Bistability, Basins of Attraction, and Predictability in a Forced Mass-Reaction Model

摘要

Bistable phenomena can occur in many physical, chemical, and biological models of natural phenomena. An important subset of problems exhibiting bistability consists of those models employing mass-reaction kinetics. Crucial to understanding any mass-reaction model is a knowledge of the parameter corresponding to the rate of contact between two or more species. In certain applications, the contact rate may be time dependent, and in fact, periodic. For example, the process of temporally increasing and decreasing the solar intensity respectively changes the probability of contact between two reacting species in the atmosphere. In addition to perturbing reactants in the atmosphere, periodic forcing of contact rates plays an important role in modelling recurrent epidemic outbreaks. It is important to note that both physical and biological phenomena exhibit oscillations which are longer than the forcing period, or not periodic at all. Furthermore, it is not uncommon for periodically forced differential equation models to exhibit two or more stable subharmonic solutions for a given set of parameters. The question we consider here is, how well can one predict the asymptotic final state given initial conditions having finite precision for a problem that exhibits two different stable periodic orbits. In particular, this document considers a simple mass-reaction model with a periodically forced contact rate. The following is shown numerically: 1) There exist parameter values for which the model exhibits at least two distinct stable subharmonic periodic orbits; 2) The basins of attraction of each orbit can be very complicated, thus affecting final state predictability as a function of precision in initial conditions.