Dynamics in the Charged Restricted Circular Three-Body Problem

Abstract

The existence and stability of periodic solutions for different types of perturbations associated to the Charged Restricted Circular Three Body Problem (shortly, CHRCTBP) is tackled using reduction and averaging theories as well as the technique of continuation of Poincaré for the study of symmetric periodic solutions. The determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the occurrence of Hamiltonian-Hopf bifurcations associated to some equilibrium points of the CHRCTBP.

Appendix

Consider the linear Hamiltonian system

$$\begin{aligned} \dot{z}=Az=\mathbb {J}\nabla \mathcal {H}(z),\quad \mathcal {H}=\frac{1}{2} z^TSz, \end{aligned}$$

(45)

where S is a symmetric constant matrix and $A=\mathbb {J}S$ is a Hamiltonian matrix.

Definition 6.1

(Strong stability) System (45) (or the matrix A) is strongly stable (or parametrically stable) if it and all sufficiently small linear constant Hamiltonian perturbations of it are stable. If system (45) is stable but not strongly stable, we say that it is weakly stable.

Let $\pm \alpha _1 i,\pm \alpha _2 i,\ldots ,\pm \alpha _s i$ be the eigenvalues of the matrix A, and let $V_j$, $j=1,\ldots , s$, be the maximal real linear subspace where A has eigenvalues $\pm \alpha _j i$. So $V_j$ is an A-invariant symplectic subspace, A restricted to $V_j$ has eigenvalues $\pm \alpha _j i$, and $\mathbb {R}^{2n}=V_1\oplus V_2\oplus \ldots \oplus V_s$. Let $\mathcal {H}_j$ be the restriction of $\mathcal {H}$ to $V_j$.

Theorem 6.1

(Krein–Gel’fand) System (45) is strongly stable if and only if

• all the eigenvalues of A are purely imaginary,

• A is nonsingular,

• A is diagonalizable over the complex numbers, and

• Hamiltonian $\mathcal {H}_j$ is positive or negative definite for each j.

See the proof in [27] or in [19].

Let $(M,\Omega )$ be a symplectic manifold of dimension 2n, $\mathcal {H}_0:M\rightarrow \mathbb {R}$ a smooth Hamiltonian which defines a Hamiltonian vector field $Y_0 = (dH_0)^\#$ with symplectic flow $\varphi _0^t$. Let $I\subset \mathbb {R}$ be an interval such that each $h\in I$ is a regular value of $\mathcal {H}_0$ and $\mathcal {N}_0(h)=\mathcal {H}_0^{-1}(h)$ is a compact connected circle bundle over a base space $\mathcal {B}(h)$ with projection $\pi :\mathcal {N}_0(h)\rightarrow \mathcal {B}(h)$. So, this is the setting of regular reduction theory. Assume that all the solutions of $Y_0$ in $\mathcal {N}_0(h)$ are periodic and have periods smoothly depending only on the value of the Hamiltonian; i.e., the period is a smooth function $T=T(h)$.

Let $\varepsilon$ be a small parameter, $\mathcal {H}_1:M\rightarrow \mathbb {R}$ be smooth, $\mathcal {H}_\varepsilon =\mathcal {H}_0+\varepsilon \mathcal {H}_1$, $Y_\varepsilon =Y_0+\varepsilon Y_1 =d\mathcal {H}_\varepsilon ^\#$, $\mathcal {N}_\varepsilon (h)=\mathcal {H}_\varepsilon ^{-1}(h)$, $\pi :\mathcal {N}_\varepsilon (h)\rightarrow \mathcal {B}(h)$ the projection, and $\phi _\varepsilon ^t$ be the flow defined by $Y_\varepsilon$.

Let the average of $\mathcal {H}_1$ be

$$\begin{aligned} \bar{\mathcal {H}}=\frac{1}{T}\int _0^T\mathcal {H}_1(\phi _0^t)dt. \end{aligned}$$

The next result provides sufficient conditions for characterising the existence of periodic solutions of the Hamiltonian system associated to $\mathcal {H}_\varepsilon$. For more information on this subject the reader is addressed to [20, 25, 28].

