Experiments have been made on an electronic implementation of a two-centre harmonic oscillator. This oscillator implementation is shown in Figure 10 and consists of a simple two-integrator loop with damping, which models a simple damped linear second order system. There is a refinement in that the offset of the centre of the system is switched from +X to -X when the displacement variable x passes downwards through zero. The opposite happens when the displacement returns and passes upwards through zero.
This two-centre system is very similar to a damped impact oscillator, where the velocity changes discontinuously when the displacement reaches the wall position X. The impacting system models a mass-spring-damping system where the mass bounces elastically on a rigid wall. There is a complete isomorphism between the impacting system, and an adaptation of the two-centre system in which the driving function is inverted every time the displacement passes through zero. Thus we have two more VSS circuits to consider; intermittency has been observed in computer simulations and theoretical studies of the impact oscillator [12].
When, in the impacting system, the velocity discontinuously reverses, the time development of the linear part of the motion may be regarded as being suddenly advanced. This is not an exact modelling, but it explains the observation that when the drive frequency is adjusted to lie a little below the non-impacting resonant frequency of the system, the impacts can keep the motion more in phase with the drive and the phase plane portraits are on average larger than they would be if the wall were moved away to larger X. Thus the impacts are sustained until, happenchance, the chaotic fluctuations lead to an impact or impacts being missed, and the motion collapses onto the limit cycle of the linear system.
Since the two-centre system only differs from the impacting system by lacking an inversion of the drive, the phase relationship in the case of the two-centre system is reversed, and the orbits are, on average, larger for the drive frequency a little above resonance.
Figure 10: A two-centre VSS equivalent circuit diagram
Thus we see in Figure 11 a time series development in which the chaotic fluctuations of the two-centre system eventually result in the subsequent zero-transitions not occuring and the system then falls onto a limit cycle (see the phase plane portrait for this system in Figure 12) on one side of the displacement origin at x=0. In the analogous case of the impact oscillator, this is seen when the orbits no longer cross the wall position, and impacts then cease.
This behaviour is a classic case of trapping in a VSS. By adjusting the amplitude of the non-impacting limit cycle, it can be arranged for the average time to trap to become very large. The experimental photo shown in Figure 11 was the tenth exposure in a continuous sequence during which the system did not trap. The time span across the photo is 35 seconds; in this case the system took about 10 minutes to trap at an angular drive frequency of 10 radians per second. The size of the limit cycle is about 3/4 of the amplitude at which switching occurs. Thus we see that trapping can be arranged to be improbable; what is certain however is that for these conditions, this system will always trap if one waits long enough.
Figure 11: Trapping in the system of figure 10; time series. The axes are time,
horizontal; displacement, vertical.
Figure 12: The phase plane portrait of trapping in the two-centre system. The axes are
displacement, horizontal; velocity, vertical.