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

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.

### MSC:

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

### Keywords:

Lyapunov stability

Zbl 1109.70308
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