On the periodic solutions of a rigid dumbbell satellite in a circular orbit

Abstract

The aim of the present paper is to provide sufficient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit.

Appendix: Basic results on averaging theory

In this appendix we present the basic result from the averaging theory that we shall need for proving the main results of this paper.

We consider the problem of the bifurcation of T-periodic solutions from differential systems of the form

$$ \dot{\mathbf{x}}(t)= G_{0}(t,\mathbf{x})+\varepsilon G_{1}(t,\mathbf{x})+\varepsilon^{2} G_{2}(t, \mathbf{x}, \varepsilon), $$

(11)

with ε=0 to ε≠0 sufficiently small. Here the functions $G_{0},G_{1}: \mathbb{R} \times\varOmega\to\mathbb{R} ^{n}$ and $G_{2}:\mathbb{R} \times\varOmega\times(-\varepsilon_{0},\varepsilon _{0})\to\mathbb{R} ^{n}$ are $\mathcal{C}^{2}$ functions, T-periodic in the first variable, and Ω is an open subset of $\mathbb{R} ^{n}$. The main assumption is that the unperturbed system

$$ \dot{\mathbf{x}}(t)= G_{0}(t,\mathbf{x}), $$

(12)

has a submanifold of periodic solutions.

Let x(t,z,ε) be the solution of the system (12) such that x(0,z,ε)=z. We write the linearization of the unperturbed system along a periodic solution x(t,z,0) as

$$ \dot{\mathbf{y}}(t)=D_{\mathbf{x}}{G_{0}}\bigl(t,\mathbf{x}(t, \mathbf{z} ,0)\bigr)\mathbf{y}. $$

(13)

In what follows we denote by M _z (t) some fundamental matrix of the linear differential system (13), and by $\xi:\mathbb {R}^{k}\times\mathbb{R}^{n-k}\rightarrow\mathbb{R}^{k}$ the projection of $\mathbb{R}^{n}$ onto its first k coordinates; i.e. ξ(x ₁,…,x _n )=(x ₁,…,x _k ).

We assume that there exists a k-dimensional submanifold $\mathcal {Z}$ of Ω filled with T-periodic solutions of (12). Then an answer to the problem of bifurcation of T-periodic solutions from the periodic solutions contained in $\mathcal{Z}$ for system (11) is given in the following result.

Theorem 6

Let V be an open and bounded subset of $\mathbb{R} ^{k}$, and let $\beta: {\mathrm{Cl}}(V)\to\mathbb{R} ^{n-k}$ be a $\mathcal{C}^{2}$ function. We assume that

• (i) $\mathcal{Z}= \{ \mathbf{z}_{\alpha}= ( \alpha, \beta(\alpha) ) ,\ \alpha\in{\mathrm{Cl}}(V) \} \subset\varOmega$ and that for each $\mathbf{z}_{\alpha}\in\mathcal {Z}$ the solution x(t,z _α ) of (12) is T-periodic;

• (ii) for each $\mathbf{z}_{\alpha}\in\mathcal{Z}$ there is a fundamental matrix $M_{\mathbf{z}_{\alpha}}(t)$ of (13) such that the matrix $M_{\mathbf{z}_{\alpha}}^{-1}(0)- M_{\mathbf{z}_{\alpha }}^{-1}(T)$ has in the upper right corner the k×(n−k) zero matrix, and in the lower right corner a (n−k)×(n−k) matrix Δ _α with det(Δ _α )≠0.

We consider the function $\mathcal{G} :{\mathrm{Cl}}(V) \to \mathbb{R} ^{k}$

$$ \mathcal{G} (\alpha)=\xi \biggl( \frac{1}{T} \int _{0}^{T} M_{\mathbf{z}_{\alpha}}^{-1}(t)G_{1} \bigl(t,\mathbf{x}(t,\mathbf {z}_{\alpha})\bigr) dt \biggr) . $$

(14)

If there exists a∈V with $\mathcal{G} (a)=0$ and ${\det ( ( {d\mathcal{G} }/{d\alpha} ) (a) ) \neq 0}$, then there is a T-periodic solution φ(t,ε) of system (11) such that φ(0,ε)→z _a as ε→0.

Theorem 6 goes back to Malkin (Malkin 1956) and Roseau (Roseau 1966), for a shorter proof see (Buică et al. 2007).