next up previous
Next: Traps in transient Up: TRAPS IN CHAOTIC SYSTEMS Previous: Two-dimensional trapping

Large dimensional traps

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.



D Jefferies
Wed May 15 22:18:30 BST 1996