Slow passage through homoclinic orbits for the unfolding of a saddle-center bifurcation and the change in the adiabatic invariant

Slowly varying, conservative, one degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation. At the bifurcation, a homoclinic orbit connects to a nonhyperbolic saddle point. Using averaging for strongly nonlinear oscillations, action is an adiabatic invariant before and after the slow passage of the homoclinic orbit. The homoclinic orbit is assumed to be crossed near to its creation in the saddle-center bifurcation, a dynamic unfolding. A large sequence of nearly homoclinic orbits with autonomous saddle approaches is matched to the strongly nonlinear oscillations valid before and after. Connection formulas are computed, determining the change in the action due to the slow passage through the unfolding of the saddle-center bifurcation. If the energy in one specific saddle region is particularly small, as occurs near the boundaries of the basin of attraction, then the solution in only that saddle region satisfies the nonautonomous Painlevé I.