Transverse bounded solutions to saddle-centers in periodically perturbed ordinary differential equations
A periodically perturbed system of ordinary differential equation is 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 included in the system correspond to damping and forcing. It is assumed that (1) the reduced equation which is obtained from the hyperbolic equation by setting to zero the variable from the center equation has a homoclinic solution; (2) the center equation is resonant in the equilibrium. Under certain additional assumptions, the Melnikov theory is used to prove the existence of a transverse bounded solution of the whole system. The techniques make use of exponential dichotomies and an averaging procedure.
scientific article; zbMATH DE number 5152791