Families of Bounded Solutions Near Perturbed Homoclinics of R.D.E.s with Symmetric Eigenvalues, Application to Retarded Power-Law Oscillations

We develop methods giving sufficient conditions for the existence of bounded solutions near homoclinic orbits of planar R.D.E.s. We provide families of bounded solutions near the homoclinic by decomposing the corresponding variational equation of the system on the space of solutions. We apply the results for controlling a power-law oscillatory system with a retarded feedback control. Our results are numerically verifiable and do not meet the restrictions of classic methods (such as Melnikov’s integrals and Poincare’s map).