# Cantor Set Structure of the Weak Stability Boundary for Infinitely Many
Cycles in the Restricted Three-Body Problem

EDWARD BELBRUNO  Yeshiva University, Department of Mathematical Sciences, New
York, NY 10016, USA<EMAIL_ADDRESS>
(Date: March 30, 2024)

###### Abstract.

The geometry of the weak stability boundary region for the planar restricted
three-body problem about the secondary mass point has been an open problem.
Previous studies have conjectured that it may have a fractal structure. In
this paper, this region is studied for infinitely many cycles about the
secondary mass point, instead of a finite number studied previously. It is
shown that in this case the boundary consists of a family of infinitely many
Cantor sets and is thus fractal in nature. It is also shown that on two-
dimensional surfaces of section, it is the boundary of a region only having
bounded cycling motion for infinitely many cycles, while the complement of
this region generally has unbounded motion. It is shown that that this shares
many properties of a Mandelbrot set. Its relationship to the non-existence of
KAM tori is described, among many other properties. Applications are
discussed.

###### Key words and phrases:

celestial mechanics, chaos, stability, restricted three-body problem, KAM,
Cantor set

##  1\. Introduction

The planar circular restricted three-body problem is considered. This model
goes back to the work of H. Poincaré [25]. In it, a particle $P$ of negligible
mass moves on the same plane as two mass-points, $P_{1},P_{2}$, as they make
mutual circular orbits about their common center of mass, with constant
angular frequency. $P_{1}$ is assumed to be much more massive than $P_{2}$.
For example, $P_{1}$ is the Sun and $P_{2}$ is the Earth.

The motion of $P$ about $P_{1}$ is well understood by the Kolmogorov-Arnold-
Moser (KAM) Theorem when $P$ starts with elliptic Keplerian initial conditions
about $P_{1}$. When the ratio of the angular frequencies of the circular
motion of $P_{1},P_{2}$ to that of the initial elliptic orbit of $P$ are
sufficiently irrational, satisfying diophantine conditions, then the motion of
$P$ about $P_{1}$ remains close to elliptical for all time, and is quasi-
periodic. It lies on two-dimensional invariant tori on a three-dimensional
fixed energy surface in the four-dimensional phase space of position and
velocity. The tori separate the energy surface into disjoint connected
components. There are gaps in the tori corresponding to approximate rational
frequency ratios where the motion of $P$ is chaotic [28, 23, 1, 8].

The motion of $P$ about $P_{2}$ is a much different situation. When $P$ starts
with Keplerian elliptic initial conditions about $P_{2}$, the resulting
dynamics is not generally understood. If $P$ starts sufficiently near to
$P_{2}$, where the Jacobi energy is sufficiently large, then KAM tori exist by
a theorem of M. Kummer [19]. This is accomplished by a special regularization.
As the distance from $P_{2}$ increases, however, this methodology is not
possible and the existence of invariant tori is not proven to exist. This is
an open problem as to how far from $P_{2}$ invariant tori can exist, although
there are interesting numerical analysis (see [29]).

A method to understand the motion about $P_{2}$ is to use weak stability
boundary methods. The weak stability boundary, $W$, defines a region about
$P_{2}$ in phase space that measures the stability of cycling motion about
$P_{2}$. It was first defined in [2] in 1987 for the more general three-
dimensional restricted three-body problem (see also [5, 6]). The motivation
for this was to find special low energy trajectories from the Earth to the
Moon that were dynamically captured at the Moon. This led to a new type of
transfer to the Moon for spacecraft which use no fuel for lunar capture.111It
was demonstrated operationally in 1991 with Japan’s Hiten spacecraft [3]. This
transfer class has been used by other lunar missions: GRAIL, CAPSTONE,
Flashlight (NASA), Denuri (KARI), Hakuto-R (ispace) Related capture transfers,
obtained by considering $W$, were used also for the SMART-1 lunar mission of
the European Space Agency (ESA) [27] as well as ESA’s BepiColombo mission to
Mercury [16] (see also, [9], [26]).

$W$ is determined numerically, by estimating a transition between stable and
unstable motion after $n$ cycles of the trajectory for $P$ about $P_{2}$,
$n=1,2,\ldots$, starting from the set of Keplerian elliptical conditions at
the periapsis positions (see Section 3). This yields a distribution of points
about $P_{2}$ depending on several parameters, including $n$, defining the
$n$th weak stability boundary, $W_{n}$. This set is a boundary for stable
cycling motion. The case of $n=1$ was defined in [2]. A more general and
rigorous analysis for any integer $n\geq 1$ was done by E. Garcia, G. Gomez in
an important paper [13]. This was further numerically and theoretically
studied by M. Gidea, F. Topputo, E. Belbruno in [7], with a general
numerically based result showing the equivalence of $W_{n}$ to a network of
intersecting invariant manifolds associated to Lyapunov orbits (see Section
3.3). $W$ does not exist on a fixed Jacobi energy surface, but rather for a
range of energies.

The general structure of $W$ has been an open problem. This paper determines
the geometry of this set in the case of infinite cycling and shows it is an
infinite family of Cantor sets. More precisely, this paper shows the
equivalency of the limit of $W_{n}$ as $n\rightarrow\infty$, labeled
$W^{\prime}$, with the boundary of a set $M^{*}$, $W^{\prime}=\partial M^{*}$,
and that $W^{\prime}$ is a Cantor set of hyperbolic points on each two-
dimensional Poincaré surface of section, $S_{\theta}$, for a Poincaré map
$\Phi$, parameterized by the polar angle $\theta\in[0,2\pi]$ with respect to
$P_{2}$, for each fixed Jacobi energy, $C$, and mass ratio, within a suitable
range. $W^{\prime}$ is an infinite union of Cantor sets of hyperbolic points
over all $\theta$, suitable Jacobi energies, and mass ratios.

The interior of $M^{*}$ corresponds to points yielding stable motion for
infinitely many cycles. $W^{\prime}$ represents points giving unstable motion
for infinitely many cycles. $W^{\prime}$ is the weak stability boundary for
infinitely many cycles of $P$ about $P_{2}$ (see Theorem 5.6).

It is remarked that $M^{*}$ shares a number of properties with that of the
classical Mandelbrot set even though they are defined much differently (see
Table 1). The main difference is that the boundary of $M$ is a continuous
fractal set, whereas the boundary of $M^{*}$ is totally discontinuous on each
two-dimensional surface of section.

It is also remarked that the existence of $W^{\prime}$ being an infinite union
of Cantor sets (see Section 3.3) clarifies a brief comment in [13](page 3)
alluding to the possibility the stable points themselves may be related to a
Cantor set, which is not entirely correct since it turns out it is the
unstable points (see Section 3.2).

The relationship of $M^{*}$ to the existence and non-existence of KAM tori is
discussed in Section 6 (Theorem 6.1). It says that for points of $\mathcal{P}$
on the boundary of $M^{*}$ then these cannot lie on KAM tori, while points on
the interior of $M^{*}$ may lie on KAM tori, but that is not known.

A general discussion of the results is given in Section 7.

The methodology of this paper is to give a theoretical framework of the
results, which is supported and motivated by numerous previous numerical
studies.

##  2\. Restricted Three-Body Problem

The two mass points $P_{1},P_{2}$, defined in the Introduction, are assumed to
move in mutual circular orbits of constant frequency, $\omega=1$, about their
common center of mass on a plane in a rotating coordinate system $y_{1},y_{2}$
of rotational frequency $\omega$, centered at $P_{1}$. The mass of $P_{1}$ is
$1-\mu$ and the mass of $P_{2}$ is $\mu$, $\mu$ is assumed to be small.
$P_{1},P_{2}$ are fixed on the $y_{1}$-axis at $(0,0),(1,0)$, respectively
(see Figure 1).222The center of mass between $P_{1},P_{2}$ is at $(\mu,0)$.
The gravitational constant is normalized to $1$. The particle $P$ of zero mass
moves on the $(y_{1},y_{2})$-plane. Its motion is given by ${\bf
y}(t)=(y_{1}(t),y_{2}(t))$ as a function of time, $t$, defined as a solution
to the system of differential equations,

(2.1) $\ddot{y}_{1}-2\dot{y}_{2}=\Omega_{y_{1}},\hskip
7.22743pt\ddot{y}_{2}+2\dot{y}_{1}=\Omega_{y_{2}},\hskip 7.22743pt,$

where

(2.2)
$\Omega=\frac{1}{2}[(y_{1}-\mu)^{2}+y_{2}^{2}]+\frac{1-\mu}{r_{1}}+\frac{\mu}{r_{2}}+\frac{1}{2}\mu(1-\mu),$

and $r_{1}^{2}=y_{1}^{2}+y_{2}^{2}$, $r_{2}^{2}=(y_{1}-1)^{2}+y_{2}^{2}$,
$\Omega_{x}\equiv\partial\Omega/\partial x$, ${}^{.}\equiv$ d/dt.

Figure 1. Rotating coordinate system $(y_{1},y_{2})$.

