Exponential stability of fast driven systems, with an application to celestial mechanics

Chen, Qinbo; Pinzari, Gabriella

doi:10.1016/j.na.2021.112306

Cited by 7 publications

(20 citation statements)

References 33 publications

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“…Our interest in H C is indeed precisely related to the Euler integral ( 16): we aim to find zones in the phase space of H C where E affords slow variations and, simultaneously, chaos is detected. A similar point of view has been taken up, on the other hand, in the published papers [22,3,9].…”

Section: Purpose Of the Papermentioning

confidence: 70%

“…For general information on chaotic phenomena, the reader may consult [6,18,7]. Closely related papers to the current one are the aforementioned [22,3,9]. Specifically, in [22] Conjecture 1.3 has been proved in the case (c), while in [3] it has been proved in the case (a), with r 1 and for motions very close to S 0 (r).…”

Section: Purpose Of the Papermentioning

confidence: 96%

“…We fix a reference frame (i, j, k) in the Euclidean space, which we identify with R 3 . In such a space we consider three masses 1, µ and κ, with µ, κ < 1, interacting through gravity only.…”

Section: Purpose Of the Papermentioning

confidence: 99%

“…[1,17,10,19,5,4]. 3 We call "Newtonian potential" the function 1 x − x . Note that the term γy • y has zero-average (being y proportional to the -derivative of x and y -independent), so it is merged in H C , together with the zero-average part U of the Newtonian potential.…”

Section: Purpose Of the Papermentioning

confidence: 99%

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Euler integral as a source of chaos in the three-body problem

Di Ruzza,

Pinzari

2022

Preprint

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In this paper we address, from a purely numerical point of view, the question, raised in [20,21], and partly considered in [22,9,3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in [13,24,23].

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confidence: 70%

Section: Purpose Of the Papermentioning

confidence: 96%

Section: Purpose Of the Papermentioning

confidence: 99%

Section: Purpose Of the Papermentioning

confidence: 99%

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Euler integral as a source of chaos in the three-body problem

Di Ruzza,

Pinzari

2022

Preprint

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Chaotic coexistence of librational and rotational dynamics in the averaged planar three-body problem

Ruzza

2023

Celest Mech Dyn Astron

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Through an appropriate change of reference frame and rescalings of the variables and the parameters introduced, the Hamiltonian of the three-body problem is written as a perturbed Kepler problem. In this system, new Delaunay variables are defined and a suitable configuration of the phase space and the mass parameters is chosen. In such a system, wide regions of librational and rotational motions where orbits are regular and stable are found. Close to the separatrix of these regions, the existence of chaotic motions presenting a double rotational and librational dynamics is proved, numerically, through Poincaré sections and the use of FLI.

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Numerical studies to detect chaotic motion in the full planar averaged three-body problem

Ruzza

2023

Boll Unione Mat Ital

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In this paper, the author deals with a well-known problem of Celestial Mechanics, namely the three-body problem. A numerical analysis has been done in order to prove existence of chaotic motions of the full-averaged problem in particular configurations. Full because all the three bodies have non-negligible masses and averaged because the Hamiltonian describing the system has been averaged with respect to a fast angle. A reduction of degrees of freedom and of the phase-space is performed in order to apply the notion of covering relations and symbolic dynamics.

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Exponential stability of fast driven systems, with an application to celestial mechanics

Cited by 7 publications

References 33 publications

Euler integral as a source of chaos in the three-body problem

Euler integral as a source of chaos in the three-body problem

Chaotic coexistence of librational and rotational dynamics in the averaged planar three-body problem

Numerical studies to detect chaotic motion in the full planar averaged three-body problem

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