Theorem 6.2

(Reeb) If $\mathcal {\bar{H}}$ has a non-degenerate critical point at $\pi (p)=\bar{p}\in \mathcal {B}(h)$ with $p\in \mathcal {N}_0(h)$, then there are smooth functions $p(\varepsilon )$ and $T(\varepsilon )$ for $\varepsilon$ small with $p(0)=p$, $T(0)=T$, and $p(\varepsilon )\in \mathcal {N}_\varepsilon$, and the solution of $Y_\varepsilon$ through $p(\varepsilon )$ is $T(\varepsilon )$-periodic. In addition, if the characteristic exponents of the critical point $\bar{p}$ (that is, the eigenvalues of the matrix $A=\mathbb {J}D^2\mathcal {\bar{H}}(\bar{p})$) are $\lambda _1, \lambda _2,\ldots ,\lambda _{2n-2}$, then the characteristic multipliers of the periodic solution through $p(\varepsilon )$ are

$$\begin{aligned} 1,1,1+\varepsilon \lambda _1 T+O(\varepsilon ^2),1+\varepsilon \lambda _2 T+O(\varepsilon ^2), \ldots ,1+\varepsilon \lambda _{2n-2} T+O(\varepsilon ^2). \end{aligned}$$

Theorem 6.3

Let p and $\bar{p}$ as in the previous theorem. If one or more of the characteristic exponents $\lambda _j$ is real or has nonzero real part, then the periodic solution through $p(\varepsilon )$ is unstable. If the matrix A is strongly stable, then the periodic solution through $p(\varepsilon )$ is elliptic, i.e., linearly stable.

The proofs of Theorems 6.2 and 6.3 appear in [28].

Consider a Hamiltonian system of the form

$$\begin{aligned} \mathcal {H}_{\varepsilon }(I,\varphi ,\varepsilon )=h_0(I^{n_0})+\varepsilon ^{m_1}h_1(I^{n_1})+\ldots + \varepsilon ^{m_a}h_a(I^{n_a})+\varepsilon ^{m_a+1}p(I,\varphi ,\varepsilon ), \end{aligned}$$

(46)

where $(I,\varphi )\in \mathbb {R}^n\times \mathbb {T}^n$ are action-angle coordinates with the standard symplectic structure $dI\wedge d\varphi$, and $\varepsilon >0$ is a sufficiently small parameter. Hamiltonian $\mathcal {H}_\varepsilon$ is real analytic, and the parameters $a,m,n_i$ ($i=0,1,\ldots ,a$) and $m_j$ ($j=1,2,\ldots ,a$) are positive integers satisfying $n_0\le n_1\le \ldots \le n_a=n$, $m_1\le m_2\le \ldots \le m_a=m$, $I^{n_i}=(I_1,\ldots ,I_{n_i})$, for $i=1,2,\ldots ,a$, and p depends on $\varepsilon$ smoothly.

Hamiltonian $\mathcal {H}_{\varepsilon }(I,\varphi ,\varepsilon )$ is taken in a bounded closed region $Z\times \mathbb {T}^n\subset \mathbb {R}^n\times \mathbb {T}^n$. For each $\varepsilon$ the integrable part of $\mathcal {H}_{\varepsilon }$,

$$\begin{aligned} X_\varepsilon (I)=h_0(I^{n_0})+\varepsilon ^{m_1}h_1(I^{n_1})+\ldots +\varepsilon ^{m_a}h_a(I^{n_a}), \end{aligned}$$

admits a family of invariant n-tori $T^\varepsilon _\zeta =\{\zeta \}\times \mathbb {T}^n$, with linear flows $\{x_0+\omega ^\varepsilon (\zeta ) t\}$, where, for each $\zeta \in Z$, $\omega ^\varepsilon (\zeta )=\nabla X_\varepsilon (\zeta )$ is the frequency vector of the n-torus $T^\varepsilon _\zeta$ and $\nabla$ is the gradient operator. When $\omega ^\varepsilon (\zeta )$ is nonresonant, the n-torus $T^\varepsilon _\zeta$ becomes quasi-periodic with slow and fast frequencies of different scales. We refer to the integrable part $X_\varepsilon$ and its associated tori $\{T^\varepsilon _\zeta \}$ as the intermediate Hamiltonian and intermediate tori, respectively.

