Quasi-Periodic Solutions of Einstein-Friedman Equations

Ershkov, Sergey V.

doi:10.12732/ijpam.v90i4.8

Cited by 2 publications

(1 citation statement)

References 8 publications

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“…The above Eq. ( 4) is known as the Abel ODE of the second kind (for research with Abel ODE, see [8]), which is a kind of generalization of the Riccati ODE (for research with Riccati ODE, see [9]). This type of equation has no general analytical solution [10].…”

Section: Appendix; Only the Resulting Formulae Are Given In The Main ...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 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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Section: Appendix; Only the Resulting Formulae Are Given In The Main ...mentioning

confidence: 99%