Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

Pinotsis, A. D.

doi:10.1016/j.pss.2009.06.028

Cited by 3 publications

(6 citation statements)

References 9 publications

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“…Each main family of the above three cases and each bifurcating families of various multiplicities presents the Feigenbaum scenario of infinite period doubling pitchfork bifurcations and each bifurcation ratio tends to approach the universal number δ 8.72 at a finite value of the Jacobian constant, which we have already found for the families of symmetric periodic orbits (Contopoulos and Pinotsis 1984, Pinotsis 1986, 1987.…”

Section: Infinite Sets Of Spirals and Self-similar Orbitssupporting

confidence: 54%

“…For the main families of simple non-symmetric periodic orbits fb1, fl2, fc1, and fz 1 that bifurcate at the points b1, l2, c1, (Hénon 1965) and z 1 (Pinotsis 1986) respectively, we found the evolution and their stability. In their first bifurcation point these families present a similar bifurcation structure.…”

Section: Discussionmentioning

confidence: 92%

“…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 ). In this paper we make a systematic study of the families of non-symmetric periodic orbits that bifurcate at the points b1, c1, l2, n 2 4 (Hénon 1965) and z 1 (Pinotsis 1986), in the planar circular restricted 3-body problem and in the case of equal masses of the primaries. We found interesting information about the evolution and termination of the bifurcating families of various multiplicities of non-symmetric periodic orbits.…”

Section: Introductionmentioning

confidence: 93%

“…Bifurcated families have in general large unstable regions and generate large chaotic regions (Contopoulos and Pinotsis 1984, Pinotsis 1986, 1987.…”

Section: Introductionmentioning

confidence: 99%

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

Pinotsis

2010

Celest Mech Dyn Astr

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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.

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Section: Infinite Sets Of Spirals and Self-similar Orbitssupporting

confidence: 54%

Section: Discussionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 93%

“…Bifurcated families have in general large unstable regions and generate large chaotic regions (Contopoulos and Pinotsis 1984, Pinotsis 1986, 1987.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Pinotsis

2010

Celest Mech Dyn Astr

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“…[25] has a study of the phase space for solutions near L 4 and [8] treats the elliptic case where one has four periods for solutions close to L 4 . The stability of the orbits close to L 4 is studied in [7] and the connection from E 3 to L 4 is explored in [23]. A very complete numerical study, [6], using AUTO, shows the many different types of periodic orbits and the connections between the Lagrange points and also the secondary bifurcations along the curves in the x, y, µ space, where µ is the mass of one of the primaries.…”

Section: Introductionmentioning

confidence: 99%

Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem

García-Azpeitia¹,

Ize²

2011

Celest Mech Dyn Astr

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Homoclinic dynamics in a spatial restricted four body problem blue skies into Smale horseshoes for vertical Lyapunov families

Murray,

Mireles-James

2020

Preprint

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The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admit certain simple homoclinic oribts which form the skeleton of the complete homoclinic intersection -or homoclinic web. In the present work the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. KeywordsGravitational 4-body problem • blue sky catastrophes • Smale tangle • vertical Lyapunov families • invariant manifolds • boundary value problems

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Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

Cited by 3 publications

References 9 publications

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

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

Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem

Homoclinic dynamics in a spatial restricted four body problem blue skies into Smale horseshoes for vertical Lyapunov families

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