Dynamics is a large area, so a one semester course will have to be selective. We begin with the study of dynamics in one dimension (including logistic and circle maps), understanding such things as local bifurcations and normal forms. We will define chaos and learn how to measure it (entropy, Lyapunov exponents and the like). We will study the onset of chaos and learn about Fiegenbaum’s theory of renormalization.

Moving to higher dimensional systems, we will study stable and center manifolds, and use them for bifurcation theory and discuss the onset of chaos through Smale’s horseshoe and the like.

An area that I would like to cover next includes some recent results in Hamiltonian & symplectic dynamics (including KAM theory, Poincaré-Birkhoff and Aubry-Mather theory, and the Anti-Integrable limit). Not all of these topics are complete in my notes; however, some of the texts listed below (e.g. Hasselblatt, Meyer and Hall, Arrowsmith and Place, or MacKay and Meiss) will be found to be helpful.

If you are interested in particular topics, let me know and we may change the course organization.

 

Grading

 

There will be no exams. Homework will be informal, often assigned as group problems.  In addition each student will give one or two presentations during the semester to the class.

 

1. 1.Homework                                                                                                   40%
   

2. 2.Class Presentations                                                                                      50%
   

3. 3.In-Class participation                                                                                  10%
   

Outside Reading Reference List

Popularizations

Gleick, J. (1987), Chaos, the Making of a New Science, (Heinemann, London).

Percival, I. C. (1989), “Chaos: a Science for the Real World,” New Scientist beginning October 21 issue, pages 20-25, continuing through December 2 issue.

Stewart, I. (1989), Does God Play Dice?, (Blackwell, Cambridge).

Introductory References

Arrowsmith, D. K. and Place, C. M. (1992) Dynamical Systems: Differential Equations, Maps and Chaotic Behavior, (Chapman & Hall, London).

Baker, G. L. and Gollub, J. P., 1990, Chaotic Dynamics, Vol (Cambridge Univ. Press, Cambridge).

Devaney, R. L. (1986). An Introduction to Chaotic Dynamical Systems. Menlo Park, Benjamin/Cummings.

Glass, L. and M. Mackey (1988), From Clocks to Chaos: The Rhythms of Life, (Princeton Univ. Press, Princeton).

Percival, I. C. and D. Richard (1982). Introduction to Dynamics. Cambridge, Cambridge Univ. Press.

Pritchard, J. (1992). The Chaos Cookbook: A Practical Programming Guide. Oxford, Butterworth-Heinemann.

Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Reading, Addison-Wesley.

Advanced Texts

Arnold, V. I. (1978). Mathematical Methods of Classical Mechanics. New York, Springer.

Arrowsmith, D. K. and C. M. Place (1990). An Introduction to Dynamical Systems. Cambridge, Cambridge University Press.

Cvitanovic, P.(1984) Universality in Chaos, (Adam Hilgar,London), 513 pages.

Cornfeld, I. P., S. V. Fomin and Y. G. Sinai (1982). Ergodic Theory. New York, Springer-Verlag.

Field, M. and M. Golubitsky (1992), Symmetry in chaos: a search for patterns in mathematics, art and nature, (Oxford University Press, New York).

Glendinning, P. (1994). Stability, Instability and Chaos. Cambridge, Cambridge Univ Press.

*Hale, J. and H. Koçak (1991). Dynamics and Bifurcations. New York, Springer-Verlag.

Hasselblatt, B. and A. Katok (2003). A First Course in Dynamics: with a Panorama of Recent Developments. Cambridge, Cambridge Univ. Press.

Hirsh, M.W. and S. Smale (1974), Differential Equations, Dynamical Systems, and Linear Algebra, (Academic Press, New York).

Lanford, O. E. (1973), “Introduction to the Mathematical Theory of Dynamical Systems,” in Chaotic Behavior of deterministic systems, G. Ioos, R. H. G. Helleman and R. Stora (ed), (North Holland, Amsterdam).

Lichtenberg, A.J. and M. A. Lieberman, Regular and Chaotic Dynamics, (Springer-Verlag, New York 1983).

MacKay, R. S. (1993) Renormalization in Area-Preserving Maps, Advanced Series in Nonlinear Dynamics, Vol 6, (World Scientific, Singapore).

MacKay, R. S. and J. D. Meiss (1987), Hamiltonian Dynamical Systems: a reprint selection, (Adam-Hilgar Press, London).

Meiss, J.D. Differential Dynamical Systems (2007) (SIAM, Philadelphia).

Ott, E. (1993) Chaos in Dynamical Systems, Vol (Cambridge Univ. Press, Cambridge).

Ott, E., T. Sauer, et al., Eds. (1994). Coping with Chaos: Analysis of Chaotic Data and the Exploration of Chaotic Systems. Wiley Series in Nonlinear Science. New York, John Wiley.

Ottino, J. M. (1989), The Kinematics of Mixing: Stretching, Chaos, and Transport, (Cambridge Univ. Press, Cambridge).

Pritchard, J. (1992). The Chaos Cookbook: A Practical Programming Guide. Oxford, Butterworth-Heinemann.

Tabor, M. (1989), Chaos and Integrability in Nonlinear Dynamics : an Introduction, (Wiley, New York).