The Jacobi integral of the system is

(2.3) $J=J({\bf y},\dot{\bf{y}})=2\Omega-|\dot{\bf{y}}|^{2},$

$|\dot{\bf{y}}|^{2}=(\dot{y}_{1}^{2}+\dot{y}_{2}^{2})$. Thus, along a solution
${{\bm{\psi}}}(t)=({\bf{y}}(t),\dot{\bf{y}}(t))$,

(2.4) $J({\bm{\psi}}(t))=C,$

for a constant $C$, the Jacobi constant. Each value of $C$ defines a three-
dimensional energy surface in the four-dimensional phase space
$({\bf{y}},\dot{\bf{y}})$ that the solutions lie on, the Jacobi surface,
$\sigma=\sigma(C)=\\{({\bf{y}},\dot{\bf{y}})\in\mathbb{R}^{4}|J=C\\}$.

As $C$ varies, these surfaces have different geometric properties that
constrain the motion of $P$ in ${\bf y}$-space. This is obtained by projecting
the Jacobi surfaces onto ${\bf y}$-space, defining the Hill’s regions where
physical motion can occur. The Hill’s regions are defined by ${H}(C)=\\{{\bf
y}\in\mathbb{R}^{2}|2\Omega\geq C\\}$. $(\ref{eq:DEs})$ has five equilibrium
points, $L_{i},1=1,2,3,4,5$, the Euler-Lagrange points, where
$\dot{\bf{y}}=\ddot{\bf{y}}=\bf{0}$, where
$C=C_{i},C_{4}=C_{5}=3<C_{3}<C_{2}<C_{1}$. $L_{1},L_{2}$ are of interest to
this paper and shown in Figure 1.

The set $Z=\\{{\bf y}\in\mathbb{R}^{2}|2\Omega=C\\}$ defines the points of
zero velocity, which is the boundary curve of the Hill’s regions. When
$C>C_{1}$, trajectories cannot pass between $P_{1}$ and $P_{2}$ from the
respective disconnected Hill’s regions about $P_{1},P_{2}$, labeled
$H_{1},H_{2}$. There is also an outer Hill’s region about both $H_{1},H_{2}$,
disconnected from them, labeled, $H_{O}$. $P$ can pass between $H_{1}$ and
$H_{2}$ when $C_{2}<C\lessapprox C_{1}$.333$a\lessapprox b$ means $a<b$ and
$a$ is slightly less than $b$. Similarly for $a\gtrapprox b$. It goes through
a small channel region between $H_{1},H_{2}$, labeled, $\mathcal{C}_{1}$.
Similarly, when $C$ reduces further to $C\lessapprox C_{2}$, $P$ can also pass
though a small channel region between $H_{2}$ and $H_{O}$, labeled
$\mathcal{C}_{2}$.

Within the $\mathcal{C}_{i}$, $i=1,2$, are unstable retrograde periodic
orbits, $\gamma_{i}$, Lyapunov orbits, whose linearized flow has eigenvalues
$\lambda_{i}=\pm a_{i}\pm\sqrt{-1}\ b_{i}$, where $a_{i},b_{i},i=1,2$, are
positive real numbers (see [10, 11]). These periodic orbits have local two-
dimensional stable and unstable manifolds,
$W^{s}(\gamma_{i}),W^{u}(\gamma_{i}),i=1,2$, topologically equivalent to
cylinders. The trajectories that pass between the Hill’s regions move through
in interior of the manifolds, as shown in Figure 2. The trajectories
asymptotically spiral to $\gamma_{i}$ on $W^{s}(\gamma_{i})$, as
$t\rightarrow\infty$, and to $\gamma_{i}$ on $W^{u}(\gamma_{i})$ as
$t\rightarrow-\infty$. In the case where $C_{2}<C<C_{1}$, only $\gamma_{1},\
W^{s,u}(\gamma_{1})$ exist.

Figure 2. $W^{s,u}(\gamma_{i})$, $i=1,2$, and the flow of the trajectories
towards or away from $\gamma_{i}$, $C\lessapprox C_{2}$. $P$ is shown moving
between $H_{1},H_{2},H_{O}$ on transit orbits. This is an illustrative figure.

##  3\. Weak Stability Boundary and Related Sets

###  3.1. Basic Definitions

The definitions of the weak stability boundary are recalled from [7, 13, 2].

The center of the $(y_{1},y_{2})$ coordinate system is translated to $P_{2}$
by ${\bf y}\rightarrow{\bf Y}:\ Y_{1}=y_{1}-1,Y_{2}=y_{2}$, where
$Y_{1}=Y_{2}=0$. $P_{1}$ is located at $Y_{1}=-1,Y_{2}=0$. Let
${\bm{\psi}}(t)=({\bf{Y}}(t),\dot{\bf{Y}}(t))$ be a solution of the restricted
problem for $P$.

The Kepler energy between $P$ and $P_{2}$ is labeled $E_{2}$. It is a function
of $({\bf{Y}},\dot{\bf{Y}})$ and given by (A.2) in Appendix A.1. We define the
Kepler energy along the trajectory as $E_{2}(t)=E_{2}(\bm{\psi}(t))$.

The Jacobi surface, $\sigma$, defined for $P_{1}$-centered coordinates, is
labeled,
$\Sigma=\Sigma(C)=\\{({\bf{Y}},\dot{\bf{Y}})\in\mathbb{R}^{4}|\tilde{J}=C\\}$
in $P_{2}$-centered coordinates, where $\tilde{J}$ is $J$ in $P_{2}$-centered
coordinates given by (A.3).

Consider a line $L(\theta)$ extending from $P_{2}$ making an angle $\theta$,
$0\leq\theta\leq 2\pi$ with the $Y_{1}$-axis (see Figure 3). Choose initial
conditions for ${\bm{\psi}}(t)$ of (2.1), in $P_{2}$-centered coordinates,
along each point of $L(\theta)$ at an initial time, $t_{0}$. It is assumed
that for each initial point on $L(\theta)$, $P$ is at the osculating periapsis
of a Keplerian ellipse, where the velocity vector is perpendicular to
$L(\theta)$.

The magnitude $v$ of the velocity ${\dot{\bf{Y}}}$ at $t=t_{0}$ is determined
at each value of $r=|{\bf{Y}}|>0$ along $L(\theta)$, so that the initial
periapsis value has a given eccentricity $e$, $0\leq e<1$. In polar
coordinates,

(3.1) $v=(\mu(1+e)/r)^{1/2}-r.$

Thus, for these initial conditions, $E_{2}<0$. The corresponding initial
osculating periapsis value is $r=a(1-e)$ and $a$ is the initial semimajor
axis. The initial value of $E_{2}=\mu(e-1)/(2r)$ at $t=t_{0}$. Thus, each of
these initial conditions along $L(\theta)$ are the periapsis of an osculating
ellipse of eccentricity $e$ where $\dot{r}=0$. The initial points lie on the
set,

$\Lambda=\\{(r,\theta,\dot{r},\dot{\theta})\in\mathbb{R}^{4}|\dot{r}=0,E_{2}<0\\}.$

We assume posigrade motion.444The retrograde case is obtained by the symmetry
of solutions.

When $t$ increases from $t_{0}$, two types of cycling motions are defined:

Stable cycling motion is defined for integer $n$ cycles, $n\geq 1$, when
${\bf{Y}}(t)$ makes $n$ complete cycles about $P_{2}$, without going around
$P_{1}$, and $E_{2}<0$ on $L(\theta)$ on each cycle. It is assumed all
intersections of the stable cycling trajectory with $L(\theta)$ are
transverse.555The assumption of transverse intersections for stable cycling
motion is generally observed based on numerical evidence (see [7]).

Unstable cycling motion for $n$ cycles occurs when stable cycling for $n$
cycles does not occur.

Unstable cycling for $n$ cycles can occur when (i) ${\bf{Y}}(t)$ makes one
full cycle around $P_{1}$ prior to $n$ full cycles about $P_{2}$, cycling
around $P_{2}$ at most $n-1$ times (see Figure 3), $n\geq 1$. Other types of
unstable motion about $P_{2}$ can also occur when: (ii) $E_{2}\geq 0$ on
$L(\theta)$ on any cycle up to the $n$th cycle about $P_{2}$, (iii)
Intersections of the cycling trajectory on $L(\theta)$ are not transverse for
all cycles, (iv) $P$ does not return to $L(\theta)$ prior to $n$-cycles and
doesn’t cycle around $P_{1}$. This can occur, for example, for those
$n$-unstable trajectories that start on $L(\theta)$ and lie on
$W^{s}(\gamma_{i})$; these will neither cycle about $P_{2}$ or $P_{1}$. They
will asymptotically approach $\gamma_{i}$. It is numerically observed that (i)
is generic.

Stable cycling motion for $n$ cycles is denoted by $n$-stable motion, and
unstable cycling motion for $n$ cycles is denoted by $n$-unstable motion.
1-stable and 1-unstable motion are shown in Figure 3.

This procedure of determining $n$-stable and $n$-unstable cycling motion
defines an numerical algorithm that can be implemented on the computer (see
[13, 30, 7]). This is referred to as the W-algorithm.

Figure 3. Line $L(\theta)$ emanating from $P_{2}$ making an angle $\theta$
with the $Y_{1}$-axis. 1-stable and 1-unstable motion for $n=1$ cycles for $P$
on the trajectory ${\bf{Y}}(t)$. This is an illustrative figure.

