Ising model on the Cayley tree with competing interactions up to the third Nearest-neighbors with spins belonging to different branches of the tree

We study the phase diagram of the Ising model on the Cayley tree with competing interactions up to the third nearest-neighbors with spins belonging to different branches of the tree. In addition to the expected ferromagnetic, anti-ferromagnetic and paramagnetic phases, we present a new paramodulated phase. Moreover, the transition lines are analyzed and they are in agreement with the lines obtained numerically. Lastly, the stability of the anti-ferromagnetic phase was studied in detail by investigating the Lyapunov exponent associated with the corresponding dynamical system.