On the Periodicity of the Rational Dynamical System Corresponding to the Vannimenus–Ising Model

Abstract

The universal behaviors of a rational dynamical system associated with the Vannimenus–Ising model having two coupling constants on a Cayley tree of order three are studied. Cobweb diagrams and related map iterates for some relevant parameters are investigated. The local stability of fixed points is discussed and illustrated through cobweb diagrams. We deal with quantitative universality, such as orbit diagrams and Lyapunov exponents for a class of rational maps. We show that our model is periodic using orbit diagrams and relevant Lyapunov exponents.