It is noted that in the definition of stable cycling and unstable cycling,
when initial conditions are chosen for trajectories ${\bm{\psi}}(t)$ along
$L(\theta)$ for the W-algorithm, $C$ will vary. For initial conditions
sufficiently close to $r=0$, $C$ is arbitrarily large, and ${\bf{Y}}(t)$
remains in $H_{2}$ for all $t>t_{0}$ since $H_{1},H_{2}$ are disjoint. Thus,
in this case, ${\bf{Y}}(t)$ cannot move about $P_{1}$ and unstable motion by
going around $P_{1}$ is not possible.

It is remarked that the open condition $E_{2}<0$ is motivated for the sake of
applications since such a condition is used for capture of objects (e.g.
spacecraft) at $P_{2}$ [2]. It serves as a mathematical condition to define
stable cycling and as a normalized initial condition for both stable or
unstable cycling. The initial condition $\dot{r}=0$ for stable or unstable
cycling is used as a consistent normalizing condition, where other values of
$\dot{r}$ could have been used. Using the periapsis of an osculating ellipse
is well defined.

Sets of all points about $P_{2}$ leading to $n$-stable and $n$-unstable
motion, for a fixed $\theta,e,n$, can be defined. These are called the
n-stable set, n-unstable set, respectively. More precisely, the $n$-stable and
$n$-unstable sets on $L(\theta)$ for each $\theta\in[0,2\pi)$, $e\in[0,1)$ are
denoted by $S_{n}(\theta,e)$, $U_{n}(\theta,e)$, respectively.

For a fixed line, $L(\theta)$, and fixed $e,n$, the set $S_{n}(\theta,e)$ is
open, since the $n$-stability condition is open. This is because the condition
$E_{2}<0$ is an open condition, and if for some $r>0$ the motion is
$n$-stable, then there exists a small $\delta>0$ where it is also $n$-stable
in a neighborhood $(r-\delta,r+\delta)$ along $L(\theta)$ by smooth dependence
of solutions of the differential equations on initial conditions. Since
$S_{n}(\theta,e)$ is an open subset of $L(\theta)$, then topologically this
implies that $S_{n}(\theta,e)$ can be written as a countable union of disjoint
open intervals,

(3.2) $S_{n}(\theta,e)=\bigcup_{k\geq
1}(r^{*}_{2k-1},r^{*}_{2k})=(r^{*}_{1},r^{*}_{2})\cup(r^{*}_{3},r^{*}_{4})\cup\ldots,$

where $r^{*}_{1}=0,r>r^{*}_{1}$. The end points of the open intervals
$r^{*}_{j}$, $j=1,2,\ldots$, are $n$-unstable. The open intervals in (3.2),
$(r^{*}_{2k-1},r^{*}_{2k})$, are referred to as $I^{n}_{k},k\geq 1$ for
reference.

If a point is $m$-stable then it is $n$-stable for $n<m$, by definition. Thus,

(3.3) $S_{m}(\theta,e)\subset S_{n}(\theta,e),\ n<m.$

This implies that the more times $P$ cycles about $P_{2}$ in a stable way,
then generally fewer points are available to support that.

A change of stability across one of the end points of
$(r_{2k-1}^{*},r_{2k}^{*})$ means that for all $r\in(r_{2k-1}^{*},r_{2k}^{*})$
the motion is $n$-stable and there exists
$r^{*},r^{**}\notin(r_{2k-1}^{*},r_{2k}^{*})$ arbitrarily close to to
$r_{2k-1}^{*}$, $r_{2k}^{*}$, respectively, for which the motion is
$n$-unstable.

The $n$-weak stability boundary, $W_{n}$, is defined to be the union of all
$r^{*}(\theta,e)$ along $L(\theta)$ where there is a change of stability
between $n$-stable and $n$-unstable motion,

(3.4) $W_{n}=\\{r^{*}(\theta,e),\ \theta\in[0,2\pi],e\in[0,1)\\}.$

$W_{n}(\theta,e)$ is a slice of $W_{n}$. (3.4) can equivalently be written as
$W_{n}=\\{W_{n}(\theta,e),\ \theta\in[0,2\pi],e\in[0,1)\\}$. Figure 3 shows a
point, $r^{*}$, of $W_{1}(\theta,e)$.

(3.4) represents a union of $r^{*}(\theta,e)$ for the range of $\theta,e$
given. This is equivalent to

$W_{n}=\bigcup_{\theta\in[0,2\pi],e\in[0,1)}r^{*}(\theta,e)$

.

The notation for union in (3.4) is used throughout the paper for other sets as
well.

$W_{n}(\theta,e)$ is a discrete set of points for each fixed $\theta,e$ since
it’s points are the boundary points of disjoint open intervals,
$W_{n}(\theta,e)=\partial S_{n}(\theta,e)$. Therefore, taking the union over
all $\theta\in[0,2\pi],e\in[0,1)$ yields,

(3.5) $W_{n}=\partial S_{n}.$

$S_{n}=\\{S_{n}(\theta,e),\ \theta\in[0,2\pi],e\in[0,1)\\}$ is the general
$n$-stable set over all $\theta,e$.

It is remarked that (3.3) does not imply a similar property for
$W_{n}(\theta,e)$.

The following sets are defined by taking unions over different parameters,

(3.6) $S_{n}(\theta)=\\{S_{n}(\theta,e),\ e\in[0,1)\\},\
S_{n}(e)=\\{S_{n}(\theta,e),\ \theta\in[0,2\pi]\\},$ (3.7) $S(e)=\\{S_{n}(e),\
n\geq 1\\},\ S(\theta)=\\{S_{n}(\theta),\ n\geq 1\\},\ S=\\{S(\theta),\
\theta\in[0,2\pi]\\}$

In a similar way, $W_{n}(\theta),W_{n}(e),W(e),W(\theta),W$ are defined, as
well as for $U_{n}(\theta,e)$. (It is remarked that also
$S=\\{S(e),e\in[0,1)\\}$.)

$W$ is the general weak stability boundary and $S$ is the general stable set.
By construction, $C$ varies over $W,S$. Thus these sets are not invariant
under the flow of the restricted three-body problem. $W$ exists over a set of
Jacobi energy surfaces, $\Sigma(C)$ in four dimensional phase space.

Note, in $W(\theta)$ the points are obtained as $e$ varies continuously.
Therefore, the points of $W(\theta)$ are not discrete, whereas the points of
$W(\theta,e)$ are discrete.

$S_{n}(\theta,e),W_{n}(\theta,e)$ have been numerically computed for a fine
grid of $\theta\in[0,2\pi]$, for a range of $e$, $n$ and different $\mu$:

Numerical Results A

I. It is numerically observed that $S_{n}(\theta,e),W_{n}(\theta,e)$ should
exist for all finite $n$ and that (3.3) is also true (see [30], Figure 6,
$\mu=.00095$; $S_{n}(e)$, $n=1,2,...,8;\ e=0$, and Figure 7, $W_{n}(e),n=1;\
e=0,.2,.3,...,.95$; $n=1,2,...,8;\ e=0$, and see [13], Figures 1,3,
$\mu=.01215;\ e=0,.3,.6,.9;\ n=3,6$).

II. It is numerically observed that $S_{n}(e)$, is bounded by $W_{n}(e)$ (see
[30], Figure 4, $\mu=.00095;\ n=1;\ e=0$; and see [7], Figure 4, $\mu=.01$;
$W_{1}(e),\ e=.02,.6,.95$; [17], Figure 4, $\mu=.0000003$;
$e=.2,.4,.6,.9,.95$).

III. It is numerically observed that $C$ varies for different points of
$W_{n},S_{n}$. For example, the variation of $C$ over $S_{1}(e)$ for
$\mu=.00095$; $e=.95$ is shown in ([30], Figure 5).

The observed numerical results in [13, 30, 7, 17] are for finite values of $n$
not exceeding $n=8$. Computing these sets is a lengthy numerical process.
Visualizing these sets for larger values of $n$ and more $e$ values is not
necessary for the purposes of this paper. The general structure and how it
changes as $n$ increases and as $e$ approaches $1$ is generally understood.
For example, as $n$ increases, the sets have fewer points.

Notation in different papers \- It is important to note that in [30] and [7],
the symbol $\mathcal{W}$ is used in place of $S$ and $\mathcal{W}^{*}$ is used
in place of $W$. In [13], $\bar{\mathcal{W}}$ is used for $S$ and
$\partial{\bar{\mathcal{W}}}$ for $W$. In [17], $W_{n}$ is used for $S_{n}$
and $\partial W_{n}$ for $W_{n}$ as in this paper.

The sets of most relevance for this paper are $S(\theta),W(\theta)$, for a
fixed value of $\theta$, and other variations, such as
$S_{n}(\theta,e),W_{n}(\theta,e),S_{n}(e),W_{n}(e)$ and also $S,W$, which will
be clear from the context.

###  3.2. Sets $W$, $S$, $\hat{S}$, $W^{\prime}$ and their Properties

Properties of $W,S$ and related sets, $\hat{S}$, $W^{\prime}$ are described.

The following property is true from the basic definition of $W,S$.

######  Property 3.1.

$W(\theta)=\partial S(\theta)$. The trajectories with initial conditions on
$L(\theta)$ in $W(\theta)$ are $n$-unstable, $n\geq 1$.

