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Chaos and Nonlinear DynamicsAn Introduction for Scientists and Engineers$

Robert C. Hilborn

Print publication date: 2000

Print ISBN-13: 9780198507239

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198507239.001.0001

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(p.579) Appendix G The Duffing Double-Well Oscillator

(p.579) Appendix G The Duffing Double-Well Oscillator

Source:
Chaos and Nonlinear Dynamics
Publisher:
Oxford University Press

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 double-well 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

(G-1) Appendix G The Duffing Double-Well Oscillator
The name “double-well” enters because the corresponding potential energy function has a double well structure. Formally, the potential energy function is written as
(G-2) Appendix G The Duffing Double-Well Oscillator
Figure G.1 shows a plot of the potential energy function. We see that there are two stable equilibrium states at Appendix G The Duffing Double-Well Oscillator. There is an unstable equilibrium point at x = 0.

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.

(p.580)

Appendix G The Duffing Double-Well Oscillator

Fig. G.1. A plot of the Duffing double-well potential energy function with k = 1 and b= 1.

The equations describing the dynamics in state space are usually written with k = 1 and b = 1 (with no loss in generality) as

(G-3) Appendix G The Duffing Double-Well Oscillator
where the y term represents damping proportional to the velocity of the particle. The motion of the particle in this situation is relatively simple. If started off with a certain amount of kinetic energy, the particle oscillates back and forth, gradually losing energy via damping and finally comes to rest at the bottom of one of the wells. What makes the oscillator interesting is that the period of the oscillations depends on the amplitude. A typical trajectory is illustrated in Fig. G.2. Note the difference in oscillation period for the initial large oscillation compared to the smaller amplitude oscillations. For the small oscillations, the oscillation period is Appendix G The Duffing Double-Well Oscillator; that is, the natural oscillation frequency ω 0 (for small oscillation amplitudes) is equal to Appendix G The Duffing Double-Well Oscillator.

Appendix G The Duffing Double-Well Oscillator

Fig. G.2. On the left, a plot of x as a function of time. On the right, a state space plot of a Duffing model trajectory from Eq. (G-3) starting with x = 1 and y = 1.5. The damping coefficient is γ = 0.5.

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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

(G-4) Appendix G The Duffing Double-Well Oscillator
We might expect, based on experience with the simple harmonic oscillator, that the particle will respond with relatively large amplitude motion when the frequency of the external force matches the natural oscillation frequency of the particle. The complication is that the natural oscillation frequency depends on the amplitude of the motion. So as the particle begins to respond to the external oscillating force, its amplitude changes and hence its natural oscillation frequency changes. Several novel features can appear:
  1. 1. The response curve of the system changes shape as the amplitude of the external oscillating force increases.

  2. 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. 3. The system can display chaotic behavior.

The first two features are discussed in some detail, supported by analytical calculations, in [Strogatz, 1994], pp. 226–7 and 238–40 and [Jackson, 1991], Vol. 1, pp. 308-314. Here, we will focus our attention on the third item, 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)

Appendix G The Duffing Double-Well Oscillator

Fig. G.3. State space trajectories for the Duffing model with F = 025, γ = 0.5, and ω = 1.0. On the left, y 0 = −1.0, x 0 = 0.5. On the right, y 0 = −0.5, x 0 = 0.5. The two trajectories lead to the two different limit cycles.

examples are shown in Fig. G.4. Figure G.5 shows a Poincaré section (stroboscopic portrait) of the state space taken at the phase when the driving force has its largest value, that is, when cos(ωt) = 1. The complex structure of the chaotic attractor is apparent. For larger values of F, the behavior is periodic with the period of the driving force.

The Duffing oscillator model, though relatively simple mathematically, yields surprisingly rich behavior. References for further reading are given in the next section.

Appendix G The Duffing Double-Well Oscillator

Fig. G.4. On the left, a period-4 attractor with F= 0.5. On the right, a chaotic attractor with F=0.7. In both cases γ = 0.5 and ω = 1.0.

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Appendix G The Duffing Double-Well Oscillator

Fig. G.5. The Poincaré section of the state space for the chaotic attractor shown on the right in Fig. G.4.

G-2. 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 Two-Well 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).

G-3. Computer Exercises

CEG-1. 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 period-doubling sequence if you can.

CEG-2. 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.

CEG-3. 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?