### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

### Prerequisites

*18.03SC Differential Equations* or *18.034 Honors Differential Equations*

### Description

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

### Outline of the Course

A rough idea follows. Some things may be covered in more detail than this implies, or the reverse. This is just to give you an idea of the “flavor”.

- One-dimensional systems and elementary bifurcations.
- Two-dimensional systems; phase plane analysis, limit cycles, Poincaré-Bendixson theory.
- Nonlinear Oscillators, qualitative and approximate asymptotic techniques, Hopf bifurcations.
- Lorenz and Rossler equations, chaos, strange attractors and fractals.
- Iterated mappings, period-doubling, chaos, renormalization, universality.
- Hamiltonian systems; complete integrability and ergodicity.
- Area preserving mappings, KAM theory.
- Other: Floquet theory, Infinite Dimensional Hamiltonians, On-Off Dissipative Systems, etc.

### Textbook

Strogatz, Steven H. *Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering*. Westview Press, 2014. ISBN: 9780813349107. [Preview with Google Books]

### References

Wiggins, S. *Introduction to Applied Nonlinear Dynamical Systems and Chaos*. Springer, 2003. ISBN: 9780387001777. (More “mathy” than Strogatz. Used for some topics.) [Preview with Google Books]

Drazin, P. G. *Nonlinear Systems*. Cambridge University Press, 2012. ISBN: 9781139172455.

Peitgen, H-O, H. Jurgens, and D. Saupe. *Chaos and Fractals: New Frontiers of Science*. Springer, 2012. ISBN: 9781468493962.

Parker, T. S., and L. O. Chua. *Practical Numerical Algorithms for Chaotic Systems*. Springer, 2011. ISBN: 9781461281214.

Jordan, D. W., and P. Smith. *Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers*. Oxford University Press, 2007. ISBN: 9780199208258.

Berge, P., Y. Pomeau, and C. Vidal. *Order Within Chaos*. Wiley-VCH, 1987. ISBN: 9780471849674.

McCuskey, S. W. *Introduction to Celestial Mechanics*. Addison-Wesley Publishing Company, Incorporation, 1963. ISBN: 9780201045703.

Guckenheimer, J., and P. Holmes. *Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields*. Springer, 2002. ISBN: 9780387908199. Requires mathematical sophistication. Subject covered at a rigorous level, with proofs requiring knowledge beyond course pre-requisites (say, at the level of Coddington, E. A., and N. Levinson. *Theory of Ordinary Differential Equation*. McGraw-Hill, 1984. ISBN: 9780070992566.)

Lichtenberg, A. J., and M. A. Lieberman. *Regular and Chaotic Dynamics, Applied Mathematical Sciences*. Springer, 1992. ISBN: 9780387977454.

MacKay, R. S., and J. D. Meiss. *Hamiltonian Dynamical Systems: A Reprint Selection*. CRC Press, 1987. ISBN: 9780852742167. [Preview with Google Book]

### Problem Sets and Exams

There are 10 problem sets. Do them all! No way you can learn the material in this course if you do not! There is no exam (neither midterm, nor final).

### Term Paper

Can be on any topic relevant to the course material (instructor pre-approval required). It does not have to be original research, but it must be original work [e.g.: Review the literature in some topic, and summarize the results in your own words, giving proper credit to the sources]. The explanations must be clear, and accessible by someone with the level of an average student in the class! You can use material from your own research, but “recycling” (e.g.: Handing a piece of your thesis) is not allowed. You must process it to follow the guidelines here. Further requirements:

- Must be typed (font size 12–14) and submitted electronically in pdf format.
- Length should not exceed about 15 pages, using standard page formatting. You can use more if you have many figures, but use judgement here!

Failure to follow all these guidelines may result in a paper being classified as “not acceptable”.

### Grading

Each problem set will contain a buried mini-quiz within it (1–3 problems). Only the quiz will be graded, but you need to do all the problems, since the quiz problems will not be identified. The course grade will be based on the cumulative quiz-grade, and a binary-grade term paper. If you hand in an acceptable term paper, then the grade is quiz-based. Else the grade is F. The term paper can be handed any time before the last day of lectures.