# A 2-variable stochastic difference equation exhibiting 2 stable orbits with switching?

I have some social science data to which I would like to fit a stochastic difference equation in two variables. I will describe the dynamics of the system that I have observed. I am hoping someone can suggest to me one or more families of models which can show the sort of dynamics I have observed, or perhaps point me toward a “bestiary” of dynamic behaviors from which I might construct such a models. I have software that I believe can do the fitting if given a model with the right structure. But I have not found a two-variable difference equation structure that has the correct dynamics.

Basically I am looking at a system that has two fairly stable regimes, well separated from each other in a two-dimensional phase space. Each regime can be described as an orbit around its own central point. For the vast majority of the time the system is in one of these two orbits. The orbits are non-overlapping. One of the orbits is larger and somewhat less regular than the other. The orbits do not appear to converge to or diverge from their respective centers.

I have observed several regime switches. The switch from the larger orbit to the smaller may be possible to explain as the combination of an unusually large swing and a well-placed random shock toward the smaller basin of attraction, followed by capture There may also be extra-model forces involved. I am exploring that.

The switch from the smaller to the larger orbit has some puzzling features that have me stymied, though the same thing occurs each time. During a regular orbit, consecutive periods usually flip to somewhere on the far side of the center – as if it were a planetary orbit observed semi-annually. In the state transitions, the orbit seems to be temporarily arrested, remaining in a small area for several periods. It then seems to leap to the far side, traveling much farther than usual – roughly double the regular orbit diameter. In all the cases I have observed, the leap has been toward the half-plane where the other orbit lies. (So I don't know what happens if the leap is in the opposite direction). The center of the larger orbit is six or seven times the small orbit diameter from the small orbit center. Over the next seven or eight periods the state transitions toward the center of the larger orbit, even though it is closer to the smaller orbit center for first several periods. Then it takes up the larger orbit, apparently in a stable manner, again not overlapping with the smaller orbit.

Actually, I would be grateful for a model which can exhibit either of the behaviors I have observed: + The first behavior is rare regime switching between two widely separated stable orbits. + The second behavior is a generally stable orbit that also displays the hesitate-and-leap phenomena that I have described – as if the usual orbit is caused by a force toward the attractive center, but that force is accumulated if the state is prevented from moving (as perhaps by a random shock).

• Am I getting no responses (so far) because this question is hard, because it is stupid, or because it is complicated? Would I be more likely to get a response if I took it apart into several questions? – andrewH Jul 22 '15 at 16:45