Competing binary and k-tuple interactions on a Cayley tree of arbitrary order

We study the phase diagram for the Ising model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J1, prolonged next-nearest-neighbor interactions Jp, and one-level k-tuple neighbor interaction Jo. The phase diagram is studied for several ranges of the competing parameters; it shows the appearance of several features and modulated phases arising from the frustration effects introduced by the one-level k-tuple neighbor interaction Jo. The variation of the wavevector with temperature in the modulate phase is studied in detail; it shows narrow commensurate steps between incommensurate regions. Finally, the Lyapunov exponent associated with the trajectory of the system is investigated.