Five oscilloscope photographs of experiments which extend the 1-dimensional simulations reported [1][2] previously are presented below. It is important to confirm the results of chaotic simulations by real experiment, as it is not always clear what the effects of noise and imprecisions in the real electronic implementations will be. Therefore we have gone to some trouble to produce this experimental evidence to support our previous ``thought experiments" and simulations on this system.
First, Figure 5 shows an experimental one-dimensional time series of crisis-induced intermittency from a snag that just extends beyond the region of the feature. This is a quite distinct phenomenon from the "noisy trapping" simulation presented [1] earlier, but nevertheless looks very similar. The release from the chaotic trap does not require added noise in this case. Since a single transfer function generator and shift register is used, it is noticeable that the snag recaptures the motion very quickly after its escape. This happens because the proportion of system space occupied by the snag is not particularly small.
Figure 5: Time series of 1-d crisis-induced intermittency
Second, Figure 6 shows
the time series for the 2-dimensional
system, consisting of the two shift registers, the two transfer
function generators, and the rotation matrix circuit. Here we observe
that it takes significantly longer for the snag to recapture the motion.
This is because the snag of size
has area
which
is of second order of smallness in the 2-d system. We also notice
a phenomenon of incomplete escape from the snag which happens from
time to time. Here ``experimentalist's license" has been used to adjust
the overlap of the snag with the region of the feature in order to
capture a telling picture. Size and offset controls have both been used.
It is in this kind of example that the superiority of direct experiment
over simulation is demonstrated.
Figure 6: Time series of one channel of a 2-d crisis-induced intermittency
Next, we show a photo of the average position in the 2 dimensions (Figure 7) for the case of system motion wholly contained within a 2-dimensional snag. Here, the snag has no part which extends beyond the region of the feature.
Figure 7: Motion inside a 2-d snag
Next, a photo of the escape (Figure 8) from the 2-dimensional snag whose size and offset have been both adjusted to provoke a crisis. Incomplete escape can be seen, as can the nearby motion on part of the wider attractor.
Figure 8: leakage from a snag in crisis
Fifth and last, a photo (Figure 9) showing the time-average position of the 2-dimensional motion encompassing the entire space which contains both the intermittent snag and the wider attractor. If one can imagine the dynamics behind this picture one gets a good idea of what a crisis-induced intermittency looks like in more than one dimension.
Figure 9: A 2-d snag and the wider attractor; intermittent behaviour