Equivariant singularity analysis of the 2 : 2 resonance

Marchesiello, Antonella; Pucacco, Giuseppe

doi:10.1088/0951-7715/27/1/43

Cited by 18 publications

(22 citation statements)

References 48 publications

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“…A prediction based on an N-th order normal form can be obtained by a suitable extension of the theory ( [16,20]). The Hamiltonian on the center manifold is of the formK…”

Section: Higher Order Theorymentioning

confidence: 99%

“…A resonant perturbation theory allows us to investigate the halo family and to determine the value of the energy at which the bifurcation from the planar Lyapunov family, namely the horizontal normal mode, takes place [24]. We also construct a normal form to perform the center manifold reduction (see [13]), which yields an integrable approximation of the dynamics (compare with [20]). The unperturbed linear dynamics on the 2-dimensional center manifold is characterized by almost equal values of the frequencies for all mass ratios.…”

Section: Introductionmentioning

confidence: 99%

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Halo orbits around the collinear points of the restricted three-body problem

Ceccaroni

Celletti

Pucacco

2016

Physica D: Nonlinear Phenomena

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“…A prediction based on an N-th order normal form can be obtained by a suitable extension of the theory ( [16,20]). The Hamiltonian on the center manifold is of the formK…”

Section: Higher Order Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Halo orbits around the collinear points of the restricted three-body problem

Ceccaroni

Celletti

Pucacco

2016

Physica D: Nonlinear Phenomena

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“…Recalling that σ and η are the angle variables conjugated to S and T , respectively, the equations of motion associated to (4.9) are given bẏ To determine the bifurcation thresholds according to the technique described, e.g., in [40,41,42], we need to constrain the variable S within an interval, which is determined on the basis of the relation of S with the integral of motion represented by T . However, in the present situation we have also a physical constraint, which leads to a bound on S on the basis of the following physical considerations (compare with [5,6]).…”

Section: Remarkmentioning

confidence: 99%

“…An alternative method for detecting bifurcations. An alternative method to find bifurcations with respect to the technique presented in Section 4.2 relies on the following procedure, which is based on the geometric approach used in [41,42] and is related to the analysis of the critical inclination in [14]. and we notice that [14]), the variables (4.15) satisfy the identity…”

mentioning

confidence: 99%

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Celletti¹,

Galeş²,

Pucacco³

2016

SIAM J. Appl. Dyn. Syst.

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Using bifurcation theory, we study the secular resonances induced by the Sun and Moon on space debris orbits around the Earth. In particular, we concentrate on a special class of secular resonances, which depend only on the debris' orbital inclination. This class is typically subdivided into three distinct types of secular resonances: those occurring at the critical inclination, those corresponding to polar orbits, and a third type resulting from a linear combination of the rates of variation of the argument of perigee and the longitude of the ascending node.The model describing the dynamics of space debris includes the effects of the geopotential, as well as the Sun's and Moon's attractions, and it is defined in terms of suitable action-angle variables. We consider the system averaged over both the mean anomaly of the debris and those of the Sun and Moon. Such multiply-averaged Hamiltonian is used to study the lunisolar resonances which depend just on the inclination.Borrowing the technique from the theory of bifurcations of Hamiltonian normal forms, we study the birth of periodic orbits and we determine the energy thresholds at which the bifurcations of lunisolar secular resonances take place. This approach gives us physically relevant information on the existence and location of the equilibria, which help us to identify stable and unstable regions in the phase space. Besides their physical interest, the study of inclination dependent resonances offers interesting insights from the dynamical point of view, since it sheds light on different phenomena related to bifurcation theory.2010 Mathematics Subject Classification. 37N05,37L10,70F15.

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“…where k, ∈ Z, and δ is a small real parameter which we refer to as the detuning (see [28,17,18]). It is important to notice that, in this generic case, the resonance is in principle absent from the unperturbed dynamics, but it can appear in the perturbed system, once it is triggered by the non-linear, higher-order coupling terms.…”

Section: 2mentioning

confidence: 99%

Accurate modelling of the low-order secondary resonances in the spin-orbit problem

Gkolias

Efthymiopoulos

Celletti

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in [10], based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to 'book-keep' the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series.Keywords. Normal form -primary and secondary resonances -spin-orbit problem. IntroductionThe study of resonant configurations is of primary importance in many astronomical problems. One of the most frequently observed commensurabilities in our Solar system is that between the orbital and the rotational period of natural satellites. Our Moon, for example, is locked in a synchronous (1:1) spin-orbit resonance and this is probably the case also for all large planetary satellites. In a simple spin-orbit coupling model, the dynamics about the synchronous resonance can be described with a pendulum approximation. The phase-space is separated by a separatrix into a rotation and a libration domain. The frequency of the librations is determined to a first-order approximation by the shape of the satellite. For particular values of the asphericity parameter, used to measure the divergence of the real shape from a sphere, this frequency can become resonant with the orbital frequency. This situation, known as a secondary resonance,

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Equivariant singularity analysis of the 2 : 2 resonance

Cited by 18 publications

References 48 publications

Halo orbits around the collinear points of the restricted three-body problem

Halo orbits around the collinear points of the restricted three-body problem

Bifurcation of Lunisolar Secular Resonances for Space Debris Orbits

Accurate modelling of the low-order secondary resonances in the spin-orbit problem

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