Linear Fractionals - Simple Models with Chaotic-like Behavior

摘要

What are the simplest systems which display chaos? We consider linear fractionals which are the ratio of two linear functions. While these systems often behave like linear systems, we show that for some parameter values these systems are chaotic-like displaying sensitive dependence on initial conditions, visiting all open sets, and having a non-attractive invariant density. Because these systems do not have cycles of every period they are not chaotic in the sense of Devaney. For other parameter values, these systems are periodic, but if the parameters are rationals, the only possible periods are 1,2,3,4, and 6. The period 2 linear fractionals can be used to prove global stability for many of the usual population dynamics models. Because of their simplicity, we are able to give a complete characterization of which linear fractional have which behavior.