CONTINUATION OF PERIODIC MOTIONS OF A REVERSIBLE SYSTEM IN NON-STRUCTURALLY STABLE CASES - APPLICATION TO THE N-PLANET PROBLEM

The problem of continuing symmetric periodic solutions of an autonomou s or periodic reversible system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not gu arantee that the solution can be continued, are considered. Three appr oaches are proposed to solving the problem: (a) particular considerati on of terms that depend on the small parameter and the selection of ge nerating solutions; (b) the selection of a generating system depending on the small parameter; (c) reduction to a quasi-linear system which is then analysed using the first approach. Within the framework of the third approach the existence of a periodic motion is also established that differs from the generating one by a quantity whose order is a f ractional power of the small parameter. The theoretical results are us ed to prove the existence of two families of periodic three-dimensiona l orbits in the N-planet problem. The orbit of each planet is nearly e lliptical and situated in the neighbourhood of its fixed plane; the an gle between the planes is arbitrary. The average motions of the planet s in these orbits relate to one another as natural numbers (the resona nce property), and at instants of time that are multiples of the half- period the planets are either aligned in a straight line-the line of n odes (the first family), or cross the same fixed plane (the second fam ily). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbits are independent. (C) 1998 Elsevie r Science Ltd. All rights reserved.