The generation of spiral characteristics

Abstract

When the Lagrangian points L4, L5 in a rotating dynamical system become unstable (at a critical perturbation e = el) the characteristics of some families of orbits bifurcating from the short and long period orbits (SPO and LPO) become spiral. For a given e, slightly larger than et, an infinity of families of multiplicities n, n + 1, n + 2 .... do not bifurcate any more from SPO or LPO but join each other into a spiral. As e increases this spiral is joined by lower multiplicity families, until the SPO-LPO family itself joins the spiral. Further spirals with the same or different focuses are formed by joining other sequences of families of order n, n + 1, n + 2,..., or n, n + 2, n + 4,.... Other spirals are generated at particular values of E, starting and terminating at the same focus or two different focuses. As the perturbation e increases such spiral characteristics join other families, away from the focus. The orbits along the spiral characteristics have an increasing number of loops (either n, n + 1, n + 2 ..... or n, n + 2, n + 4... ) around, or close to L 4. In the first case the loops are along the symmetry axis, passing through L4. In the second case the loops appear in pairs outside the symmetry axis. The focuses correspond to homoclinic, or heteroclinic orbits, spiralling around/-,4 and/or L 5.

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

“…The values of x (for each branch of a family) were calculated as the average of the extreme -closest to the focal point-values of the corresponding final spiral, which was found numerically. Contopoulos (1991) investigated periodic orbits of a galactic type potential and explained how the spirals are generated as soon as the Langrangian points L 4 , L 5 become unstable. Figure 2 shows a set of some simple periodic orbits.…”

“…The stability of motions in the neighborhood of the Lagrangian points L 4 and L5 has been investigated for some values of the mass ratio μ by a number of investigators (Pinotsis 1988, Contopoulos 1991, Sandor et. al.…”

Systems with Escapes

“…The escape regions E(O i , n) (i = 1, 2, n = 0, 1) in the Hamiltonian (1) for ω 1 = ω 2 = 1, ε = 1, and h = 0.15. 24 the areas on a surface of section are not conserved, but also the volumes in phase space are not conserved.…”

“…23 More recently we have attacked the problem of escapes in more detail. 24 We have separated the basins of escape into independent zones according to the time required for escape. Most of these zones have a spiral form, which we could explain theoretically.…”

“…29 More recently we have studied in detail the asymptotic curves from the unstable periodic orbits and their homoclinic and heteroclinic intersections. 30,31 The corresponding orbits are doubly asymptotic orbits, asymptotically approaching the various unstable periodic orbits.…”