This paper is part three in a series on the dynamics of two coupled, p
arametrically driven pendulums, In the previous parts Banning and van
der Weele (1995) and Banning et al. (1997) studied the case of linear
coupling; the present paper deals with the changes brought on by the i
nclusion of a nonlinear (third-order) term in the coupling. Special at
tention will be given to the phenomenon of mode competition. The nonli
near coupling is seen to introduce a new kind of threshold into the sy
stem, namely a lower limit to the frequency at which certain motions c
an exist. Another consequence is that the mode interaction between 1 a
lpha and 2 beta (two of the normal motions of the system) is less dege
nerate, causing the intermediary mixed motion known as MP to manifest
itself more strongly.