Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

We consider the Vannimenus model on a Cayley tree of arbitrary order k with competing nearest-neighbour interactions J(1) and next-nearest-neighbour interactions J(2) and J(3) in the presence of an external magnetic field h. In this paper we study the phase diagram of the model using an iterative scheme for a renormalized effective nearest-neighbour coupling K, and effective field per site X, for spins on the rth level; it recovers, as particular cases, previous works by Vannimenus, Inawashiro et al, Mariz et al and Ganikhodjaev and Uguz. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second- or first- order phase transitions. We present a few typical attractors and at finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into few disconnected pieces, etc) are exhibited for typical values of parameters.