A method for studying stability domains in physical models

We present a method for investigating the simultaneous movement of all zeros of equations of motions defined by discrete mappings. The method is used to show that knowledge of the interplay of all zeros is of fundamental importance for establishing periodicities and relative stability properties of the various possible physical solutions. The method is also used (i) to show that the Frontière set of Fatou is defined primarily by zeros of functions leading to an entire invariant limiting function which underlies every dynamical system, (ii) to identify cyclotomic polynomials as components of the limiting function obtained for a parameter value supporting a particular superstable orbit of the quadratic map, (iii) to describe highly symmetric periodic cycles embedded in these components, and (iv) to provide an unified picture about which mathematical objects form basin boundaries of dynamical systems in general: the closure of all zeros not belonging to “stable” orbits.