A previous paper [2] introduced the behaviour of simple low-dimensional non-linear mappings under iteration. The mappings are designed to contain small Features which we called traps [1] or snags [2]. The trapping behaviour was discussed and reported [1] theoretically and by simulation in our paper at COMPLEX96. Here, we present the supporting experiments.
The Features give rise to the following behaviours: If the state variable enters a trap, the chaotic time series ceases, and the system remains at a fixed point. Thus a trap is an attracting fixed point of the system which is the end point of a chaotic transient; this can, in principle, last an arbitrarily long time. Normally, one expects some kind of progressive approach to a fixed point. In our systems, on the other hand, trapping is sudden, and happens without warning.
A snag consists of a chaotic attractor embedded in the larger chaotic environment. In the one-dimensional case considered in the earlier paper [2] , there is a small probability of entering or leaving the snag from the larger domain. Thus the motion appears always chaotic, but ``bursty'' with clearly visible (Figure 5) differences between the bounds of the two competing attractors. Again, the transitions are sudden and happen without warning. In that paper, we attributed the bursty behaviour to the interaction of the added noise in the simulation with the features. Below, we show clear experimental evidence of true intermittency brought about by the overlap of the chaotic attractors. In the two-dimesional experiments presented here for the first time, it is not so easy to picture the structure of the 2-d features and so the experiments provide a valuable insight into the dynamical possibilities.
The term ``crisis'' refers to a point in the gradual change of a parameter which causes the snag to grow or move until its structure extends beyond the feature, and ejection (Figure 8) into the wider chaotic attractor becomes possible.