Although the complexity of the electronics precludes experiments in large numbers of dimensions, it becomes clear that if this system is generalised to increasing numbers of dimensions, the probablity of trapping will decrease rapidly with the dimension. Therefore, reasonably large traps (large per dimension) can be introduced, compared to the noise level, and one can in principle construct an electronic circuit which will have a very large mean time to trapping, and a negligibly small chance of being released from the trap by noise. Alternatively, in a noisy real-world environment, one can envisage sizeable traps larger than the noise which nevertheless are entered with low probability.
Suppose one calls the trap size, with respect to
the range of the variable,
. Typically in our
simulations and experiments
was of the order
of 0.001. In an N- dimensional system the probability
of trapping, per iteration, will be of the order
. Thus in our system of examples in figures 2
and 5 where it is possible to have
iterations
to trapping, in a 10-dimensional system we would
find experiments taking
iterations to trapping.
Thus in the real world examples, considering the global behaviour of
high dimensional systems which are intrinsically
quite noisy, traps might be seen only rarely.
However if they do exist; as we have seen they may be
entered suddenly and without warning; and therefore
sudden naturally occurring events (extinctions?)
may be a consequence of the natural dynamics and
have no prime cause other than the nature of the dynamics.