The coupled nonlinear oscillating systems studied do not possess linear normal modes. The nonlinearly related modes of

m_{1}ẍ_{1} + k = 1n a_{k}x_{1}^{k} + k = 1n A_{k}(x_{1} − x_{2})^{k} = 0m_{2}ẍ_{2} + k = 1n b_{k}x_{2}^{k} − k = 1n A_{k}(x_{1} − x_{2})^{k} = 0 (1)

with k = 1, 3, 5, . . . . . .

are expressed as superpositions of the linear normal modes of each of the homogeneous systems of the form

m_{1}ẍ_{1} + a_{k}x_{1}^{k} + A_{k}(x_{1} − x_{2})^{k} = 0m_{2}ẍ_{2} + b_{k}x_{2}^{k} − A_{k}(x_{1} − x_{2})^{k} = 0 (2)

whose force fields are superimposed to yield the force field of equations (1).