Existence of a Center Manifold in a Practical Domain around L1 in the Restricted Three-Body Problem

We present a method of proving existence of center manifolds within specified domains. The method is based on a combination of topological tools, normal forms, and rigorous computer-assisted com- putations. We apply our method to obtain a proof of a center manifold in an explicit region around the equilibrium point L1 in the Earth-Sun planar restricted circular three-body problem. 1. Introduction. In this paper we give a method for proving existence of center manifolds for systems with an integral of motion. The aim of the paper is not to give yet another proof of the center manifold theorem, but to provide a practical tool which can be applied to nontrivial systems. There are a number of advantages to the method. First, the method is not perturbative. We thus do not need to start with an invariant manifold and then perturb it. All that is required is a good numerical approximation of the position of a center manifold. The conditions required in order to ensure existence of the manifold in the vicinity of the numerical approximation are such that it is possible to verify them using (rigorous, interval-based) computer-assisted computations. This is another advantage, since it allows for application to problems which cannot be treated analytically. The method gives explicit bounds on the position and on the size of the manifold. Moreover, under appropriate assumptions we can also prove that the manifold is unique. Our proof of existence of the center manifold is performed using purely topological ar- guments. This means that it can be applied to treat nonanalytic invariant manifolds. The main disadvantage of using topological tools, though, is that the proof ensures only Lipschitz continuity of the manifold even for manifolds with higher order regularity.