It is numerically observed that $S_{n},W_{n}$ lie in a region about $P_{2}$
out roughly to the $L_{1,2}$ distance, then along the approximate distance of
$P_{2}$ to $P_{1}$, up to $90^{o}$ ahead and behind $P_{2}$ (see [30, 13, 7,
17]).

######  Property 3.2.

The trajectories of initial points on $L(\theta)$ in $S(\theta)$ cycle around
$P_{2}$ in an ’$n$-stable manner’ for $n\geq 1$, i.e. up to the $n$th cycle
about $P_{2}$ the trajectory intersects $L(\theta)$ transversally with
$E_{2}<0$.

Let $C_{3}\leq C_{a}<C<C_{2}$, where $C_{a}$ is the largest value of $C$ where
the Lyapunov orbits, $\gamma_{i}$, $i=1,2$, don’t collide with $P_{2}$ and the
channels $\mathcal{C}_{i}$ exist to allow $P$ to escape into $H_{1}$ or
$H_{O}$. For example, in [7], where $\mu=.01215$ the value of $C_{a}=3.15$ is
assumed, where, in that case, $C_{1}\approx 3.20034,C_{2}\approx
3.18416,C_{3}\approx 3.02415$.

######  Property 3.3.

$S(\theta)$ is open and bounded on $L(\theta)$ for $C_{a}<C<C_{1}$.

Proof -

Case 1: $C_{2}\leq C<C_{1}$ . $H_{2}$ has one channel $\mathcal{C}_{1}$ about
$L_{1}$ connecting $H_{2}$ with $H_{1}$ of width $\mathcal{O}(\mu^{1/3})$ for
$\mu$ sufficiently small [10, 20]. This enables unstable cycling about $P_{2}$
where $P$ can escape $H_{2}$ and eventually cycle about $P_{1}$.
$S_{n}(\theta)$ is bounded. This follows by first noting that $L(\theta)\cap
H_{2}$ is bounded for all $\theta\in[0,2\pi]$. Thus for any trajectory
starting on $L(\theta)$ on $\Lambda$, it will always be bounded, which
includes $n$-stable cycles. For $L(\theta)\cap H_{O}$, all trajectories with
initial conditions on $\Lambda$ that cycle about $P_{2}$ are $n$-unstable.
This follows since for any cycle about $P_{2}$, $P$ automatically makes a full
cycle about $P_{1}$ when it returns to $L(\theta)$.666If $C_{a}$ were allowed
to go below $C_{3}$, $P$ need not make a full cycle about $P_{1}$. This
implies $P$ cannot make $n$-stable cycles about $P_{2}$ in this case. Thus,
$S_{n}(\theta)$ is bounded for all $n$, and so is $S(\theta)$. This set is
also open since it is the infinite union over $n\geq 1$, $e\in[0,1)$ of open
sets, $S_{n}(\theta,e)$, and therefore open.

Case 2: $C_{a}\leq C<C_{2}$ . The only difference in this case is that another
channel opens up, $\mathcal{C}_{2}$, connecting $H_{2}$ to $H_{O}$. This proof
that $S(\theta)$ is bounded and open follows in a similar way to Case 1.

This proves Property 3.3.

It is noted that numerical results show $S$ is bounded from figures for the
stable sets for different $n,\ e$ (see, Numerical Results A, and, for example,
[30, 13]).

Also, numerically, it would be difficult to discern $S_{n}(\theta,e)$ on
$L(\theta)$ for large $n$. This follows since $S_{n}(\theta,e)$ can be written
as a countable union of open intervals and because of the property (3.3) of
$S_{n}(\theta,e)$, the $r^{*}$ values would be difficult to numerically
discern. This situation becomes more complicated for different values of
$e\in[0,1)$. Since the solutions of the differential equations vary
continuously as a function of the initial conditions on $L(\theta)$, then near
the intervals for a fixed $e=e_{0}$, as $e$ varies
$(e_{0}-\delta,e_{0}+\delta)$, $\delta$ small, the open intervals defining
$S(\theta,e)$ vary by a small amount. Thus, the $r^{*}(e)$ vary by a small
amount. This yields overlapping open intervals and boundary points for $e$
near $e_{0}$, which would be difficult to discern.

Infinite Cycling Stable Motion, $\hat{S}(\theta)$

It is of interest in this study to find a subset of points of
$S_{n}(\theta,e)$ on $L(\theta)$ where the trajectories cycle about $P_{2}$ in
a stable manner for all time, i.e. for all infinitely many cycles. This is
defined by,

(3.8) $\hat{S}(\theta,e)=\lim_{n\rightarrow\infty}S_{n}(\theta,e)\
=\bigcap_{n\geq 1}S_{n}(\theta,e).$

This is a set of all points starting on $L(\theta)\cap\Lambda$ where
trajectories through these points cycle infinitely often about $P_{2}$ in a
stable manner for all time. $C$ varies over this set.
$\hat{S}(\theta)=\\{\hat{S}(\theta,e),\ e\in[0,1)\\}$.

######  Property 3.4.

$\hat{S}(\theta,e)$ need not be open. Assuming it is non-empty, then it is
bounded for $C_{a}<C<C_{1}$.

Proof -

$S_{n}(\theta,e)$ form a sequence of open sets satisfying (3.3) that are
descreasing in size, $n=1,2,\ldots$. As $n\rightarrow\infty$, the limiting
intersection need not be open since the intersection of infinitiely many open
sets need not be open.

The boundedness follows from Property 3.3. This proves Property 3.4.

If $C$ is sufficiently large, then it can be analytically proven
$\hat{S}(\theta)$ is non-empty.

######  Theorem 3.5.

If $C$ is sufficiently large ($C\in C_{L}$) then $\hat{S}(\theta)$ is non-
empty.

Proof -

The existence of two-dimensional KAM tori about $P_{2}$ on each three-
dimensional energy surface $\Sigma(C)$ is proven in [19] for $C$ sufficiently
large, in a set $C_{L}=\\{C\ |\ C>M\\}$ for $M$ sufficiently large. For $C\in
C_{L}$, $H_{2}$ shrinks down to $r\gtrapprox 0$. Say, $0<r<\delta(M)$,
$\delta$ is small.

The tori separate the phase space on $\Sigma(C)$. Trajectories on the tori,
$\bm{\psi}(t)=({\bf Y}(t),\dot{\bf{Y}}(t))$, with initial conditions at say
$t=t_{0}$, have two frequencies of motion. One is rotational frequency
$\omega=1$, and the other is the osculating frequency $\omega^{*}(t,\mu)$
along the trajectory. To first order, $\omega^{*}(t,0)=\tilde{\omega}$ is a
constant, representing the elliptic Kepler frequency of motion for an ellipse
of constant semimajor axis $a\gtrapprox 0$ and constant eccentricity
$e\in[0,1)$. $\omega/\tilde{\omega}$ are sufficiently irrational satisfying
diophantine conditions. The motion between the tori is not sufficiently
irrational and chaotic.

Since the tori separate phase space, the trajectory $\bm{\psi}(t)$ for $P$ is
trapped between the tori for all time, where $a,e$ vary by $\mathcal{O}(\mu)$.
$a,e$ also vary by $\mathcal{O}(\mu)$ when $\bm{\psi}(t)$ lies on the
invariant tori. Thus, in the full three-dimensional region on $\Sigma(C)$
where the tori exist near $P_{2}$, the periapsis and apoapsis distances of $P$
from $P_{2}$ vary as $a(1-e)+\mathcal{O}(\mu),a(1+e)+\mathcal{O}(\mu)$,
respectively, for all time. In particular, $E_{2}(\bm{\psi}(t))<0$. The family
of tori transversally cut a surface of section $S_{\theta}$ (see (3.11)), in a
family of invariant curves.

The line $L(\theta)$ extending from $P_{2}$ is given by the $r$-axis of the
section. The trajectories transversally cut $L(\theta)$ as they cycle about
$P_{2}$ on or between the KAM tori.

This structure implies that trajectories moving about $P_{2}$ for all time for
$C\in C_{L}$ starting on $\Lambda$ will lie on or between the KAM tori. In
particular, trajectories with initial conditions on $\hat{S}(\theta)$ will
move on or betwen KAM tori. Thus, $\hat{S}(\theta)$ is non-empty.

This proves Theorem 3.5.

It is noted that numerical results show that $\\{S_{n}(\theta,e)\ |\ n\geq
1\\}$ is non-empty for the ranges of $C$ given in Property 3.4. (See Numerical
Results A). Although the numerical investigations only go as far as $n=8$, it
is seen that the figures of $S_{n}(\theta,e)$ appear consistent as $n$ varies,
and points of $S_{n}(\theta,e)$ are close to $P_{2}$. These suggest that
$S_{n}(\theta,e)$ is nonempty as $n\rightarrow\infty$, although this is not
proven. It is assumed that $\hat{S}(\theta,e)$ is nonempty for this paper. It
is out of the scope of this paper to numerically investigate the structure of
$\hat{S}(\theta,e)$ for this range of $C$, and is an interesting topic to
study further.

