Two Potts Models on Cayley Tree of Arbitrary Order

On a Cayley tree of arbitrary order k we consider two different Potts models with competing nearest-neighbour interactions J₁ and next-nearest-neighbour interactions J_p and J_o, where coupling J_p corresponds to interaction of spins belonging to the same branch of the tree (prolonged) and coupling J_o corresponds to interaction of spins belonging to the same shell of the tree (one-level) and find for each model in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of coupling values where the local magnetization has chaotic oscillatory glass-like behaviour. We also show that the ranges corresponding two models are different.