General Melnikov Approach to Implicit ODE’s

Abstract

Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed differential equation has a solution of the same kind. By a suitable, nonlinear, change of coordinates these kind of solutions are associated to homoclinic solutions to a fixed point of an ordinary differential equation with a one-dimensional centre manifold. Then we obtain a Melnikov condition for the persistence of homoclinic orbits which is simpler than the one obtained in Battelli and Feckan (J Differ Equ 256:1157–1190, 2014). This difference is due to the fact that this method does not distinguish solutions according to the rate of convergence to the fixed point and then another assumption on the perturbation term is needed.