The rectilinear three-body problem using symbol sequence I. Role of triple collision

Saito, Masaya; Tanikawa, Kiyotaka

doi:10.1007/s10569-007-9070-0

Cited by 13 publications

(9 citation statements)

References 13 publications

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“…The same is true with their mirrored intersectable counterparts MR N fp −i and MR N fp −i+1 . These properties are assured by the following facts studied in Saito and Tanikawa (2007) and McGehee (1974): on the TCM smoothly connect with each other. Therefore, foot-points are symmetrically aligned with respect to θ = 90 • on the θ -axis.…”

Section: Applicability Of Results: Cylinder Of Persistent Archesmentioning

confidence: 88%

“…This problem, called the rectilinear three-body problem, has been extensively studied (Mikkola and Hietarinta 1989;Hietarinta and Mikkola 1993;Tanikawa and Mikkola 2000a;Tanikawa and Mikkola 2000b) and recent progress is reviewed in Orlov et al (2009). The present paper is the third of our works on this system (Saito and Tanikawa 2007;Saito and Tanikawa 2009) referred to as Papers I and II). We have been interested in the behaviour of periodic orbits when the mass parameter is changed.…”

Section: Introductionmentioning

confidence: 93%

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Non-schubart periodic orbits in the rectilinear three-body problem

Saito

Tanikawa

2010

Celest Mech Dyn Astr

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In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and end at triple collision. This may give us some insight into the nature of periodic orbits in the N -body problem.

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Section: Applicability Of Results: Cylinder Of Persistent Archesmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 93%

Non-schubart periodic orbits in the rectilinear three-body problem

Saito

Tanikawa

2010

Celest Mech Dyn Astr

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“…Then col i (S G JB) = S G Jc i where c i is the i th column of B = Y (π/4). This and Equation (11) imply that the (i, j) entry of W is then c T i S G Jc j . But Equation (9) implies that the (6, 6) entry of W is the (2, 2) entry of K. Continuing in this manner we find the remaining entries of the lower right 3 × 3 submatrix of K to be given as prescribed.…”

Section: 2mentioning

confidence: 96%

Linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

Bakker

Mancuso

Simmons

2012

Journal of Mathematical Analysis and Applications

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We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass m ∈ (0, 1] as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a 3 × 3 matrix for each m ∈ (0, 1] obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over (0, 1], with linear instability for m close or equal to 0.01, and linear stability for m close or equal to 1.

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“…Further developments have been carried out by Saito (2005) and Saito and Tanikawa (2007;hereafter ST2007). M.M.…”

Section: Poincaré Sections and Symbolic Dynamicsmentioning

confidence: 99%

The rectilinear three-body problem

Орлов

Петрова

Tanikawa

et al. 2008

Celestial Mech Dyn Astr

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The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time.In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.Keywords Three-body problem · Rectilinear three-body problem · Triple approaches · Schubart periodic orbit · Escapes · Ejections A. I. Martynova State Forest Technical Academy, Institutskij per. 5, St. Petersburg 194021, Russia 123 94 V. V. Orlov et al.

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The rectilinear three-body problem using symbol sequence I. Role of triple collision

Cited by 13 publications

References 13 publications

Non-schubart periodic orbits in the rectilinear three-body problem

Non-schubart periodic orbits in the rectilinear three-body problem

Linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

The rectilinear three-body problem

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