Synchronization of oscillators arising from second-order, and higher, nonlinear couplings

We explore a system of two coupled nonlinear pendula, with the lowest-order coupling term being quadratic, rather than the more typical linear coupling term. We show that synchronization can occur in this system, but with the synchronization frequency lying below either of the original independent frequencies, in contrast to the usual case where the synchronization frequency is between the two frequencies of the independent pendula. Both phase synchronization and amplitude synchronization are found, and one can have situations, for example, where phase synchronization exists, but not amplitude synchronization. We develop analytic formulas to understand the frequency synchronization.