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Long term evolution and chaotic diffusion of the insolation quantities of Mars
, 2004
"... As the obliquity of Mars is strongly chaotic, it is not possible to give a solution for its evolution over more than a few million years. Using the most recent data for the rotational state of Mars, and a new numerical integration of the Solar System, we provide here a precise solution for the evol ..."
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Cited by 115 (1 self)
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As the obliquity of Mars is strongly chaotic, it is not possible to give a solution for its evolution over more than a few million years. Using the most recent data for the rotational state of Mars, and a new numerical integration of the Solar System, we provide here a precise solution for the evolution of Mars’ spin over 10 to 20 Myr. Over 250 Myr, we present a statistical study of its possible evolution, when considering the uncertainties in the present rotational state. Over much longer time span, reaching 5 Gyr, chaotic diffusion prevails, and we have performed an extensive statistical analysis of the orbital and rotational evolution of Mars, relying on Laskar’s secular solution of the Solar System, based on more than 600 orbital and 200 000 obliquity solutions over 5 Gyr. The density
Geometric Integrators for ODEs
 J. Phys. A
, 2006
"... Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, ..."
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Cited by 37 (6 self)
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Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within roundoff error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, timereversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.
Splitting and composition methods in the numerical integration of differential equations
, 2008
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On the construction of the Kolmogorov normal form for the Trojan asteroids
 Nonlinearity
, 2005
"... In this paper we focus on the stability of the Trojan asteroids for the planar Restricted ThreeBody Problem (RTBP), by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on the numerical determination of the freque ..."
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Cited by 11 (8 self)
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In this paper we focus on the stability of the Trojan asteroids for the planar Restricted ThreeBody Problem (RTBP), by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on the numerical determination of the frequencies of the asteroid and the effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalog, and it has worked successfully for 23 of them. The construction of this normal form allows for computerassisted proofs of stability. To show it, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.
New families of symplectic splitting methods for numerical integration in dynamical astronomy
 Appl. Numer. Math
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On the accuracy of Restricted ThreeBody Models for the Trojan motion. Discrete Contin
 Dynam. Systems
, 2003
"... Abstract. In this note we compare the frequencies of the motion of the Trojan asteroids in the Restricted ThreeBody Problem (RTBP), the Elliptic Restricted ThreeBody Problem (ERTBP) and the Outer Solar System (OSS) model. The RTBP and ERTBP are wellknown academic models for the motion of these as ..."
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Cited by 6 (4 self)
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Abstract. In this note we compare the frequencies of the motion of the Trojan asteroids in the Restricted ThreeBody Problem (RTBP), the Elliptic Restricted ThreeBody Problem (ERTBP) and the Outer Solar System (OSS) model. The RTBP and ERTBP are wellknown academic models for the motion of these asteroids, and the OSS is the standard model used for realistic simulations. Our results are based on a systematic frequency analysis of the motion of these asteroids. The main conclusion is that both the RTBP and ERTBP are not very accurate models for the longterm dynamics, although the level of accuracy strongly depends on the selected asteroid. 1. Introduction. The Restricted ThreeBody Problem models the motion of a particle under the gravitational attraction of two point masses following a (Keplerian) solution of the twobody problem (a general reference is [17]). The goal of this note is to discuss the degree of accuracy of such a model to study the real motion of an asteroid moving near the Lagrangian points of the SunJupiter system.
The resonant structure of Jupiter’s Trojan asteroids II: Evolution during the planetary migration
, 2006
"... We study the global dynamics of the jovian Trojan asteroids by means of the Frequency Map Analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the longterm dynamics of the Trojans. Bes ..."
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Cited by 6 (1 self)
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We study the global dynamics of the jovian Trojan asteroids by means of the Frequency Map Analysis. We find and classify the main resonant structures that serve as skeleton of the phase space near the Lagrangian points. These resonances organize and control the longterm dynamics of the Trojans. Besides the secondary and secular resonances, that have already been found in other asteroid sets in mean motion resonance (e.g. Main belt, Kuiper belt), we identify a new type of resonance that involves secular frequencies and the frequency of the Great Inequality, but not the libration frequency. Moreover, this new family of resonances plays an important role in the slow transport mechanism that drives Trojans from the inner stable region to eventual ejections. Finally, we relate this global view of the dynamics with the observed Trojans, identify the asteroids that are close to these resonances and study their longterm behaviour. 1
Invariant tori in the SunJupiterSaturn system
 Discrete Contin. Dyn. Syst. Ser. B
"... (Communicated by Àngel Jorba) Abstract. We discuss the applicability of Kolmogorov’s theorem on existence of invariant tori to the real Sun–Jupiter–Saturn system. Using computer algebra, we construct a Kolmogorov’s normal form defined in a neighborhood of the actual orbit in the phase space, givin ..."
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Cited by 6 (1 self)
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(Communicated by Àngel Jorba) Abstract. We discuss the applicability of Kolmogorov’s theorem on existence of invariant tori to the real Sun–Jupiter–Saturn system. Using computer algebra, we construct a Kolmogorov’s normal form defined in a neighborhood of the actual orbit in the phase space, giving a sharp evidence of the convergence of the algorithm. If not a rigorous proof, we consider our calculation as a strong indication that Kolmogorov’s theorem applies to the motion of the two biggest planets of our solar system. 1. Introduction. We
Local spectral time splitting method for first and secondorder partial differential equations
, 2005
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Comparing the efficiency of numerical techniques for the integration of variational equations
, 2011
"... We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positi ..."
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Cited by 3 (3 self)
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We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positions. We apply the various techniques to the wellknown HénonHeiles system, and use the Smaller Alignment Index (SALI) method of chaos detection to evaluate the percentage of its chaotic orbits. The accuracy and the speed of the integration schemes in evaluating this percentage are used to investigate the numerical efficiency of the various techniques.