Homoclinic Bifurcations with Nonhyperbolic Equilibria

A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong λ-lemma, and Lyapunov–Schmidt reduction for the Poincaré maps under Sil’nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.