On the dynamic unfolding of a saddle-center bifurcation and the change in the action

Conservative Hamiltonian systems with slow variations are considered in the case of a dynamic saddle-center bifurcation. Using the method of averaging, action is an adiabatic invariant before and after the slow passage of the homoclinic orbit. The bifurcation is unfolded by assuming that the time that the method of averaging predicts that the homoclinic orbit is crossed is near to the time of the saddle-center bifurcation. The slow passage through homoclinic orbits associated with the unfolding of a saddle-center bifurcation is analyzed and the change in the adiabatic invariant is computed.