Abstract
Melnikov's method is a well-established technique for detecting homoclinic bifurcation of perturbed autonomous or forced systems. This method uses a regular perturbation expansion in terms of a small parameter in the system. Whilst the approach correctly estimates the parameter values for the bifurcation and transverse intersections of separatrices and manifolds, it does not correctly represent solutions near the associated fixed point of the homoclinic orbit. For the autonomous case, a multiple scales method using matched inner and outer solutions is developed in this exposition which corrects this deficiency, whilst still confirming Melnikov's result to leading order.
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