can do.
This course shall investigate dynamical systems, with applications to various fields of science, and how chaos arises.
There will be weekly homework sets during the semester, assigned on Friday and due the following Friday.
Homework 1, due February 1.
Homework 2, due February 8.
Homework 3, due February 15.
Homework 4, due February 22.
Homework 5, due March 8.
Homework 6, due March 22.
Homework 7, due March 29.
Homework 8, due April 5.
Homework 9, due April 12.
Homework 10, due April 19.
Homework 11, due April 26.
Basis for Grading
Attendance and class participation will be 5% of your grade, homework will be 30%, the midterm will be 30%, and the final will be 40%.
Homework is an extremely important part of the course. Whilst talking to
other people about it is not dis-allowed, too often this degenerates into one
person solving the problem, and other people copying them (often justified to
themselves by saying "I provide the ideas, X does the details" - but
the details are the key. If you can't translate the idea into a real proof, you
don't understand the material well enough). So I shall introduce the following
(a) You can only talk to some-one else about a problem if you have made a genuine effort to solve it yourself.
(b) You must write up the solutions on your own. Suspiciously similar write-ups will receive 0 points.
I expect you to come to class every day, and to participate in class
I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.
Text Nonlinear dynamics and Chaos, Second Edition, by S.H. Strogatz (CRC 2015).
Other books Chaos: Making a New Science by James Gleick is one
of the best general interest science books ever written,
capturing the excitement of the discoveries and describing the people who made them.
Websites Here are some videos: http://www.clausewitz.com/mobile/chaosdemos.htm
Verhulst and the logistic equation: http://webpages.fc.ul.pt/~mcgomes/aulas/dinpop/Mod13/Verhulst.pdf
Proof of the Poincare-Bendixson theorem: http://www.math.harvard.edu/library/sternberg/slides/11809PB.pdf
Supercritical Hopf bifurcation: https://www.youtube.com/watch?v=2XGIKPUZc2U
Subcritical Hopf bifurcation: https://www.youtube.com/watch?v=zclp8vLKJzU
Briggs-Rauscher reaction: https://www.youtube.com/watch?v=-RFb8T2ED5E
Belousov-Zhabotinsky reaction: https://www.youtube.com/watch?v=PpyKSRo8Iec
Chaotic Lorenz Water Wheel: https://www.youtube.com/watch?v=7A_rl-DAmUE
Period 3 implies chaos: https://www.jstor.org/stable/2318254?seq=1#metadata_info_tab_contents
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