Lorenz strange attractor ( Ed Lorenz, maybe the most famous discoverer of chaos, graduated in math from Dartmouth in 1938!)
Chaotic dynamical systems are everywhere: weather patterns, swinging pendula, population dynamics, even human heart rhythms. With a balance of theory and applications, this course will introduce: flows, fixed points, bifurcations, Lorenz equations, Lyapunov exponent, one-dimensional maps, period-doubling, Julia sets, fractal dimension. Optional topics may include: Hamiltonian systems, symbolic dynamics. Numerical explorations will form an integral part of the course; I recommend (and will be using and demonstrating) Matlab (or its free cousin Octave). In the final 3 weeks you will research and present a final project investigating a topic beyond the taught material.
The goal is to introduce you to the recent, exciting, and rapidly-growing area, and to strike a balance between theoretical analysis, concepts, computer-aided exploration, and applications. The impact of nonlinear dynamical systems on science has been far-reaching, including the physical, life and social sciences, engineering, finance and mathematics, and therefore we expect this course to be of interest to all such students with a mathematical background in linear algebra and differential equations.
Here are the write-ups (plus some codes and movies) from the Fall 2011 student projects. Enjoy!
Lectures / OH: Kemeny 108, T+Th 10-11:50am ("10A" slot), important to attend since we'll do lots of worksheets together. I strongly recommend you read the material in the book in advance of the lecture. X-hour is W 3-3:50pm, and I will use intermittently for: Matlab sessions, review, problem-solving sessions, etc. Do not schedule anything regular in this X-hr. I encourage you to come to office hours: 5-6 M, 3-4 Tu, 5-6 W
Required book: Chaos: An Introduction to Dynamical Systems, First Edition, by Kathleen Alligood, Tim Sauer and James Yorke (Published by Springer, 1996). Available at Wheelock Books, etc.
Homework: about 7 weekly HWs due Thursdays at start of lecture. I strongly encourage you to collaborate, and to try at least some of the relevant homework problems before the next lecture (leaving it all for Wednesday night risks you getting left behind in this fast-paced course.) Please make your working/reasoning as clear as you can, write clearly, don't be scared of using lots of space on the page, and staple your work. Late homework will not be accepted (unless by prior arrangement for a valid, and exceptional, reason). Your lowest HW score will be dropped.
Exams: I will try to give you ample time to complete exam questions. This course has no final exam so the 2 midterms are important. Practise is key (also read this).
Project: Mainly in the last 3 weeks you will work (possibly in groups of 2-3) and research in detail a topic, usually a mix of background reading and computer experimentation, and present it in class, along with a short (5-page) write-up. It is a sizeable fraction of your course grade. Here's a preliminary list of topics and details. Project topics should be chosen by Tues Oct 25, a 1-2 page description with references is due Tues Nov 1, and in-class presentations Tues Nov 29 (continued Wed Nov 30). Final write-up will be due later that week.
Honor principle. Exams: no help given or received. Homework: group discussion and collaboration on problem techniques is great and helpful. Write-ups must be done individually (ie no copying).
Grades: Will be based on HW 25%, Midterms 2*20%, Project 35%. Note the HW is quite heavily weighted, and is the main chance you get to learn the concepts, practise and get feedback, so stay on top of it. Grades in Math 53 are not curved; other students' good performance will not hurt your grade. (So please work together and help each other out!)
Special needs: I encourage students with disabilities, including "invisible" disabilities like chronic diseases and learning disabilities, to discuss with us any appropriate accommodations that might be helpful. Let me know asap, certainly in first 2 weeks. Also stop by the Academic Skills Center in 301 Collis to register for support services.
Private tutoring: Tutor Clearinghouse may have private one-on-one tutors available for Math 53 (they would have to have taken the course or be advanced math majors). If a student receives financial aid, the College will pay for three hours of tutoring per week. If you would like to have a tutor, please go to 301 Collis and apply as early as possible.
Religious observance: Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me before the end of the second week of the term to discuss appropriate accommodations.