Periodic solutions of twist type of an Earth satellite equation. (English) Zbl 1068.70027

Núñez, Daniel; Torres, Pedro J.

Discrete Contin. Dyn. Syst. 7, No. 2, 303-306 (2001).

The authors consider the ordinary differential equation of the form \[ x''+\alpha\sqrt{1+\sin^2t}\sin x=6\sin {2t}(1+\sin^2t)^{-2}, \] where \(\alpha\) is a positive physical parameter related to magnetic intensity. The equation is a simple model of the motion of a satellite (a material point) in the orbit around the Earth. Earlier by A. A. Zevin and M. A. Pinsky [Discrete Contin. Dyn. Syst. 6, No. 2, 293–297 (2000; Zbl 1109.70308)], it was proved that for \(\alpha \leq1/2\) there exists a stable solution (in Lyapunov sense) in the linear case. The goal of this paper is to prove the Lyapunov stability of satellite motion under more restrictive bound on the parameter \(\alpha \leq 1/18\) by the use of some new results on the so-called twist solutions and taking into account the bilateral bounds found in the above-mentioned paper by Zevin and Pinsky.

Reviewer: Sergei Georgievich Zhuravlev (Moskva)

MSC:

70M20	Orbital mechanics
34D20	Stability of solutions to ordinary differential equations

Keywords:

Lyapunov stability

Citations:

Zbl 1109.70308