Transverse bounded solutions to saddle–centers in periodically perturbed ordinary differential equations

Ordinary differential equations are considered consisting of two equations with nonlinear coupling where the linear parts of the two equations have equilibria which are, respectively, a saddle and a center. Perturbation terms are added which correspond to damping and forcing. A reduced equation is obtained from the hyperbolic equation by setting to zero the variable from the center equation with a homoclinic structure. The center equation is resonant in the equilibrium. Melnikov theory is used to obtain a transverse bounded solution of the whole equation. The techniques make use of exponential dichotomies and an averaging procedure.