Quasi-Periodic Attractors in Celestial Mechanics

Celletti, Alessandra; Chierchia, Luigi

doi:10.1007/s00205-008-0141-5

Cited by 59 publications

(64 citation statements)

References 11 publications

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“…On the other hand, quasi-periodic solutions, i.e. solutions of the form x(t) = ωt + u(ωt, t) with ω irrational and u(θ 1 , θ 2 ) 2π-periodic in each variables θ i , do not always exist: they exist only for Diophantine frequencies ω and provided the driving frequency ν is sharply tuned with ω so that, in particular, ν = ω + O(ε 2 ) (see Celletti and Chierchia 2008); however, when such quasi-periodic attractors exist, they are unique and seem to have a large BAM, as shown below.…”

Section: The Modelmentioning

confidence: 99%

“…In the weakly dissipative regime, periodic attractors may be easily analytically shown to coexist (Biasco and Chierchia 2008). The quasi-periodic case is more difficult in view of the appearance of small divisors and it was proved in Celletti and Chierchia (2008) that KAM tori smoothly bifurcate into quasi-periodic attractors for the dissipative system, provided that the "driving frequency" is suitably tuned with the parameters of the model.…”

mentioning

confidence: 98%

“…In the purely conservative regime, periodic orbits exist for all rational periods and many quasi-periodic tori exist thanks to KAM theory, 1 while, when the dissipation is turned on, at most one quasi-periodic attractor may exist, provided the "driving frequency" is suitably related to the angular velocities; (Celletti and Chierchia 2008) and co-existence with many periodic orbits (spin-orbit resonances) is possible (Broer et al 1996(Broer et al , 1998.…”

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confidence: 99%

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Measures of basins of attraction in spin-orbit dynamics

Celletti

Chierchia

2008

Celest Mech Dyn Astr

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We consider a dissipative spin-orbit model where it is assumed that the orbit of the satellite is Keplerian, the obliquity is zero, and the dissipative effects depend linearly on the relative angular velocity. The measure of the basins of attraction associated to periodic and quasi-periodic attractors is numerically investigated. The results depend on the interaction among the physically relevant parameters, namely, the orbital eccentricity, the equatorial oblateness and the dissipative constant. In particular, it appears that, for astronomically relevant parameter values, for low eccentricities (as in the Moon's case) about 96% of the initial data belong to the basin of attraction of the 1/1 spin-orbit resonance; for larger values of the eccentricities higher order spin-orbit resonances and quasi-periodic attractors become dominant providing a mechanism for explaining the observed state of Mercury into the 3/2 resonance

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Section: The Modelmentioning

confidence: 99%

mentioning

confidence: 98%

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confidence: 99%

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Measures of basins of attraction in spin-orbit dynamics

Celletti

Chierchia

2008

Celest Mech Dyn Astr

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“…Full details are given in Ref. 6. Actually, the above Nash-Moser approach is rather robust and general; indeed it can be easily adapted to cover dissipative maps such as the "fattened Arnold family" studied in Ref.…”

Section: Resultsmentioning

confidence: 99%

Quasi–periodic Attractors and Spin–orbit Resonances

Celletti

Chierchia

2007

Symmetry and Perturbation Theory

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“…They argue that this model, among other things, is incompatible with the physically plausible rheology of terrestrial planets, and can also give rise to quasi-periodic solutions for the spin-orbit problem (Correia and Laskar 2004;Celletti and Chierchia 2009;Bartuccelli et al 2015). Tidal dissipation is modelled in Noyelles et al (2014) by expanding both the tide-raising potential of the star, and the tidal potential of the planet, as Fourier series.…”

Section: The Tidal Torquementioning

confidence: 99%

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

Bartuccelli

Deane

Gentile

2017

Celest Mech Dyn Astr

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We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is C 1 in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of 10 6 -10 7 years. The proposed algorithm is based on the high-order Euler method which was described in Bartuccelli et al. (Celest Mech Dyn Astron 121(3):233-260, 2015), and it requires computer algebra to set up the code for its implementation. The payoff is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.

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Quasi-Periodic Attractors in Celestial Mechanics

Cited by 59 publications

References 11 publications

Measures of basins of attraction in spin-orbit dynamics

Measures of basins of attraction in spin-orbit dynamics

Quasi–periodic Attractors and Spin–orbit Resonances

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

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