Duffing (1918) introduced a nonlinear oscillator describing the hardening spring effect along with forced vibrations given by the equation:

This system can produce very erratic or chaotic behavior. Your project is to discuss the physical model, then perform some of the mathematical analyses associated with this problem. (You can't exhaustively research this problem as hundreds of papers have been written on the subject.) Perform a local analysis finding where Hopf bifurcations occur and do some numerical studies, showing how simple bifurcations become more complicated. Select one of the mathematical tools like averaging or Melnikov's method and show how it applies to this problem. Compare different graphical techniques like Poincare maps and 3-D solutions to determine how you can best explain the phenomenon. There is an excellent website describing the model and providing applets (and even the source code) for further investigation.

http://www.apmaths.uwo.ca/~bfraser/version1/dwintro.html

References:

1. Guckenheimer and Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag 1983 Chap. 2.2 (pp. 82-91), 173-177, 191-193, 198-201.
2. Arrowsmith and Place: An Introduction to Dynamical Systems, Cambridge 1990.
3. Perko: Differential Equations and Dynamical Systems, Springer-Verlag 1991.