Normal form of the satellite oscillation equation in the singular case (Q1922310)

A system of ordinary differential equations of the form \[ dX/dt=\varepsilon F(X,t),\tag{1} \] is considered, where \(X=(x_1,x_2,\dots,x_n)\), the vector function \(F=(f_1,f_2,\dots,f_n)\) is analytic in \(X\) in a domain \(Q\) and \(2\pi\)-periodic in \(t\), and \(\varepsilon\) is a real small parameter. Using a change of variables \[ X=Y+\sum^\infty_{k=1} \varepsilon^kH_k(Y,t),\tag{2} \] the author reduces the system (1) to a so-called normal form (an autonomous system) \[ \dot Y=\sum^\infty_{k=1} \varepsilon^kG_k(Y)\equiv G(Y,\varepsilon),\tag{3} \] where the vector functions \(H_k(Y,t)\), \(G_k(Y)\) are analytic in \(Y\in Q\) and \(H_k\) are \(2\pi\)-period in \(t\). It was proved by other authors that the set of the stationary points of (3) \[ A=\{Y,\varepsilon: G(Y,\varepsilon)=0\}\tag{4} \] is analytic, the series (2) converges on \(A\) and the corresponding functions \(X(t)\) given by (2) are the \(2\pi\)-periodic solutions of the system (1). Considering then the problem of the motion of a satellite relative to its center of mass (moving itself in the plane of an elliptic orbit with eccentricity \(e\)) the author introduces an additional parameter \(e\) in the procedure considered. Finally, the structure of the set \(A\) in the case \(\varepsilon\to0\), \(e\to1\) is investigated.