Slow Passage through a Saddle-Center Bifurcation

Slowly varying conservative one-degree of freedom Hamiltonian systems are analyzed in the case of a saddle-center bifurcation. Away from unperturbed homoclinic orbits, strongly nonlinear oscillations are obtained using the method of averaging. A long sequence of nearly homoclinic orbits is matched to the strongly nonlinear oscillations before and after the slow passage. Usually solutions pass through the separatrix associated with the double homoclinic orbit before the saddle-center bifurcation occurs. For the case of slow passage through the special homoclinic orbit associated with a saddle-center bifurcation, solutions consist of a large sequence of nearly saddle-center homoclinic orbits connecting autonomous nonlinear saddle approaches, and the change in the action is computed. However, if the energy at one specific saddle approach is particularly small, then that one nonlinear saddle approach instead satisfies the nonautonomous first Painlevé transcendent, in which case the solution usually either passes through the saddle-center homoclinic orbit in nearly the same way or makes a transition backwards in time to small oscillations around the stable center (though it is possible that the solution approaches the unstable saddle).