The Global Symplectic Integrator: an efficient tool for stability studies of dynamical systems. Application to the Kozai resonance in the restricted three-body problem

Libert, Anne-Sophie; Hubaux, Ch.; Carletti, Timotéo

doi:10.1111/j.1365-2966.2011.18431.x

Cited by 4 publications

(2 citation statements)

References 32 publications

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“…Actually, in Sect. 3 we show that, for negative energies E close to zero, the phase space acquires a structure reminiscent to the one of the non-integrable Lidov-Kozai case of the restricted three-body problem (i.e., non-intersecting trajectories examined in a higher than quadrupolar development of the disturbing function, see Libert et al (2011)). The typical behavior of the trajectories in the Lidov-Kozai regime is to (quasi-)periodically exchange eccentricity with mutual inclination.…”

Section: Introductionmentioning

confidence: 69%

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

Mastroianni

Efthymiopoulos

2023

Celest Mech Dyn Astron

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We revisit the secular 3D planetary three-body problem, aiming to provide a unified formalism representing all basic phenomena in the phase space as the mutual inclination between the planetary orbits increases. We propose a ‘book-keeping’ technique allowing to decompose the Hamiltonian as $$\mathcal {H}_\textrm{sec}=\mathcal {H}_\textrm{planar}+\mathcal {H}_\textrm{space}$$ H sec = H planar + H space , with $$\mathcal {H}_\textrm{space}$$ H space collecting all terms depending on the planets mutual inclination $$i_\textrm{mut}$$ i mut . We numerically compare several models obtained by multipole (Legendre) or Laplace–Lagrange expansions of $$\mathcal {H}_\textrm{sec}$$ H sec , aiming to define suitable truncation orders for these models. We explore the transition, as $$i_\textrm{mut}$$ i mut increases, from a ‘planar-like’ to a ‘Lidov–Kozai’ regime. Using a numerical example far from hierarchical limits, we find that the structure of the phase portraits of the (integrable) planar case is reproduced to a large extent also in the 3D case. A semi-analytical criterion allows to estimate the level of $$i_\textrm{mut}$$ i mut up to which the dynamics remains nearly integrable. We propose a normal form method to compute the basic periodic orbits (apsidal corotation orbits A and B) in this regime. We explore the sequence of saddle-node and pitchfork bifurcations by which the A and B families are connected to the highly inclined periodic orbits of the Lidov–Kozai regime. Finally, we perform a numerical study of phase portraits for different planetary mass and distance ratios and qualitatively describe the approach to the corresponding hierarchical limits.

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Section: Introductionmentioning

confidence: 69%

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

Mastroianni

Efthymiopoulos

2023

Celest Mech Dyn Astron

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“…The method makes it possible for the Lyapunov exponent to be consequently recovered. For these reasons, MEGNO has been largely used in the framework of planetary systems [58,59], satellites and spatial debris [60][61][62] and also generic nonlinear dynamical systems [57]. The method overcomes the above mentioned limitation by performing a sort of time average of the norm of the deviation vector (see Appendix A).…”

Section: Master Stability Function On Hypergraphsmentioning

confidence: 99%

Dynamical systems on Hypergraphs

Carletti¹,

Fanelli²

2020

Preprint

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Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the Master Stability Function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions. I. INTRODUCTION.Network science [1,2] has proved successful in describing many real-world systems [3-5], which, despite inherent differences, share common structural features. Even more interestingly, dynamical processes and hosting networks are indissolubly entangled with the ensuing patterns that reflect in fact the complex topology of the supports to which they are anchored [6][7][8].Networks constitute abstract frameworks, where pairwise interactions among generic agents, represented by nodes, are schematised by edges. Stated simply, two agents are connected if they interact. Hence by their very first definition, networks encode for binary relationships among units. This descriptive framework is sufficiently accurate in many cases of interest, although several examples exist of systems for which it holds true just as a first order approximation [9,10]. The relevance of high-orders structures has been indeed emphasised in the context of functional brain networks [11,12], in applications to protein interaction networks [13], to the study of ecological communities [14] and co-authorship networks [15,16].Starting from this observation, higher-order models have been developed so as to capture the many body interactions among interacting units. The most notable examples are simplicial complexes [17][18][19] and hypergraphs [20][21][22], non trivial mathematical generalisations of ordinary networks that are currently attracting a lot of interest. The concept of simplicial complexes has been for instance invoked to address problems in epidemic spreading [23,24] or synchronisation phenomena [25,26]. Our work is positioned in the framework of hypergraphs, a domain of investigation which is still in its infancy. In this respect, we mention applications to social contagion model [27,28], to the modelling of random walks [16] and to the study of synchronisation [29,30] and diffusion [28].Hypergraphs constitute indeed a very flexible paradigm: an arbitrary number of agents are allowed to interact, thus extending beyond the limit of binary interactions of conventional network models. On the other hand, hypergraphs define a leap forward as compared to simplicial com...

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