Reply to the commentary written by M. Zurru on the paper “Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification”, by Arthur Givois, Jin-Jack Tan, Cyril Touzé and Olivier Thomas, http://doi.org/10.1007/s11012-020-01132-2

The commentary sent by M. Zurru on our paper is perfectly right and of course we were aware of that point. The fact that the internal force does not derive from a potential when \(C_1 \ne C_2\) was indeed undertoned in our introductory remarks. Also, in the identification procedure from the experiments, the two options had been investigated and commented. In the first option, the four coefficients are let free and each of them is identified from specific points of the solution curves. In the second option, one enforces equal values for the four coefficients, see Table 2. We agree that even in the case of an imperfect structure, the model needs to refer to a conservative field force, enforcing \(C_1 = C_2\). We also note that our results can be used in very different contexts, since such a mathematical model of coupled oscillator is not necessarily connected to a mechanical problem. Finally, we point out that our recent experience shows that the main assumption that needs to be taken into account is a full normal form with eight coefficients as starting point. In this case, the typology of the solutions drastically changes, mostly because of the appearance of resonant invariant-breaking monomial terms. …