Coupled nonlinear oscillators: dynamics with a period matching the forcing one

We consider rather generic coupled nonlinear damped oscillators that are in some of their specifications frequently encountered in modeling mechanical and biological systems, subject to periodic external forcing. Combining a global co-bifurcation result based on topological degree theory with “local” properties, we establish sufficient conditions granting the emergence from equilibrium of nontrivial periodic solutions with a period exactly matching that of the forcing term. Also, under appropriate but still general structural assumptions, we derive quantitative estimates for the amplitude of such solutions. Recourse to topological degree theory allows us to leave aside explicit expressions of the nonlinear coupling. Particular choices of it have, indeed, different effects, which emerge, for example, by looking at the modulation of energy transfer across the system, as we show numerically, in order to clarify the matter. These aspects, however, do not avoid the possibility of a global bifurcation, as it is clear from the proof.