On the Periodic Solutions Emerging from the Equilibria of the Hill Lunar Problem with Oblateness

Abstract

In this paper, using the averaging theory of first order, we obtain sufficient conditions for computing periodic solutions in the 3D Hill problem with oblateness.

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 a differential system of the form

$$\begin{aligned} \dot{\mathrm {x}}(t)=G_{0}(t, \mathrm {x})+\epsilon G_{1}(t, \mathrm {x})+\epsilon ^{2}G_{2}(t, \mathrm {x},\epsilon ), \end{aligned}$$

(11)

with $\epsilon \ne 0$ sufficiently small. Here the functions $G_{0}, G_{1}:\mathbb {R}\times \Omega \rightarrow \mathbb {R}^{n}$ and $G_{2}:\mathbb {R}\times \Omega \times (-\epsilon _0,\epsilon _0) \rightarrow \mathbb {R}^{n}$ are ${{\mathcal {C}}}^{2}$ functions, T-periodic in the first variable, and $\Omega$ is an open subset of $\mathbb {R}^{n}$. The main assumption is that the unperturbed system

$$\begin{aligned} \dot{\mathrm {x}}(t)=G_{0}(t, \mathrm {x}), \end{aligned}$$

(12)

has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory.

Malkin (see [3] and references therein) studied the bifurcation of T-periodic solutions in the T-periodic system $\dot{x}=G_0(t,x) + \varepsilon G_1(t,x,\varepsilon )$, whose unperturbed system has a family of T-periodic solutions with initial conditions given by a smooth function $\beta :{\mathbb {R}}^k\rightarrow {\mathbb {R}}^n$, and proved that if the bifurcation function

$$\begin{aligned} \mathcal {G}(\alpha )=\int _0^T\begin{pmatrix}<u_1(t, \alpha ), g(t, x(t,\beta (\alpha )),0)>\\ \cdots \\ <u_k(t, \alpha ), g(t, x(t,\beta (\alpha )),0)> \end{pmatrix}dt, \end{aligned}$$

where $u_i$, $i=1,\dots ,k$, are k linearly independent T-periodic solutions of the adjoint linearized differential system, has a simple zero $\alpha _0$ such that $\det (\mathcal {G})|_{\alpha =\alpha _0}\ne 0$, then for any $\varepsilon >0$ sufficiently small, the system $\dot{x}=G_0(t,x) + \varepsilon G_1(t,x,\varepsilon )$ has an unique T-periodic solution $x_\varepsilon$ such that $x_\varepsilon (0)\rightarrow \beta (\alpha _0)$ as $\varepsilon \rightarrow 0$.

This can be rephrased as follows: Let $\mathrm {x}(t, \mathrm {z},\epsilon )$ be the solution of the system (12) such that $\mathrm {x}(0,\mathrm {z},\epsilon )=\mathrm {z}$. We write the linearization of the unperturbed system along a periodic solution $\mathrm {x}(t,\mathrm {z},0)$ as

$$\begin{aligned} \dot{\mathrm {y}}=D_{\mathrm {x}}G_{0}(t, \mathrm {x}(t, \mathrm {z},0))\mathrm {y}. \end{aligned}$$

(13)

In what follows we denote by $M_{\mathrm {z}}(t)$ 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. $\xi (x_{1}, \ldots , x_{n})=(x_{1}, \ldots ,x_{k})$.

We assume that there exists a k-dimensional submanifold $\mathcal {Z}$ of $\Omega$ 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 7

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

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

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

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

$$\begin{aligned} \mathcal {G}(\alpha )=\xi \left( \frac{1}{T}\int _{0}^{T}M_{\mathrm {z}_{\alpha }}^{-1}(t)G_{1}(t, \mathrm {x}(t,\mathrm {z}_{\alpha },0))dt\right) . \end{aligned}$$

(14)

If there exists $a\in V$ with $\mathcal {G}(a)=0$ and $\det ((d\mathcal {G}/d\alpha )(a))\ne 0$, then there is a T-periodic solution $\mathrm {x}(t,\epsilon )$ of system (11) such that $\mathrm {x}(0,\epsilon )\rightarrow \mathrm {z}_{a}$ as $\epsilon \rightarrow 0$.

For a proof of Theorem 7 see Malkin [7] and Roseau [16], or [2] for shorter proof.