When $C\notin C_{L}$, the results from [19] on the use of the KAM theorem
cannot be used to study bounded motion for all time from initial values on
$\hat{S}$. However, interesting properties of $\hat{S}$ can be obtained for
$C_{a}<C<C_{1}$ without the use of KAM tori. This described in Section 3.4.

Notation Since $C$ varies over $\hat{S}(\theta)$, then points of
$\hat{S}(\theta)$ on $\tilde{J}=C$ can be denoted as,
$\hat{S}_{C}(\theta)=\hat{S}(\theta)\cap\Sigma(C)$. Similarly,
$W_{C}(\theta)=W(\theta)\cap\Sigma(C),\ S_{C}(\theta)=S(\theta)\cap\Sigma(C)$,
etc. The subscript $C$ can be omitted if the context is clear.

Properties of $W(\theta)$ and Infinite Cycling Unstable Motion,
$W^{\prime}(\theta)$

The set $W_{n}(\theta,e)$ is considered.

######  Property 3.6.

$W_{n}(\theta,e)$ has a countable set of points. $W(\theta,e)$ contains a
countably infinite number of points, and $W(\theta)$ has an uncountably
infinite number of points. $\dim W_{n}(\theta,e)=\dim W(\theta,e)=0$

Proof -

$W_{n}(\theta,e)$ contains a countable set of points, as follows from (3.2).
Thus, $W(\theta,e)$ contains a countably infinite number of points since it is
the union over $n\geq 1$.

Taking the union over all $e$, $W(\theta)=\\{W(\theta,e),\ e\in[0,1)\\}$,
implies that $W(\theta)$ contains an uncountably infinite number of points.

The dimension of $W_{n}(\theta,e)$ is zero since it contains a countable set
of points. Taking the union over all $n=1,2,\ldots$, $W(\theta,e)$ also has
zero dimension. This proves Property 3.6.

It is remarked that $\dim W(\theta)$ is not clear at this stage. Although
$\dim W_{n}(\theta,e)=0$, when the union is taken over all $e\in[0,1)$ and $C$
is restricted to a given range, further analyisis is needed.

It is noted that $W_{n}(\theta,e)$ may contain a countably infinite number of
points. This can happen when the lengths of the open sets,
$I^{n}_{j},j=1,2,\ldots$ comprising $S_{n}(\theta,e)$ converge to $0$.

######  Lemma 3.7.

$W(\theta)$ is bounded for $C_{a}<C<C_{1}$. $W(\theta)$ is closed assuming
uniform convergence of all sequences $r^{*}_{k}(\theta,e),\
e\in[0,1),k=1,2,\ldots.$

Proof -

$W(\theta)$ is bounded since $S(\theta)$ is bounded by Property 3.3 and
$W(\theta)=\partial S(\theta)$ by Property 3.1.

Since $W_{n}(\theta,e)$ is a countable discrete set, it is closed since its
boundary itself. This is also true for $W(\theta,e)$. It is not automatically
the case that $W(\theta)$ is closed by taking the infinite union over
$e\in[0,1)$ of closed sets, since that need not be closed.

$W(\theta)$ is closed if it contains all of its limit points. This is shown to
be the case by considering the closed set $W(\theta,e)$. Since this set is
closed, any convergent sequence of points $\\{r^{*}_{k}(\theta,e)\\}$ in
$W(\theta,e)$ converges to a limit point $\alpha(\theta,e)\in W(\theta,e)$, as
$k\rightarrow\infty$.

More precisely, given an $\epsilon>0$ there exists a $K(\theta,e,\epsilon)>0$,
bounded for each $e\in[0,1)$, such that for $k>K$,
$|r^{*}_{k}(\theta,e)-\alpha(\theta,e)|<\epsilon$, where $\alpha(\theta,e)\in
W(\theta,e)$. This is a pointwise convergence for each $e\in[0,1)$. For
$W(\theta)$ to be closed, this convergence should be uniform in $e$. That is,
$K$ is independent of $e$. This is achieved by setting
$\tilde{K}(\theta,\epsilon)=\sup_{e\in[0,1)}K(\theta,e,\epsilon)$ and assuming
$K(\theta,e,\epsilon)$ is uniformly bounded for all $e\in[0,1)$.

Thus, any convergent sequence of points $\\{r^{*}_{k}(\theta)\\}$ in
$W(\theta)$ converges to a limit point $\alpha(\theta)\in W(\theta)$ as
$k\rightarrow\infty$. This implies $W(\theta)$ is closed.

This proves Lemma 3.7.

The assumption that the sequences $r^{*}_{k}(\theta),\ k=1,2,\ldots$ converge
uniformly for $e\in[0,1)$ is suggested from numerical studies, but proving
this numerically requires computing $W_{n}(\theta,e)$ for high values of $n$
for $e\uparrow 1$, which is out of the scope of this study.

In an analogous manner to the definition of $\hat{S}(\theta,e)$, points common
with $W_{n}(\theta,e)$ for all $n\geq 1$ can be defined as,

(3.9)
$W^{\prime}(\theta,e)=\lim_{n\rightarrow\infty}W_{n}(\theta,e)=\bigcap_{n\geq
1}W_{n}(\theta,e)$

It is remarked that points in $W^{\prime}(\theta,e)$ give rise to
trajectories, ${\bm{\psi}}(t)$, that cycle about $P_{2}$ infinitely often as
$t\rightarrow\infty$. Since these initial points yield unstable cycling
motion, this means that, by definition, in the process of cylng about $P_{2}$,
${\bm{\psi}}(t)$ can also cycle about $P_{1}$, before returning to
$L(\theta)$, or $E_{2}\geq 0$ on at least one intersection with $L(\theta)$.
The possibility of $P$ not returning to $L(\theta)$ cannot occur in this case,
whether cycling around $P_{1}$ or not.

######  Property 3.8.

$W^{\prime}(\theta)=\partial\hat{S}(\theta)$. $W^{\prime}(\theta,e)$ is
countable. $W^{\prime}(\theta)$ is bounded. $W^{\prime}(\theta)$ is closed,
assuming uniform convergence of all sequences $r^{*}_{k}(\theta,e),\
e\in[0,1),k=1,2,\ldots$. $C_{a}<C<C_{1}$.

Proof -

$W^{\prime}(\theta)=\partial\hat{S}(\theta)$: Consider the line $L(\theta)$
for some $\theta\in[0,2\pi]$ and fixed $e\in[0,1)$. For a fixed $n\geq 1$,
$W_{n}(\theta,e)=\partial S_{n}(\theta,e)$ (see (3.5)). Taking the limit as
$n\rightarrow\infty$, yields

(3.10) $W^{\prime}(\theta,e)=\partial\hat{S}(\theta,e)$

as follows from (3.8), (3.9). (The boundary points of $\hat{S}(\theta,e)$,
that comprise $W^{\prime}(\theta,e)$, represent boundary points of the
remaining open intervals, $I^{n}_{j},j\geq 1$, as $n\rightarrow\infty$ of
$S_{n}(\theta,e)$. These points are the boundary points of the initial points
for trajectories that cycle about $P_{2}$ for all time in a stable manner.)

(3.10) is true for each $e\in[0,1)$. Thus,
$W^{\prime}(\theta)=\partial\hat{S}(\theta)$.

$W^{\prime}(\theta,e)$ is a countable set by Property 3.6 since
$W_{n}(\theta,e)$ is countable and the countable intersection of countable
sets is countable.

$W_{n}(\theta,e)$ is closed and bounded by Lemma 3.7. Thus,
$W^{\prime}(\theta,e)$ is closed since (3.9) is a countable intersection of
closed sets which is therefore closed. Property 3.4 implies it is bounded.
This is true for each $e\in[0,1)$. Thus, $W^{\prime}(\theta,e)$ is closed and
bounded for each $\theta\in[0,2\pi],e\in[0,1)$. $W^{\prime}(\theta)$ is closed
following a similar argument as in the proof of Lemma 3.7 for $W(\theta)$,
under the same assumption, and the boundedness of $W(\theta)$ implies the same
for $W^{\prime}(\theta)$.

This proves Property 3.8.

It is noted that $\dim W^{\prime}(\theta)$ is not clear. This is the case,
since from Property 3.6, $\dim W_{n}(\theta,e)=0$, and (3.9) implies $\dim
W^{\prime}(\theta,e)=0$. However, when the union is taken over $e\in[0,1]$ and
for $C_{a}<C<C_{1}$, the dimension of $W^{\prime}(\theta)$ requires further
analysis. The dimension of $W^{\prime}(\theta)$ is determined in Section 3.3.

Similar to $\hat{S}(\theta)$, $W^{\prime}(\theta)$ is defined over a range of
$C$. It is noted that the value of $C$ of a point $w\in W^{\prime}(\theta)$ is
in general different than the values of $C$ for points in $\hat{S}(\theta)$
adjacent to it that it bounds.

Definition The set of points of $W^{\prime}(\theta)$ that lie on the energy
surface $\Sigma(C)$ are given by
$W^{\prime}_{C}(\theta)=W^{\prime}(\theta)\cap\Sigma(C)$.

It is noted that $\hat{S}\subset\Lambda,\ W^{\prime}(\theta)\subset\Lambda$.

