Math 350, Spring 2019

 Dynamics and Chaos 

 

Instructor                      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
Phone                          935-6753

Prerequisites               Math 217 (Differential Equations) and 233 (Multivariable Calculus)

 

Description
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.

Homework

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.

Midterm. 

Content

  1. Flows on the line:  Chapter 2
  2. Bifurcations : Chapter 3
  3. Flows on the circle (skipped for now)
  4. Linear Systems : Chapter 5
  5. Phase Plane : Chapter 6
  6. Limit Cycles  : Sections 7.0- 7.3
  7. Bifurcations Revisited: Sections 8.0-8.3
  8. Lorenz equations : Sections  9.2-9.5
  9. Chaos : Section 9.3
  10. Strange attractors : Section 9.3
  11. One dimensional systems: Chapter 10
  12. Fractals: Chapter 11
  13. Strange Attractors: Chapter 12

 

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

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 rules:
(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.

Class

I expect you to come to class every day, and to participate in class discussions.
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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