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.