Classes of Families of Generalized Periodic Solutions to the Beletsky Equation

Bruno, A D; Варин, В. П.

doi:10.1023/b:cele.0000023390.25801.f9

Cited by 13 publications

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“…We can derive from Eqs. (2) and (16) and recall that in Eq. (6), as well, that the units have been chosen so that the expression below is equal to one (a = 1):…”

Section: Resultsmentioning

confidence: 99%

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Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation

Ershkov

2017

Theoretical and Applied Mechanics Letters

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The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet.First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ODE of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time.In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions.In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e → 0, p = 1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii's equation.

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“…We can derive from Eqs. (2) and (16) and recall that in Eq. (6), as well, that the units have been chosen so that the expression below is equal to one (a = 1):…”

Section: Resultsmentioning

confidence: 99%

“…We neglect the tidal torque in the derivation of Eqs. (1) and (2), which can be estimated as shown below [7], [12]:…”

Section: Revolving Scheme For Solving Cascades Of Abel Equations For mentioning

confidence: 99%

Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation

Ershkov

2017

Theoretical and Applied Mechanics Letters

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“…The motion (1.2) in the problem of the rotation of a satellite in an elliptic orbit (the Beletskii problem 3 ) has been called a a generalized periodic solution (also, see Ref. 7).…”

Section: Symmetric Periodic Motions and Their Stabilitymentioning

confidence: 99%

Periodic motions of a reversible second-order mechanical system

Тхай¹

2006

Journal of Applied Mathematics and Mechanics

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“…Also, spiral characteristics of families of periodic solutions have been found in the case of a satellite oscillations (Bruno, A.D. and Varin, V.P., 1997;Bruno, A.D., 2002 ). In this paper we make a systematic study of the families of non-symmetric periodic orbits that bifurcate at the points b1, c1, l2, n 2 4 (Hénon 1965) and z 1 (Pinotsis 1986), in the planar circular restricted 3-body problem and in the case of equal masses of the primaries.…”

Section: Introductionmentioning

confidence: 99%

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

Pinotsis

2010

Celest Mech Dyn Astr

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We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem. We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities. We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating point in x-at which the Jacobi constant is C ∞ =3 and their periodic orbits terminate at the corotation (at the Lagrangian point L 4 or L 5 ). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate large chaotic regions near the corotation points L 4 , L 5 , which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal point, infinite loops, one inside the other, surrounding the unstable triangular points L 4 or L 5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals into the corotation points L 4 , L 5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us to comprehend better the motions around the points L 4 and L 5 of Lagrange.

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Classes of Families of Generalized Periodic Solutions to the Beletsky Equation

Cited by 13 publications

References 1 publication

Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation

Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation

Periodic motions of a reversible second-order mechanical system

Infinite Feigenbaum sequences and spirals in the vicinity of the Lagrangian periodic solutions

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