Multiple transverse homoclinic solutions near a degenerate homoclinic orbit

Consider an autonomous ordinary differential equation in RnRn that has a homoclinic solution asymptotic to a hyperbolic equilibrium. The homoclinic solution is degenerate in the sense that the linear variational equation has 2 bounded, linearly independent solutions. We study bifurcation of the homoclinic solution under periodic perturbations. Using exponential dichotomies and Lyapunov–Schmidt reduction, we obtain general conditions under which the perturbed system can have transverse homoclinic solutions and nearby periodic or chaotic solutions.