Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions |
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| Authors: | Antonis D Pinotsis |
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| Institution: | 1.Section of Astrophysics, Astronomy and Mechanics, Department of Physics,University of Athens,Athens,Greece |
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| Abstract: | We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem.
We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities.
We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation
pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating
point in x- at which the Jacobi constant is C
∞ = 3 and their periodic orbits terminate at the corotation (at the Lagrangian point L4 or L5). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa
occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum
sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite
spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite
sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate
large chaotic regions near the corotation points L4, L5, which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal
point, infinite loops, one inside the other, surrounding the unstable triangular points L4 or L5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals
into the corotation points L4, L5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us
to comprehend better the motions around the points L4 and L5 of Lagrange. |
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