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

Bakker, Lennard F.; Ouyang, Tiancheng; Yan, Duokui; Simmons, Skyler

doi:10.1007/s10569-011-9358-y

Cited by 17 publications

(30 citation statements)

References 27 publications

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“…To analyze the round-off effects a convenient approach is based on the Reversibility Error Method (REM), which consists in computing the divergence of the initial point from its image after n forward and n backward iterations of the perturbed map, avoiding the exact computation of the map. This method is usually known as Reversibility test [13] and it is routinely used to analyze the regularization method for close encounters between two massive objects [14], or in the electromagnetic problems [15], or to study the collisions in the few body gravitational problem [16]. It has also been used to investigate the dynamic stability of the Hamiltonian model H = p 2 /2 − 1/x + x cos(ωt) [17].…”

Section: Introductionmentioning

confidence: 99%

Fidelity and reversibility in the restricted three body problem

Panichi

Ciotti

Turchetti

2016

Communications in Nonlinear Science and Numerical Simulation

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We use the Reversibility Error Method and the Fidelity to analyze the global effects of a small perturbation in a non-integrable system. Both methods have already been proposed and used in the literature but the aim of this paper is to compare them in a physically significant example adding some considerations on the equivalence, observed in this case, between round-off and random perturbations.As a paradigmatic example we adopt the restricted planar circular three body problem. The cumulative effect of random perturbations or round-off leads to a divergence of the perturbed orbit from the reference one. Rather than computing the distance of the perturbed orbit from the reference one, after a given number n of iterations, a procedure we name the Forward Error Method (FEM), we measure the distance of the reversed orbit (n periods forward and backward) from the initial point. This approach, that we name Reversibility Error Method (REM), does not require the computation of the unperturbed map. The loss of memory of the perturbed map is quantified by the Fidelity decay rate whose computation requires a statistical average over an invariant region. Two distinct definitions of Fidelity are given. The asymptotic equivalence of REM and FEM is analytically proved for linear symplectic maps with random perturbations. For a given map, the REM plot provides a picture of the dynamic stability regions in the phase space, very easy to obtain for any kind of perturbation and very simple to implement numerically. The REM and FEM for linear symplectic maps are proved to be asymptotically equivalent. The global error growth follows a power law in the regions of integrable (or quasi integrable) motion and an exponential law in the regions of chaotic motion. We prove that the power law exponent is 3/2 for a generic anisochronous system, but drops down to 1/2 if the system is isochronous. Correspondingly the Fidelity F (t) exhibits an exponential decay and − ln F (t) grows just as the square of the FEM or REM error. The Reversibility Error and Fidelity can be used for a quantitative analysis of dynamical systems and are suited to investigate the transition regions from chaotic to regular motion even for Hamiltonian systems with many degrees of freedom such as the N -body problem.

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Section: Introductionmentioning

confidence: 99%

Fidelity and reversibility in the restricted three body problem

Panichi

Ciotti

Turchetti

2016

Communications in Nonlinear Science and Numerical Simulation

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“…One of these potentially hazardous asteroids, named 2012 DA14, was discovered in 2012. This asteroid will pass by Earth on February 15, 2013, coming closer to the Earth than satellites in geostationary orbit 3 . How close will 2012 DA14 pass by Earth?…”

Section: Phenomenonmentioning

confidence: 99%

Understanding the Dynamics of Collision and Near-Collision Motions in the N-Body Problem

Bakker

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Although rare, collisions of two or more bodies in the N-body problem are apparent obstacles at which Newton's Law of Gravity ceases to make sense. Without understanding the nature of collisions, a complete understanding of the N-body problem can not be achieved. Historically, several methods have been developed to probe the nature of collisions in the N-body problem. One of these methods removes, or regularizes, certain types of collisions entirely, thereby relating not only analytically, but also numerically, the dynamics of such collision motions with their near collision motions. By understanding the dynamics of collision motions in the regularized setting, a better understanding of the dynamics of near-collision motions is achieved.

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“…Owing to the equality of the masses, m 1 = m 2 , this periodic solution displays remarkable symmetries: ξ 1 (τ ) and ξ 2 (τ ) behave like the symmetrically distorted graphs of the trigonometric functions − sin(π/2 · τ/q) and − cos(π/2 · τ/q). The stability of similar orbits with simultaneous binary collisions has been investigated by Bakker et al (2010Bakker et al ( , 2011.…”

Section: Periodic Solutions and Resonancementioning

confidence: 99%

The rhomboidal symmetric four-body problem

Waldvogel

2012

Celest Mech Dyn Astr

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We consider the planar symmetric four-body problem with two equal masses m 1 = m 3 > 0 at positions (±x 1 (t), 0) and two equal masses m 2 = m 4 > 0 at positions (0, ±x 2 (t)) at all times t, referred to as the rhomboidal symmetric four-body problem. Owing to the simplicity of the equations of motion this problem is well suited to study regularization of the binary collisions, periodic solutions, chaotic motion, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role.

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

Cited by 17 publications

References 27 publications

Fidelity and reversibility in the restricted three body problem

Fidelity and reversibility in the restricted three body problem

Understanding the Dynamics of Collision and Near-Collision Motions in the N-Body Problem

The rhomboidal symmetric four-body problem

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