Differential Equations and Dynamical Systems
Instructor John E. McCarthy
Class TuTh 10.00-11.20, Busch 100
Office Hours M: 3.00-3.50 Tu: 3.00-3.50 Th: 2.00-2.50, in 105 Cupples I
Exams There will be two exams in the course:
1) Exam 1 Midterm. Thursday March 7, in class
2) Exam 2 Final exam. Tuesday May 7, 6.00-8.00.
Homework
There will be weekly homework sets during the semester, assigned on Tuesday and due the following Tuesday.
Prerequisites
Math 233 (Multivariable Calculus), and 309 (Matrix Algebra).
Note: 217 (Differential Equations) is not a prerequisite for this course, but 309 is. There will be some overlap with 217, but enough differences that students can choose to take both courses. 312 will also serve instead of 217 as a prerequisite for more advanced courses on differential equations, such as 415.
Description
“Newton has shown us that a law is only a necessary relation between the present state of the world and its immediately subsequent state. All the other laws since discovered are nothing else; they are in sum, differential equations.” Henri Poincaré
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 differential equations and dynamical systems, with applications to various fields of science, and how chaos arises.
Content
We shall cover the following topics:
- Linear systems of differential equations.
- Flows on the line: Chapter 2
- Bifurcations : Chapter 3
- Linear Systems : Chapter 5
- Phase Plane : Chapter 6
- Limit Cycles : Sections 7.0- 7.3
- Bifurcations Revisited: Sections 8.0-8.3
- Lorenz equations : Sections 9.2-9.5
- Chaos : Section 9.3
- Strange attractors : Section 9.3
- One dimensional systems: Chapter 10
- Fractals: Chapter 11
- Strange Attractors: Chapter 12
Basis for Grading
Attendance and class participation will be 20% of your grade, homework will be 30%, the midterm will be 20%, and the final will be 30%.
Homework Policy
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 rigorous argument, 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.
(c) You may not get solutions from the internet.
Academic Integrity
For the academic integrity policy on homework, see above. The exams will be closed book, and you may not get assistance from another person or any other source.
Class
I expect you to attend class every day, and to participate in class discussions. If you cannot attend class because of illness or quarantine, please let me know by email. If you have more than two unexcused absences, this will affect your grade.
It is important to stay abreast of the material. At the beginning of class I will call on people to give definitions or state theorems; this will be part of your grade. I may call on you at any time to answer a question.
Text Nonlinear dynamics and Chaos, Second Edition, by S.H. Strogatz (CRC 2015). Note that there is a much cheaper Indian edition available, which has identical content.
Additional ReadingChaos: 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.mahttps://legacy-www.math.harvard.https://web.stanford.edu/~jluk/math63CMspring17/Periodic.170529.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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