PERIODIC SOLUTIONS OF TWIST TYPE OF AN EARTH SATELLITE EQUATION

We study Lyapunov stability for a given equation modelling the motion of an earth satellite. The proof combines bilateral bounds of the solution with the theory of twist solutions.


Introduction.
In the recent paper [10], it is studied the following equation as a model of the motion of a satellite in orbit around the earth x" + a J 1 + sin2 t sin x = 6 sin 2t( 1 + 3 sin 2 t) -2, where a is a positive parameter related with magnetic intensity. Due to the invariance of the equation under transformation (x, t) -+ (-x, -t), it is natural to look for and study the properties of odd 1!'-periodic solutions x(t) = -x( -t) = x(t + 1!'), Vt.
In particular, the question of stability of such a solution is of major importance because of the type of application. In this direction, in [10] it is proved that for a ::; 1/2, there exists a stable solution in the linear sense (also called elliptic), that is, such that the linearized equation is stable. As it is known, this fact does not assure stability in the sense of Lyapunov of the mentioned equation, since it depends usually on the nonlinear term of the equation. The aim of this note is to prove Lyapunov stability under the more conservative bound a ::; 1/18, by using a new result [6] on twist solutions and taking into account the bilateral bounds found in [10].
A periodic solution <pof a general periodic equation of second order is said to be 4-elementary if it is elliptic with Floquet's multipliers A, >: verifying: AP =J 1, 1 ::; p ::; 4. The solution <pis called of twist type ( [7,8]) if it is 4-elementary and the first coefficient of Birkhoff of the associated Poincaré map is different to zero. From the Moser's invariant curve theorem it follows that a periodic solution of twist type is always Lyapunov stable ( [4,9] with n -+ 00 as well as many quasi-periodic solutions. This is a consequence oí the Poincaré-Birkhoff's fixed point theorem and the KAM theorem ( [1,9]).
In a forthcoming paper, the authors plan to extend the ideas of this note to a wider class of periodic differential equations combining the results in [6] with upper and lower solutions for the Dirichlet problem.
2. Stability by the third approximation.
Let xo(t) be a periodic solution of a general1r-periodic equation where f is continuous and 1r-periodic in the variable t and of class 04 respect to x.
After a translation to the origin and a Taylor expansion, equation (2.1) can be written as where a, b, e are 1r-periodic and T denote the remaining terms.
The relation between Lyapunov stability and this expansion up to the third term is classical and at least goes back to Birkhoff. More recently, the mentioned relation is studied in the papers of Ortega [7,8]. In these papers the a.uthor always assumes that the nonlinear coefficients b and e have constant sign because the technique employed to prove the stability relies on the maximum principIe. Related results can be seen in [3,5].
AIso, the recent paper [6]study some cases which are not covered by the previous ones. Concretely, the main result of [6] is the following. Theorem 2.1. Assume that there exist (J"" positive numbers such that (J"2 :S a(t) :S,2 :S 1/9. Then, the equilibrium of (2.2) is of twist type if the following condition holds (J"617r c-(t)dt -(J"2,417r c+(t)dt > 2,517r b+(t)dt 17r b-(t)dt. (

2.3)
Let us note that the main advantage of this result is that the nonlinear coefficients are allowed to change signo 3. Main resulto Our main result is the following. It is easy to verify that the function A(a) = max{x;t(t) : tE (-11"/2, O)} is increasing with a, so in consequence for any a :S 1/18 we have the bound In order to apply Theorem 2.1, let us fix, =~and (j = vacosA(1/18). Then,  except possibly near to the strong resonances given by a = 1/8, 2/9. However actually our theoretical approach do not allow to prove this. Only we can reach stability up to a = 1/18. Note that the main assumption a :S 1/18 is needed essentially in (3.1). An improvement of our result would require a similar study as in [8] about the comparison of some coefficients of the 2-jet of the Poincaré map associated to the equation (2.2) (see [8,Lemma 4.7]). It seems to us a difficult and . delicate question.