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 autonomous or periodic system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not guarantee that the solution can be continued, are considered. Three approaches are proposed to solving the problem: (a) particular consideration of terms that depend on the small parameter and the selection of generating 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 fractional power of the small parameter. The theoretical results are used to prove the existence of two families of periodic three-dimensional orbits in the N-planet problem. The orbit of each planet is nearly elliptical and situated in the neighbourhood of its fixed plane; the angle between the planes is arbitrary. The average motions of the planets in these orbits relate to one another as natural numbers (the resonance property), and at instants of time that are multiples of the half-period the planes are either aligned in a straight line—the line of nodes (the first family), or cross the same fixed plane (the second family). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbit are independent.