The most important book, which we recommend that you purchase, is:
Strogatz, S. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering. Westview Press, 2001. ISBN: 9780738204536.
Strogatz will be especially valuable for our discussion of stability of ordinary differential equations.
Among the many books on chaos, fractals, pattern formation, and related mathematics, you may find it interesting to consult, either during or after the course, the following:
Beltrami, E. Mathematics for Dynamic Modeling. Academic Press, 1987. ISBN: 9780120855551.
An elementary presentation of many of the mathematical techniques useful in studying dynamics. Especially useful for a review of ordinary differential equations, stability, fixed points, and phase space representations.
Hilborn, Robert C. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press, 1994. ISBN: 9780195057607.
One of the more elementary books on the subject, oriented more for the physicist than for the mathematician. Big (over 600 pages). Lots of good exercises.
Kaplan, D., and L. Glass. Understanding Nonlinear Dynamics. Springer-Verlag, 1995. ISBN: 9780387944401.
A nice undergraduate-level introduction, oriented towards scientists, and especially biologists.
Manneville, P. Dissipative Structures and Weak Turbulence. Academic Press, 1990. ISBN: 9780124692602.
Covers the low-dimensional theory discussed in the course but also includes considerable discussion of research on spatially-extended systems. Advanced.
Thompson, J. M. T., and H. B. Stewart. Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. John Wiley and Sons, 1986. ISBN: 9780471909606.
A textbook of moderate difficulty with lots of nice pictures and a fair amount of material of engineering interest.
Turcotte, D. Fractals and Chaos in Geology and Geophysics. 2nd ed. Cambridge University Press, 1997. ISBN: 9780521567336. [Preview with Google Books].
Provides a number of nice examples of nonlinear phenomena in the earth sciences.