In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits bifurcations. In this method, one employs the Melnikov distance function to measure the separation of stable and unstable manifolds.
Based on the above, is it correct to write that the bifurcation values found through this method are only an approximation since the outlined approach is a perturbative method?
Yes, the Melnikov distance function is an approximation based on order $\epsilon $ perturbation calculations.