The Duffing Oscillator

The original Duffing oscillator (Duffing, 1918) was introduced in relation to the single (spatial) modes of vibration of a steel beam subjected to external periodic forces. It has now become one of the standard prototype of forced systems thanks to its simplicity and yet the richness of its solution. This is specially true of the modified model with a double well potential.

The model is described in terms of a potential well

$$V(x) = \beta x^2 / 2 + \alpha x^4 / 4$$

$\beta$ defines the shape of the potential. It is usually chosen to be +1 or -1. $\alpha$ is a measure of the non-linearity. The latter is usually taken to be 1. The steel beam is assumed to oscillate harmonically for small displacement $x$, nonlinearity setting in at larger $x$ values. For $\beta = 1$, the potential exhibits one minimum at $x = 0.0$. For $\beta = -1$, this potential is a symmetric two well potential corresponding to the physics of a steel beam influenced by two magnets as in the figure above.

From this potential, the following force is derived: $F_{beam}(x) = - \frac{ dV } { dx } = - \beta x - \alpha x^3$. The steel beam is also subjected to a dissipation term, $F_{dissipation} = - \mu v$, proportional to the velocity. A driving term is also applied, $F_{driving} = A cos( \omega t )$, which corresponds to a back-and-forth motion of the point of suspension of the beam.

The model is summarized in the two following ODEs:

$$\frac{dx}{dt} = v$$

$$\frac{dv}{dt} = \left( - \beta x - \alpha x^3 - \mu v + A cos( \omega t ) \right) / m$$

The parameters of the model are $\alpha, \beta, \mu, \omega,$ and $A$. The mass is $m$. The time is represented by $t$.

References abound for the Duffing Oscillator. For instance, you can look online at Double Well Duffing. Google shows thousands of links.

You can also read Nonlinear Dynamics: A Two-way Trip from Physics to Math by Hernan G. Solari, Mario A. Natiello, and Gabriel B. Mindlin, published by The Institute of Physics, September, 1996. Dr. Solari and Mindlin were at Drexel in the past as a researcher and graduate student respectively.

An understanding of the Duffing Oscillator in term of topology of the orbits can be found in The topology of Chaos, Robert Gilmore and Marc Lefrand, Wiley, 2002.