Bifurcations and Instabilities in the Restricted Three-Body Problem

Pinotsis, A. D.

doi:10.1007/978-94-009-3053-7_42

Cited by 7 publications

(6 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As C decreases the orbits approach more and more the peripherals 2 and 7 and move away from peripheral 1, while the curving smoothens. We should note that the orbits 19a, 20a have a form similar to some of the classes f, b of the restricted 3-body problem (Szebehely 1967;Pinotsis 1988); -in Fig. 21a we give two orbits.…”

Section: Orbits Among the Peripheralsmentioning

confidence: 84%

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

Pinotsis

2005

A&A

Self Cite

View full text Add to dashboard Cite

Abstract. We theoretically investigate the existence of families of periodic orbits in the planar N-body ring problem and we give a qualitative picture of the motion of a small particle. This study yields four families of periodic orbits which we also found numerically: two families of periodic orbits around the central body and two families around all the peripherals. These results are valid for different values of N and β. Also we investigate the evolution of simple periodic motions as well as their stability. We found stable and unstable orbits around the central body, around all the peripherals and around one or more peripherals which form rings of stability. Some families present other types of bifurcations, such as bifurcations of families of non-symmetric periodic orbits of the same period and period-doubling bifurcations.

show abstract

Section: Orbits Among the Peripheralsmentioning

confidence: 84%

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

Pinotsis

2005

A&A

Self Cite

View full text Add to dashboard Cite

show abstract

“…The well-known Feigenbaum sequence of infinite period doubling bifurcations is a particular route to chaos in dissipative and conservative dynamical systems (Feigenbaum 1978;Coullet and Tresser, 1978). Some other routes for the transition from order to chaos in conservative dynamical systems of 2-degrees of freedom have been found by Contopoulos and Zikides (1980), Contopoulos (1983aContopoulos ( ,b, 1991, Heggie (1983), Bier and Bountis (1984) and Pinotsis (1988). These scenarios of routes to chaos are mentioned by Contopoulos (1993Contopoulos ( , 2002.…”

Section: Introductionmentioning

confidence: 93%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…We made a preliminary study for the non-symmetric short period orbits (SPO) of the family b (Pinotsis 1988) and we found infinite spiral characteristics around infinite focal points. Also, spiral characteristics of families of periodic solutions have been found in the case of a satellite oscillations (Bruno, A.D. and Varin, V.P., 1997;Bruno, A.D., 2002 ).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Pinotsis

2010

Celest Mech Dyn Astr

View full text Add to dashboard Cite

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 L 4 or L 5 ). 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 L 4 , L 5 , 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 L 4 or L 5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L 4 , L 5 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 L 4 and L 5 of Lagrange.

show abstract

“…But if we also add the non-symmetric orbits we have evidence that there is an infinity of spiral characteristics and of homoclinic or heteroclinic orbits (Pinotsis, 1988). These heteroclinic orbits are limits of families of periodic orbits (Henrard, 1973).…”

Section: Introductionmentioning

confidence: 99%

The generation of spiral characteristics

Contopoulos

1991

Celestial Mech Dyn Astr

View full text Add to dashboard Cite

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.

show abstract

Bifurcations and Instabilities in the Restricted Three-Body Problem

Cited by 7 publications

References 1 publication

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

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

The generation of spiral characteristics

Contact Info

Product

Resources

About