Variable Time Step Dynamics with Choice

We are interested in systems that can and possibly do switch between different regimes; the durations of staying in each regime (dwell times) may vary. We work with the deterministic picture and study global attractors both in the state space and, one of the novelties, in the hyperspace. We give an example of a simple ODE system where the existence or non-existence of attractors depends on the intervals from which the dwell times are chosen. Another novelty is the description of restricted iterated function systems and their attractors. An unexpected corollary is that for the restricted to a subshift hyperbolic IFS, its attractor is the image under Hutchinson’s map \(p\) of a different, in general, subshift (we call it dual to the original one).