Dynamical Summary A Trajectories, $\bm{\psi}(t)$, with initial conditions at
$t=0$ on $\hat{S}(\theta),W^{\prime}(\theta)$ satisfy,

a.) $\hat{S}(\theta)$ defines initial points at $t=0$ on
$L(\theta)\cap\Lambda$ for trajectories cycling $P_{2}$ infinitely often, for
all $t>0$, in a ’stable manner’ ($E_{2}<0$ on all intersections with
$L(\theta)$ and are transversal where $\dot{r}$ need not be zero).

b.) $W^{\prime}(\theta)$ defines points on $L(\theta)\cap\Lambda$ that bound
$\hat{S}(\theta)$, and initial points for trajectories cycling $P_{2}$ for all
$t>0$ in an ’unstable manner’ (For example, $E_{2}\geq 0$ on an intersection
with $L(\theta)$), or intersections with $L(\theta)$ can be nontransversal, or
the trajectory cycles about $P_{1}$ before returning to $L(\theta)$.)

It is noted that the open sets $I^{n}_{k}$ comprising $S_{n}(\theta,e)$ change
for different $n$ and $e$. It was briefly noted in [13] that $S_{n}(\theta,e)$
suggestive of a Cantor set. It turns out that the points of $S_{n}(\theta,e)$
are not in a Cantor set, but rather the boundary points $W^{\prime}(\theta)$
of $\hat{S}(\theta)$ . Topologically, these are the boundary points of
$\lim_{n\rightarrow\infty}S_{n}(\theta)$. This is proven in Section 3.3 using
a main result in [7] on relating $W_{n}(\theta)$ to the invariant manifolds
$W^{s}(\gamma_{i})\ i=1,2$.

###  3.3. $\bm{W_{n}(\theta)}$, $\bm{W^{\prime}(\theta)}$, Invariant
Manifolds, Cantor Sets

The relationship of $W_{n}(\theta)$ to invariant manifolds is described.

Three cases for the range of the Jacobi integral are considered.

Case 1: $C_{1}\leq C$, Case 2: $C_{2}\leq C<C_{1}$, Case 3: $C_{a}<C<C_{2}$.
It is assumed that $\mu>0$ is small, $\mu\gtrapprox 0$.

A Poincaré surface of section $S_{\theta}$ of two dimensions can be defined on
$\Sigma(C)\cap H_{2}$ for the flow of the differential equations in Cases
1,2,3. $S_{\theta}$ is defined on the surface $\Sigma(C)$ for each fixed
$\theta\in[0,2\pi]$ and for each fixed $C$. The section has coordinates
$(r,\dot{r})$, where $\dot{\theta}>0$. It is given by,

(3.11) $S_{\theta}=\\{(r,\dot{r})|\ \theta=\theta_{0},\
\dot{\theta}=\dot{\theta}(r,\dot{r},\theta)>0,r>0\\}$

(see Appendix A.3).777$S_{\theta}$ not to be confused with $S(\theta)$.

Notation $S_{\theta}$ is used for a range of $C$ values or a single $C$ value
which will be clear from the context. Sometimes the notation, $S_{\theta}(C)$,
is used for a particular $C$ value.

The flow of the differential equations defines a two-dimensional Poincaré map,
$\Phi$, on $S_{\theta}\cap\Sigma(C)$ for each $C$ in the given range, and each
$\mu\gtrapprox 0$,

$\Phi:S_{\theta}\rightarrow S_{\theta}.$

Case 1 - $H_{2}$ lies within the distance to $L_{1},L_{2}$. $\gamma_{i},\
i=1,2$ do not exist, and $\mathcal{C}_{i}$ do not exist for each fixed energy
surface $\Sigma(C)$ in the $C$ range. The range of $C\in C_{L}$ in Theorem 3.5
is a subset of this case.

Case 2 - $P$ can escape $H_{2}$ and move between $H_{1},H_{2}$ through
$\mathcal{C}_{1}$. For each fixed $C$ in the range, the invariant manifolds
$W^{s,u}(\gamma_{1})$ lie on $\Sigma(C)$ and they extend into $H_{2}$.

The first intersection (cut) of $W^{s}(\gamma_{1})$ and $W^{u}(\gamma_{1})$ on
$S_{\theta}(C)$ are topological circles. If the manifolds have a tranvserse
intersection, then the topological circle breaks up on the section (see [14])
under the flow of the differential equations. It is assumed the trajectories
satisfy topological properties (see Section A.4) and for details, see [7].

After infinitiely many intersections of $W^{s}(\gamma_{1})$ and
$W^{u}(\gamma_{1})$ on $S_{\theta}(C)$, a hyperbolic invariant set is obtained
resulting from the intersections. The existence of the hyperbolic invariant
set on $S_{\theta}(C)$ follows by the Smale-Birkhoff theorem (see [24, 15]).
This hyperbolic set is a Cantor set of hyperbolic points. This set is labeled,
$\mathcal{C}_{C}(\mu)$, or just $\mathcal{C}_{C}$ for brevity. If the union is
taken over a range of $C$ in $[C_{2},C_{1})$ and $\mu\gtrapprox 0$, in the
$(C,\mu)$-plane where the manifolds have transverse intersection, then the
symbol $\mathcal{C}$ is used. This represents a union of Cantor sets lying on
a set of energy levels $\Sigma(C)$, for $(C,\mu)$.

Let $p\in\mathcal{C}_{C}$. $p$ has one-dimensional stable and unstable
manifolds on $S_{\theta}(C)$. These manifolds intersect at all the hyperbolic
points. This creates a complicated network of hyperbolic points and manifolds
(tangle). $\mathcal{C}_{C}$ is closed, self-similar, totally disconnected, and
has zero topological dimension on $\Sigma(C)$, for each $(C,\mu)$ where
transversality occurs.

$\Phi:\mathcal{C}_{C}\rightarrow\mathcal{C}_{C}$ is chaotic on
$\mathcal{C}_{C}$ for each $C$ with a dense set of periodic points ($q$ a
periodic point implies $\Phi^{m}(q)=q$, for some $m>0$). There also exists a
dense orbit, $\\{\Phi^{k}(\tilde{p}),\ k\geq 0\\}$ for some $\tilde{p}$
($\Phi^{0}(p)=p$, $\Phi^{2}(p)=\Phi\circ\Phi(p)$).

What can be said about the occurance of transversality of
$W^{u}(\gamma_{1}),W^{s}(\gamma_{1})$ in $H_{2}$ in this case?

An analytic proof (see [20]) answers this in the $H_{1}$ region, where it is
proven that for a discrete set of $C\lessapprox C_{1},\ \mu\gtrapprox 0$,
$\\{(C_{k},\ \mu_{k}),\ k=1,2,\ldots\\}$,
$W^{u}(\gamma_{1}),W^{s}(\gamma_{1})$ indeed have transverse homoclinic
intersection. $(C_{k},\mu_{k})$ are numerically determined for a set of $k$ in
[20].

It is reasonable to assume that these manifolds have transversal intersection
in $H_{2}$. In fact, as is described in Case 3, transversal heteroclinic
intersections of invariant manifolds are proven to exist when both channels
exist about $L_{1},L_{2}$, which is a more general situation than considered
in Case 2.

Case 3 - When $C$ is less than $C_{2}$, $P$ can move through
$\mathcal{C}_{1},\mathcal{C}_{2}$. Also, $\gamma_{1},\gamma_{2}$ grow in size.

Instead of considering a transverse homoclinic intersection of
$W^{s}(\gamma_{1}),W^{u}(\gamma_{1})$ in $H_{2}$, it can be proven that there
is a transverse heteroclinic intersection between $W^{s}(\gamma_{1})$ and
$W^{u}(\gamma_{2})$ in $H_{2}$, which by symmetry of solutions under the map,
$Y_{2}\rightarrow-Y_{2},\ \dot{Y}_{1}\rightarrow-\dot{Y}_{1}$, also gives
transverse intersection between $W^{u}(\gamma_{1})$ and $W^{s}(\gamma_{2})$.

This heteroclinic intersection is proven to exist, in a semi-analytic proof
(numerically assisted) in the interesting paper by J. Marsden, et. al. [18].
It is done in the case of $P_{1}=$Sun, $P_{2}=$Jupiter, where $\mu=.00095$,
and for specific $C\lessapprox C_{2}$ ($C=3.037$, $C_{2}=3.03836$). It is
shown that $W^{s}(\gamma_{1})$ and $W^{u}(\gamma_{2})$ transversally intersect
on a surface of section. This choice of $C,\mu$ is motivated by the trajectory
for the resonance transitioning comet Oterma studied in [18]. As noted in
[18], their analysis is valid for other resonant transition comets about the
Sun, for the same $\mu$, but different $C\lessapprox C_{2}$.888A list of many
other similar resonance comets about the Sun in resonance with Jupiter is
given in [4, 6]. This yields other values of $C$, for the same $\mu$, where
transversality of $W^{s}(\gamma_{1}),W^{u}(\gamma_{1})$ would similarly exist.
More generally, it is stated in [18] that it was numerically observed that
transverse intersection of the manifolds occurs for many values
$C_{a}<C<C_{2},\mu\gtrapprox 0$.

Thus, it can be assumed, based on semi-analytic analysis, that heteroclinic
intersections exist for set of $C_{a}<C<C_{2},\mu\gtrapprox 0$ in the
$(C,\mu)$-plane.

