We generalize the phase-plane method approach introduced in  to the case of higher dimensions. To this aim, the phase-space is assumed to be decomposable as a product of planes, where the corresponding components of the solutions can be controlled by means of suitable plane curves. We then apply our general result to the periodic problem associated to a system of coupled oscillators, with retraction forces having a linear growth, or with one-sided superlinear nonlinearities.
"Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces." Differential Integral Equations 25 (11/12) 993 - 1010, November/December 2012.