THE MOVING SINGULARITIES OF THE PERTURBATION EXPANSION OF THE CLASSICAL KEPLER-PROBLEM

The convergence properties of the perturbation expansion for the perio dic solution of the classical Kepler problem are studied as a function of the perturbation parameter a, corresponding to ''linear'' amplitud e. We find the convergence to be directly affected by an infinite numb er of movable singularities in the complex a-plane. The singularities occur at locations in the complex plane where the differential equatio n under consideration is singular. These singularities explain the non uniform convergence of the perturbation series that Melvin [SIAM J. Ap pl. Math., 33 (1977), pp. 161-194] noted. They occur in complex conjug ate pairs and move in the complex plane as a function of the independe nt variable, causing divergence of the perturbation series solution. O ur analysis near one of these singularities indicates that distinct br anches of solutions occur there. These solutions undergo transition at the singularity to develop into new solution branches with distinctiv ely different properties. Moving singularities similar to those discus sed here were also shown to affect the convergence of the perturbation expansion for the limit cycle of van der Pol's equation [SIAM J. Appl . Math., 44 (1984), pp. 881-895; SIAM J. Appl. Math., 50 (1990), pp. 1 764-1779]. The recurrence of this phenomenon in the present simpler pr oblem suggests that it is very likely to affect the convergence of per turbation expansions for periodic solutions of other problems.