As in Case 2, there exists an invariant Cantor set for $\Phi$, also labeled
$\mathcal{C}_{C}$, with the properties given in Case 2. $\mathcal{C}_{C}$
exists on each three-dimensional energy surface, $\Sigma(C)$ for set of
$C_{a}<C<C_{2},\mu\gtrapprox 0$.

Definition Let $I_{C}\subset I=\\{C\ |\ C_{a}<C<C_{1}\\}$ and $I_{\mu}\subset
I_{\delta}=\\{\mu\ |\ 0<\mu<\delta\\}$, where $\delta\ll 1$ is sufficiently
small, be sets in the $(C,\mu)$-plane, where, within $C_{2}\leq C<C_{1}$,
$W^{s}(\gamma_{1}),W^{u}(\gamma_{1})$ in $H_{2}$ have transverse homoclinic
intersection; and for those values within $C_{a}<C<C_{2}$,
$W^{s}(\gamma_{1}),\ W^{u}(\gamma_{2})$ in $H_{2}$ have transverse
heteroclinic intersection. This set of $(C,\mu)$ values is labeled
$V_{C,\mu}=\\{C\in I_{C},\mu\in I_{\mu}\\}$.

Relationship of $W_{n}$ to $W^{s}(\gamma_{i}),i=1,2$

The main result of [7] is that $W_{n}$ lies on $W^{s}(\gamma_{i}),i=1,2$,
under some conditions. Assumptions are necessary on the trajectories space,
described in Appendix A.4, referred to as Hypothesis A. The main technical
result is recalled,

Main result in [7]

Let $W_{n}^{A}$ be the union of the $n-1$ stable manifold intersections with
$S_{\theta}$ on $\Lambda$, for all $\theta\in[0,2\pi]$ for each $C\in I,\mu\in
I_{\delta}$, then taking the union over all $C\in I$ for each fixed $\mu\in
I_{\delta}$, where it assumed the trajectories satisfy Hypothesis A. Then,
$W_{n}^{A}=W_{n}$.

The restriction of the trajectories satisfying Hypothesis A is necessary since
this set of trajectories has the same properties as those used in the
$W$-algorithm to define $W_{n}$. This result in [7] does not necessarily
assume $C\in I_{C},\mu\in I_{\mu}$. More precisely,

(3.12)
$W_{n}=W_{n}^{A}=\\{(r,\dot{r},\theta,\dot{\theta})\in\\{W^{s}_{\theta,n-1}(\gamma_{1})\cup
W^{s}_{\theta,n-1}(\gamma_{2})\\}\cap\Lambda,\ C\in I,\ \theta\in[0,2\pi]\\},$

for each fixed $\mu\in I_{\delta}$, where $W^{s}_{\theta,n-1}(\gamma_{i})$ is
the $(n-1)$st intersection of $W^{s}(\gamma_{i})$ with $S_{\theta}$. Fixing
$\theta$ defines $W_{n}(\theta)$. All trajectories satisfy Hypothesis A.

It is noted that there are trajectories that don’t satisfy Hypothesis A (see
[7]. It is therefore assumed that these are omitted from the trajectory space
in (3.12).

Fixing $\theta$, (3.12) implies,

######  Lemma 3.9.

(3.13) $W_{n}(\theta)=W_{n}^{A}(\theta).$

for $C\in I$ for each fixed $\mu\in I_{\delta}$.

This equates the $W_{n}(\theta)$ points from the $W$-algorithm with the points
$W_{n}^{A}$ of the stable manifolds.

Since $W(\theta)=\\{W_{n}(\theta),\ n\geq 1\\}$, and
$W^{A}(\theta)=\\{W_{n}^{A}(\theta),n\geq 1\\}$ then (3.13) implies
$W(\theta)=W^{A}(\theta)$.

######  Corollary 3.10.

Assume $C\in I_{C}$, then for each $\mu\in I_{\mu}$,

(3.14)
$W^{\prime}(\theta)=\lim_{n\rightarrow\infty}W^{A}_{n}(\theta)=\mathcal{C}\cap\Lambda$

on $S_{\theta}$. $\mathcal{C}$ is a union for Cantor sets for each $\mu$.

Proof - (3.13) relates $n$ cycles of the manifolds $W^{s}(\gamma_{i}),\
i=1,2$, about $P_{2}$, intersecting $S_{\theta}$, on the right hand side of
(3.13), to $W_{n}(\theta)$ for $n$ cycles of trajectories about $P_{2}$
starting on $\Lambda$ on the left hand side of (3.13). This is true for each
$\mu\in I_{\delta}$ and for all $C\in I$.

Letting $n\rightarrow\infty$ for infinite cycling, the left hand side of
(3.13) yields $W^{\prime}(\theta)$ and the right hand side of (3.13) yields
the limit set $\mathcal{C}$ on $\Lambda\cap S_{\theta}$ assuming $C\in
I_{C}\subset I,\ \mu\in I_{\mu}\subset I_{\delta}$. That is, for $(C,\mu)\in
V_{C,\mu}$.

This proves Corollary 3.10.

It is remarked that by Corollary 3.10, for a fixed $C$ and $\mu$,
$W_{C}^{\prime}(\theta)=\mathcal{C}_{C}\cap\Lambda\cap S_{\theta}(C)$.

Since $\mathcal{C}_{C}$ is a Cantor set, and
$\mathcal{C}_{C}\cap\Lambda\subset\mathcal{C}_{C}$, then $\dim
W^{\prime}_{C}(\theta)=0$. This asnswers the question on the dimension of
$W^{\prime}(\theta)$ after the proof of Property 3.8.

Numerical results demonstrate that $W_{n}^{A}(\theta)$ is non-empty [7]. Thus,
by Corollary 3.10, $W^{\prime}(\theta)$ is assumed to be non-empty. It would
be interesting to numerically visualize the Cantor set structure of
$W^{\prime}(\theta)$. This is beyond the scope of this paper.

Definition $W^{\prime}(\theta)$ is the weak stability boundary for infinitely
many cycles of $P$ about $P_{2}$.

Topological picture

The Cantor set $\mathcal{C}_{C}$ exists on $\Sigma(C)$ for each $C$ in $I_{C}$
and for each $\mu\in I_{\mu}$. It exists on the two-dimensional section
$S_{\theta}$, where both $r,\dot{r}$ vary. The section lies in the three-
dimensional space $\Sigma(C)$. $\mathcal{C}_{C}$ is zero-dimensional. Taking
the union for $C\in I_{C}$, yields a set of Jacobi energy surfaces and a set
$\mathcal{C}$ of Cantor sets, belonging to $S_{\theta}\cap\Sigma$,
$\Sigma=\\{\Sigma(C),C\in I_{C}\\}$.

$W_{C}^{\prime}(\theta)\subset\mathcal{C}_{C}$,
$W^{\prime}(\theta)\subset\mathcal{C}$. The points of $W_{C}^{\prime}(\theta)$
and $W^{\prime}(\theta)$ lie on the $r$-axis of $S_{\theta}$, with $E_{2}<0$.
$W_{C}^{\prime}(\theta)$ is zero-dimensional.

The next result shows that $W_{C}^{\prime}(\theta)$ is a Cantor set,

######  Lemma 3.11.

$W_{C}^{\prime}(\theta)$ is a Cantor set for each $\mu\in I_{\mu}$.
$W^{\prime}(\theta)$ is a union of Cantor sets over $C\in I_{C}$.
$W^{\prime}=\\{W^{\prime}(\theta),\theta\in[0,2\pi]\\}$ is an infinite union
of Cantor sets.

Proof-

From the remark following Corollary 3.10, $W^{\prime}_{C}(\theta)$ is a subset
of a Cantor set. $W_{C}^{\prime}(\theta)$ is closed by Property 3.8. This
implies $W^{\prime}_{C}(\theta)$ is a Cantor set. Thus $W^{\prime}(\theta)$ is
a union of Cantor sets. $W^{\prime}$ is an infinite union of Cantor sets taken
over $\theta\in[0,2\pi]$.

######  Lemma 3.12.

$W^{\prime}_{C}(\theta)$ is not invariant under $\Phi$ .

This is proven by first noting that since
$W^{\prime}_{C}(\theta)\subset\mathcal{C}_{C}$, it consists of hyperbolic
points on $S_{\theta}(C)$ for $\Phi$. These points lie on the $r$-axis, where
$E_{2}<0$. Although $\mathcal{C}_{C}$ is an invariant set for $\Phi$,
$W_{C}^{\prime}(\theta)$ need not be an invariant set for $\Phi$. This is the
case since if $p\in W_{C}^{\prime}(\theta)$, then the orbit of iterates,
$\\{\Phi^{k}(p),\ k\geq 0\\}$ may remain in $W^{\prime}_{C}(\theta)$ for a
finite number of iterations, where $\dot{r}=0,E_{2}<0$ and then leave
$W^{\prime}_{C}(\theta)$ with $\dot{r}\neq 0$ or $E_{2}\geq 0$ and lie in
$\mathcal{C}-W^{\prime}_{C}(\theta)$.

Assumption It is assumed $\mu\in I_{\mu}$ is fixed for the remainder of this
paper, unless otherwise indicated.

