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

► We clarify the role of order k of the Cayley tree and one-level k-tuple neighbor interaction. ► It will be seen that the role of k is rather significant. ► The study of the variation of the wavevector with temperature in the modulated phase is presented. ► The Lyapunov exponent associated with the trajectory of the system is analyzed.