Results of some simulations will be shown at the meeting. Various parameters can be altered. A variant possible in simulation is to incorporate a restitution coefficient to the equations. This is a factor which alters the velocity at impact, or after crossing, to add loss or gain. One can then investigate the effects of various amounts of damping, restitution, and drive. Setting the drive amplitude to zero reduces the third order system to a second order system and the motion becomes uninteresting; impacting limit cycles have one impact per orbit. (Two crossings per orbit occur in the case of the extended impact system or the double well system).
Most of the simulations performed looked at the effect of positive damping, a small drive amplitude, and frequency of the drive somewhat larger than the natural frequency of one half of the oscillator. Unity restitution coefficient was used. A little thought shows that the dynamics of a system with unity restitution coefficient, but positive damping to provide contraction of the mapping, will be different in detail from the dynamics of a system with zero damping, in which energy loss is provided by having restitution coefficient less than unity. This is particularly easy to see in the case of the thresholds for grazing bifurcations. We shall show phase plane portraits (Figure 4) which develop over time, and an associated time series (Figure 5) which seems to indicate the symmetry of the attractor changing over time, interspersed with intermittent episodes [6,7] of almost periodic behaviour. This kind of behaviour is also seen in noisy iterating systems containing traps, which we shall present elsewhere [4] this summer. A piecewise linear system such as this may be entirely described by a discrete iterating system in which the variables being iterated are the initial conditions at the breakpoints. Thus it is not surprising that the intermittencies seen in the ``continuous variable" double well systems are intimately connected with the trapping behaviours which can be induced in the ``discrete step" iterating systems. It is quite possible that the artefact of adding noise by trapezoidal integration is disturbing the unperturbed ideal system which has a long chaotic transient followed suddenly by periodic motion. This computational noise results in the system being knocked off its periodic solution into transient motions having various different symmetries. Symmetry breaking in this system is commonplace; in a perfect implementation there would be nothing to choose between the two half spaces of the oscillator and LH and RH biased motions should appear with equal probability. There also seem to be attracting solutions with L-R symmetry; this is certainly true in the case of periodic orbits, where all the classes of symmetry discussed are seen.
Figure 4: Consecutive developing phase plane portraits
Figure 5: Time series of fig 4 showing trapping
Another observation of the simulated system is that a double computation of motions starting from separated but close initial conditions results often in trajectories (not just the phase plane portraits) which converge over time. This does not always occur; If for example a grazing bifurcation affects one of the trajectories but not the other the motions diverge. Thus we have the curious situation that for many slight perturbations of initial state, the motions appear chaotic and yet often are not displaying sensitivity to initial conditions. The phenomenon is best observed in real time on a computer simulation; I can supply an example in software for any who are interested in seeing it.
This behaviour is reminiscent of the ``fibers" reported by Wiggins in his lecture course [8] at the Maths department, University of Surrey. Fibers are lines or regions in phase space of localised extent; perturbations of the system along a fiber all converge to the same final trajectory. It appears that in continuous variable systems, taken to the limit in the absence of noise, there are many exotic behaviour patterns which can be predicted mathematically. Another example is the so-called ``riddled basins of attraction'' [9] seen in coupled systems of chaotic oscillators. It is possible that the circuits reported here may be well adapted to investigating such behaviour experimentally.