Computation of coefficients of the normal form in a singular case

S. Yu. Sadov,

Institute of Applied Mathematics, Moscow, Russia

Satellite motion about its center of mass in the plane of its elliptic orbit is described by the second order ordinary quasilinear equation containing two parameters, the eccentrity of orbit e, and the inertial parameter μ, which is small for the satellite being almost dynamically symmetric [1]. We are interested in finding periodic solutions. As μ = 0, the equation is explicitly integrable. As e = 1, it is singular. A numerical study of the equation for e close to 1 is very difficult (see, e.g. [2]).

In the case of small μ the analysis of periodic orbits is possible by applying the method of averaging, or in the other words, computing the normal form of the equation in the neigbourhood of periodic solutions of the equation with μ = 0. This approach has been proposed in [3] and developed in [4]. Coefficients of the normal form are to be found numerically as some integrals.

As e → 1, there arises a problem in computing the coefficients, because the straight algorithms of computation are not regular in this case. We propose a regular method of the computation, based on integrating by a path in the complex plane and factorizing integrands into functions with close but diverse poles. Apart from the poles we have a regular expression, which one can calculate numerically. Near the poles we use analytic regularization methods.

References

[1] V.V. Beletskii. Motion of an Artificial Satellite about Its Center of Mass, NASA TT F-429, pp. 257–261.

[2] V.A. Sarychev. Topics in satellite orientation. Itogi nauki i tehniki, ser. ”Issledovanije kosmicheskogo prostranstva”, Vol. 11. Moscow: VINITI, 1978. (Russian)

[3] F.L. Chernous’ko. Resonance Phenomena in the Motion of a Satellite Relative to Its Mass Center. USSR Computational Mathematics and Mathematical Phys., 3 (1964), 699-713.

[4] A.D. Bruno. Local Methods in Nonlinear Differential Equations. Springer-Verlag: Berlin-Heidelberg-New York-London-Paris-Tokyo,1989. (Chapter V).