Let $\bar{I}^{n_i}=(I_{n_{i-1}+1},\ldots ,I_{n_i})$, $i=0,1,\ldots ,a$ (where $n_{-1}=0$, hence $\bar{I}^{n_0}=I^{n_0}$), and define

$$\begin{aligned} \Omega =\left( \nabla _{\bar{I}^{n_0}}h_0(I^{n_0}),\ldots , \nabla _{\bar{I}^{n_a}}h_{n_a}(I^{n_a})\right) , \end{aligned}$$

such that, for each $i=0,1,\ldots ,a$, $\nabla _{\bar{I}^{n_i}}$ denotes the gradient with respect to $\bar{I}^{n_i}$.

We assume the following high-order degeneracy-removing condition of Bruno–Rüssman type (so named by Han, Li and Yi), giving credit to Bruno and Rüssman, who provided weak conditions on the frequencies guaranteeing the persistence of invariant tori, the so-called (A) condition: there is a positive integer s such that

$$\begin{aligned} Rank\{\partial ^\alpha \Omega (I)\,:\,0\le |\alpha |\le s\}=n,\quad \forall \,I\in Z. \end{aligned}$$

For the usual case of a nearly integrable Hamiltonian system of the type

$$\begin{aligned} \mathcal {H}_{\varepsilon }(I,\varphi ,\varepsilon )=X(I)+\varepsilon p(I,\varphi ,\varepsilon ), \quad (I,\varphi )\in Z\times \mathbb {T}^n\subset \mathbb {R}^n\times \mathbb {T}^n. \end{aligned}$$

(47)

Condition (A) given above generalises the classical Kolmogorov non-degenerate condition that $\partial \Omega (I)$ be nonsingular over Z, where $\Omega (I)=\nabla X(I)$; Bruno’s non-degenerate condition that $Rank\{\Omega (I),\partial \Omega \}=n$, $\forall \, I\in Z$; and the weakest non-degenerate condition guaranteeing such persistence provided by Rüssman, that $\omega (Z)$ should not lie in any $(n-1)$-dimensional subspace. Rüssman condition is equivalent to condition (A) for systems like (47). However, Bruno or Rüssman conditions do not apply to Hamiltonian (46), as it is too degenerate.

The following theorem gives the right setting in which one can ensure the persistence of KAM tori for a Hamiltonian like (46).

Theorem 6.4

(Han, Li and Yi) Assume the condition (A), and let $\delta$ with $0<\delta <1/5$ be given. Then there exists an $\varepsilon _0>0$ and a family of Cantor sets $Z_\varepsilon \subset Z$, $0<\varepsilon <\varepsilon _0$, with $|Z\setminus Z_\varepsilon |=O(\varepsilon ^{\delta /s})$, such that each $\zeta \in Z_\varepsilon$ corresponds to a real analytic, invariant, quasi-periodic n-torus $\bar{T}^\varepsilon _\zeta$ of Hamiltonian (46), which is slightly deformed from the intermediate n-torus $T^\varepsilon _\zeta$. Moreover, the family $\{\bar{T}^\varepsilon _\zeta \,:\,\zeta \in Z_\varepsilon ,\, 0<\varepsilon <\varepsilon _0\}$ varies Whitney smoothly.

See the proof in [13].

Consider the Hamiltonian in 1:$-1$ resonance

$$\begin{aligned} \mathcal {H}=\frac{\omega }{2}(x_1y_2-x_2y_1)+\delta (x_1^2+x_2^2)+ \Gamma (y_1^2+y_2^2)^2+\ldots , \end{aligned}$$

(48)

where $\omega \ne 0$ and $\delta =\pm 1$.

Theorem 6.5

Given the parametric family defined by (48) there are periodic solutions emanating from the origin when $\delta \Gamma >0$. One family exists for $\mathcal {H}>0$ and one for $\mathcal {H}<0$. There are no nearby $2\pi$-periodic solutions when $\delta \Gamma <0$.

See the proof in [21].