Introduction

**The Duffing equation is a differential equation used to model a double
well oscillator such as the magneto-elastic mechanical system. This system
consists of a beam positioned vertically between two magnets, with the
top end fixed, and the bottom end free to swing.**

**The beam will be attracted to one of the two magnets, and given some
velocity will oscillate about that magnet until friction stops it. Each
of the magnets creates a fixed point where the beam may come to rest above
that magnet and remain there in equilibrium. However, when this whole system
is shaken by a periodic forcing term, the beam may jump back and forth
from one magnet to the other in a seemingly random manner. Depending on
how big the shaking term is, there may be no stable fixed points and no
stable fixed cycles in the system. This system can be modeled mathematically
by the equation,**

**as shown by Moon and Holmes (1979). This equation is known as the
Duffing equation. The behavior of this equation will be the topic of study
in this document.**