Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

Cincotta, Pablo Miguel; Giordano, C.; Simó, Carles

doi:10.1016/s0167-2789(03)00103-9

Cited by 336 publications

(335 citation statements)

References 24 publications

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“…We reconstruct the structure of the phase-space in terms of the maximum Lyapunov exponent (Benettin et al 1976) which is expressed through the so called fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO, see e.g. Cincotta & Simó 2000;Cincotta et al 2003;Goździewski et al 2001 for details). This dynamical characteristic helps us to classify given sets of initial conditions as regular (leading to quasi-periodic evolution of the system, stable over infinite period of time) or chaotic (leading to irregular phase-space trajectories).…”

Section: Dynamical Stabilitymentioning

confidence: 99%

Multi-site campaign for transit timing variations of WASP-12 b: possible detection of a long-period signal of planetary origin

Maciejewski

Dimitrov

Seeliger³

et al. 2013

A&A

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Aims. The transiting planet WASP-12 b was identified as a potential target for transit-timing studies because a departure from a linear ephemeris has been reported in the literature. Such deviations could be caused by an additional planet in the system. We attempt to confirm the claimed variations in transit timing and interpret their origin. Methods. We organised a multi-site campaign to observe transits by WASP-12 b in three observing seasons, using 0.5-2.6-metre telescopes. Results. We obtained 61 transit light curves, many of them with sub-millimagnitude precision. The simultaneous analysis of the best-quality datasets allowed us to obtain refined system parameters, which agree with values reported in previous studies. The residuals versus a linear ephemeris reveal a possible periodic signal that may be approximated by a sinusoid with an amplitude of 0.00068 ± 0.00013 d and period of 500 ± 20 orbital periods of WASP-12 b. The joint analysis of timing data and published radial velocity measurements results in a two-planet model that explains observations better than do single-planet scenarios. We hypothesise that WASP-12 b might not be the only planet in the system, and there might be the additional 0.1 M Jup body on a 3.6-d eccentric orbit. A dynamical analysis indicates that the proposed two-planet system is stable on long timescales.

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Section: Dynamical Stabilitymentioning

confidence: 99%

Multi-site campaign for transit timing variations of WASP-12 b: possible detection of a long-period signal of planetary origin

Maciejewski

Dimitrov

Seeliger³

et al. 2013

A&A

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“…First-order variational equations have been widely discussed and applied in the literature (e.g., Mikkola & Innanen 1999;Tancredi et al 2001;Cincotta et al 2003). In this paper we derive second-order variational equations for the N-body problem for the first time.…”

Section: Introductionmentioning

confidence: 99%

“…For a long time, first-order variational equations have been widely used to calculate Lyapunov exponents and the Mean Exponential Growth of Nearby Orbits (MEGNO, Cincotta et al 2003) in the astrophysics community (Tancredi et al 2001;Hinse et al 2010). We speculate that higher-order variational equations may be able to improve such chaos indicators.…”

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confidence: 96%

Second-order variational equations forN-body simulations

Rein

Tamayo

2016

Mon. Not. R. Astron. Soc.

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First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO).In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton's method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection.We provide an implementation of first and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

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“…and can therefore be used by various members of the dynamical or geodesian community. The second originality of our software is the availability of two efficient tools of analysis: the indicator of chaos MEGNO (Cincotta & Simó 2000;Cincotta et al 2003;Goździewski et al 2001) and the frequency analysis NAFF 1 The name of this software has been fixed at this time. (Laskar 1999(Laskar , 2003(Laskar , 2005 are directly connected to the numerical integrations.…”

Section: Introductionmentioning

confidence: 99%

NIMASTEP: a software to modelize, study, and analyze the dynamics of various small objects orbiting specific bodies

Delsate

Compère

2012

A&A

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NIMASTEP is a dedicated numerical software developed by us, which allows one to integrate the osculating motion (using Cartesian coordinates) in a Newtonian approach of an object considered as a point-mass orbiting a homogeneous central body that rotates with a constant rate around its axis of smallest inertia. The code can be applied to objects such as particles, artificial or natural satellites, or space debris. The central body can be either any terrestrial planet of the solar system, any dwarf-planet, or even an asteroid. In addition, very many perturbations can be taken into account, such as the combined third-body attraction of the Sun, the Moon, or the planets, the direct solar radiation pressure (with the central body shadow), the non-homogeneous gravitational field caused by the non-sphericity of the central body, and even some thrust forces. The simulations were performed using different integration algorithms. Two additional tools were integrated in the software package; the indicator of chaos MEGNO and the frequency analysis NAFF. NIMASTEP is designed in a flexible modular style and allows one to (de)select very many options without compromising the performance. It also allows one to easily add other possibilities of use. The code has been validated through several tests such as comparisons with numerical integrations made with other softwares or with semi-analytical and analytical studies. The various possibilities of NIMASTEP are described and explained and some tests of astrophysical interest are presented. At present, the code is proprietary but it will be released for use by the community in the near future. Information for contacting its authors and (in the near future) for obtaining the software are available on the web site http://www.fundp.ac.be/en/research/projects/ page_view/10278201/

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Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

Cited by 336 publications

References 24 publications

Multi-site campaign for transit timing variations of WASP-12 b: possible detection of a long-period signal of planetary origin

Multi-site campaign for transit timing variations of WASP-12 b: possible detection of a long-period signal of planetary origin

Second-order variational equations forN-body simulations

NIMASTEP: a software to modelize, study, and analyze the dynamics of various small objects orbiting specific bodies

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