ABSTRACT
Melnikov’s Theorem, gives a criterion for the occurrence of the homoclinic tangle in dynamical systems of a certain class, and hence for the occurrence of chaos. The theorem is applied to the basic ideal resonance problem, as subject to a very simple perturbation, as an indication of the implications of the theorem in Celestial Mechanics. Melnikov’s theorem relates to dynamical systems which can be expressed as resulting from perturbations of an integrable dynamical system. The chapter is concerned with the case in which the integrable system is of Hamiltonian form. It considers perturbed orbital motion, in which two orbiting bodies (planets or satellites), have a ratio of orbital periods close to the ratio of two small integers. Evidence of chaos is sometimes found in association with such cases. Examples of such pairs occur, for example, within the satellite system of Saturn, as well as relating some minor planets to Jupiter.
