(p.579) Appendix G The Duffing DoubleWell Oscillator
(p.579) Appendix G The Duffing DoubleWell Oscillator
G.1. The Model
Many systems in nature have several stable states separated by energy barriers. When the system can move among the stable states, the dynamics can become quite complex. A simple model that illustrates some of these features is the Duffing doublewell oscillator. This model was first introduced to understand forced vibrations of industrial machinery [Duffing, 1918]. In this model, a particle is constrained to move in one spatial dimension. An external force acts on the particle. The force is described by
To build a simple mechanical model of the Duffing oscillator, mount a flexible metal strip vertically with the base rigidly clamped. Then place a movable mass on the flexible strip. If the mass is mounted low enough, the stable position will occur with the strip directly vertical. Small deviations from this position will result in oscillations around the vertical position. If the mass is moved further up the strip, eventually the vertical position becomes unstable, and the mass will “flop” to one side or the other. There are now two stable positions—one on each side of the vertical—with a “barrier” in between. See BEN97 for the details of setting up such a system.
The equations describing the dynamics in state space are usually written with k = 1 and b = 1 (with no loss in generality) as
The behavior becomes much more interesting if we “jiggle” the particle with another external force that varies, say, periodically in time. In that case, the state space equations become

1. The response curve of the system changes shape as the amplitude of the external oscillating force increases.

2. The response curve of the system shows hysteresis: the response amplitude depends on whether we increase the frequency through the resonance region or decrease the frequency through the resonance region.

3. The system can display chaotic behavior.
Let's begin with a relatively small value of the amplitude of the external oscillating force. In that case the system behaves, after initial transients die out, much like a simple harmonic oscillator with the oscillations confined around x = ±1. Fig. G.3 illustrates some possibilities.
We see that there are two attractors: a limit cycle centered on x = 1 and a limit cycle centered on x = −1. Which initial conditions (in state space) lead to which attractor? The answer turns out to be rather complicated because the two basins of attractions are thoroughly intertwined and their boundaries form a fractal structure in state space (MOL85). The frontispiece of this book illustrates the complex basins of attractions for F = 0.25 and γ= 0.25.
For values of F between 0.38 and 0.84 (for γ = 1 and ω = 1.0), we get a complex mix of chaotic behavior interspersed with periodic windows. Two (p.582)
The Duffing oscillator model, though relatively simple mathematically, yields surprisingly rich behavior. References for further reading are given in the next section.
G2. Further Reading
G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz and ihre technische Bedeutung (Friedr. Vieweg & Sohn, Braunschweig, 1918).
F. C. Moon and G.X. Li, “Fractal Basin Boundaries and Homoclinic Orbits for Periodic Motion in a TwoWell Potential,” Phys. Rev. Lett. 55, 1439–42 (1985).
[Moon, 1992]. Contains a good discussion of the Duffing model.
C. L. Olson and M. G. Olsson, “Dynamical symmetry breaking and chaos in Duffing's equation,” Am. J. Phys. 59, 907–11 (1991). This paper gives a detailed analysis of the single well, hardening spring version of the Duffing model.
J. E. Berger and G. Nunes, Jr., “A mechanical Duffing oscillator for the undergraduate laboratory,” Am. J. Phys. 65, 841–846 (1997).
G3. Computer Exercises
CEG1. Use Chaotic Dynamics Workbench to explore the dynamics of the Duffing oscillator. For a fixed value of c and ω find the range of F that leads to chaotic dynamics. Locate a perioddoubling sequence if you can.
CEG2. Use Dynamics: Numerical Explorations [Nusse and Yorke, 1998] (or, more challenging, write your own program) to explore the basin of attraction for the two types of limit cycles for the Duffing model as illustrated in Fig. G.3. Note: this computation may take a long time if you want a high resolution picture of the basins.
CEG3. Use Dynamics: Numerical Explorations [Nusse and Yorke, 1998] (or, more challenging, write your own program) to generate a bifurcation diagram for the Duffing oscillator with F the variable parameter. How are you sure that you have got all the attractors in the bifurcation diagram?