###  3.4. An Invariant Set for $\bm{\Phi}$

In this section an invariant set for $\Phi$ is obtained by considering
$\hat{S}(\theta)$.

$W^{\prime}(\theta)$ and $\hat{S}(\theta)$ exist on the $r$-axis of
$S_{\theta}\cap\Lambda$, for $C\in I_{C}$ and $C\in\hat{I}_{C}\subset I$,
respectively. $\hat{S}(\theta)$ is bounded by Cantor points of
$W^{\prime}(\theta)$, where $C$ varies throughout the points of
$\hat{S}(\theta)$ and $W^{\prime}(\theta)$. Thus, the points of these sets lie
on different energy surfaces $\Sigma(C)$.

Since the points of $W^{\prime}(\theta)$ are the boundaries of points in
$\hat{S}(\theta)$, then there exists values of $C$ within $\hat{I}_{C}$ that
will lie arbitrarily close to values of $C$ in $I_{C}$. It would be
interesting to numerically explore the distribution of points of
$\hat{S}(\theta)$ and $W^{\prime}(\theta)$ on the $r$-axis of the section.
This is beyond the scope of this paper.

It is recalled that if a trajectory ${\bm{\psi}}(t)$ has an initial condition
on $\hat{S}(\theta)$ (where $\dot{r}=0,E_{2}<0$) then it cycles around $P_{2}$
for all $t>0$. On each cycle, it transversally intersects $S_{\theta}$ with
$E_{2}<0$, but not necessarily $\dot{r}=0$. This implies,

######  Lemma 3.13.

$\hat{S}(\theta)$ is not invariant under $\Phi$.

It is noted that ${\bm{\psi}}(t)$ has two frequencies of the motion when
cycling about $P_{2}$. One is $\omega=1$ for the rotating frame. The other is
$\tilde{\omega}(t)$ along the trajectory. $\tilde{\omega}(t)$ can have
significant variation as a function of $t$.

Definition The set of trajectories for initial conditions in
$\hat{S}_{C}(\theta)$ move in a bounded region $T_{C}$ on $\Sigma(C)$ about
$P_{2}$, for the given values of $C$. $T=\\{T_{C},\ C\in\hat{I}_{C}\\}$ is the
union of the $T_{C}$.

Although $\hat{S}(\theta)$ isn’t invariant under $\Phi$, an invariant map can
be constructed for a set related to $\hat{S}(\theta)$.

An Invariant Set $S^{*}(\theta)$ for $\Phi$

The set is defined,

(3.15) $S_{C}^{*}(\theta)=\left\\{\\{\Phi^{k}(p),k\geq 0\\},\
p\in\hat{S}_{C}(\theta)\right\\},$

$\hat{S}_{C}(\theta)\subset S_{C}^{*}(\theta)$. Each point $p\in
S_{C}^{*}(\theta)$ lies on $\Sigma(C)\cap S_{\theta}$.
$S^{*}(\theta)=\\{S^{*}_{C}(\theta),\ C\in\hat{I}_{C}\\}$.

For a point $p\in\hat{S}(\theta)$, $\dot{r}=0,E_{2}<0$ and for all subsequent
iterates of $p$, $E_{2}<0$ since the trajectory has stable motion relative to
$L(\theta)$.

Thus,

######  Lemma 3.14.

$S^{*}(\theta)$ is invariant under $\Phi$,

(3.16) $\Phi:S^{*}(\theta)\rightarrow S^{*}(\theta),$

defined on $S_{\theta}$ for each $C\in\hat{I}_{C}$, on $\Sigma(C)$.
$S^{*}(\theta)$ consists of points $p=(r,\dot{r})$ with $E_{2}<0$ on
$S_{\theta}$. $\hat{S}(\theta)$ consists of those points of $S^{*}(\theta)$
with $\dot{r}=0$. $T$ is formed from all the trajectories starting in
$S^{*}(\theta)$.

It is noted that $W^{\prime}(\theta)$ bounds $\hat{S}(\theta)$ but
$\mathcal{C}(\theta)$ does not necessarily bound $S^{*}(\theta)$. The points
of $W^{\prime}(\theta)$ bound the points of $\hat{S}(\theta)$ since the points
of $W_{n}(\theta)$ bound the points of $S_{n}(\theta)$. The points of both of
these sets satisfy $\dot{r}=0,E_{2}<0$ and lie on the $r$-axis of
$S_{\theta}$. This is not satisfied in general for $S^{*}(\theta)$. The points
of $S^{*}(\theta),\ \mathcal{C}$ are generally not on the $r$-axis of
$S_{\theta}$. Thus, the bounding of $S^{*}(\theta)$ by $\mathcal{C}$ doesn’t
necessarily follow. The relationship between $S^{*}(\theta)$ and
$\mathcal{C}(\theta)$ is not clear and whose study is beyond the scope of this
paper.

##  4\. Mappings and Properties

The following assumptions are made,

Assumptions A The energy surface $\Sigma(C)$ is considered and a section
$S_{\theta}$, $S_{\theta}\cap\Sigma(C)$, $C\in I$ is fixed,
$\theta\in[0,2\pi]$ is fixed, $\mu\in I_{\mu}$ is fixed.

######  Lemma 4.1.

$\Phi$ is real analytic on $S_{\theta}$. In particular it is real analytic on
$\mathcal{C},S^{*}(\theta)$, $W^{\prime}(\theta),\hat{S}(\theta)$,

Proof -

Under Assumptions A, $\Phi$ is a real analytic map at each point of
$S_{\theta}$ and in particular at each point of $\mathcal{C}\subset
S_{\theta}$, $C\in I_{C}$, and therefore on
$W^{\prime}(\theta)\subset\mathcal{C}$. This follows from the real analyticity
of the solutions $\bm{\psi}(t)$ for the differential equations as a function
of initial conditions on $S_{\theta}$ and $t$. Similarly, $\Phi$ is real
analytic at each point of $S^{*}(\theta)$, $C\in\hat{I}_{C}$, and therefore at
each point of $\hat{S}(\theta)\subset S^{*}(\theta)$.

Mapping of $\mathcal{C}(\theta),S^{*}(\theta)$

$S^{*}(\theta)$ is invariant under $\Phi$ on $S_{\theta}\cap\Sigma(C)$,
$C\in\hat{I}_{C}$. $\mathcal{C}$, is invariant under $\Phi$, on
$S_{\theta}\cap\Sigma(C)$, $C\in I_{C}$ (see Figure 4). Trajectories with
initial values in $S^{*}(\theta)$ cycle about $P_{2}$ for all time in a stable
manner. By Lemma 3.14, this set of trajectories is given by $T$,
$S^{*}(\theta)=T\cap S_{\theta}$. The trajectories of $T$ lie in an three-
dimensional annular region about $P_{2}$ on $\Sigma(C)$. Trajectories that
have initial points in $\mathcal{C}$ are dynamically unstable since they have
two-dimensional stable and unstable manifolds.

Figure 4.  $\hat{S}(\theta),\ S^{*}(\theta),\ W^{\prime}(\theta),\
\mathcal{C},\text{on}\ S_{\theta}\subset\Sigma(C)$, $C\in I$.

Mapping of $W^{\prime}(\theta),\hat{S}(\theta)$

The points of $W^{\prime}(\theta),\hat{S}(\theta)$ lie on
$S_{\theta}\cap\Lambda$, or equivalently on $L(\theta)$ for all $C\in
I_{C},C\in\hat{I}_{C}$, respectivly.

For each fixed $C$, the iterates of $\Phi$ on
$W^{\prime}(\theta),\hat{S}(\theta)$, are mapped to points of
$\mathcal{C}(\theta),S^{*}(\theta)$, respectively, as seen in Figure 4.

Thus, in summary,

$\mathcal{C},S^{*}(\theta)$ are invariant under $\Phi$ and
$W^{\prime}(\theta)\subset\mathcal{C},\ \hat{S}(\theta)\subset S^{*}(\theta)$
are not invariant under $\Phi$. $C\in I_{C}$ for points in $\mathcal{C}$, and
$C\in\hat{I}_{C}$ for points in $S^{*}$. $\mu\in I_{\mu}$ is fixed.

##  5\. Bounded Regions Defined by $\bm{W^{\prime}},\bm{\hat{S}}$

$W^{\prime}(\theta)$, $\hat{S}(\theta)$ are previously described on
$S_{\theta}\cap\Lambda$ using both manifolds and the W-algorithm. These sets
equivalently exist on the line $L(\theta)$ satisfying $\dot{r}=0,E_{2}<0$.
That is, they can be described on the $Y_{1},Y_{2}$-plane for the given value
of $\theta$ along $L(\theta)$. As previously described one gets the limit of
the sets $W_{n}(\theta)=\\{W_{n}(\theta,e),\ e\in[0,1)\\}$ and
$S_{n}(\theta)=\\{S_{n}(\theta,e),\ e\in[0,1)\\}$ for $n\rightarrow\infty$ as
$W^{\prime}(\theta),\ \hat{S}(\theta)$, respectively. Taking the union over
$\theta\in[0,2\pi]$, yields $W^{\prime},\ \hat{S}$, respectively.

$W_{n}(e),S_{n}(e)$ are described in Numerical Results A for various $n$, $e$.
They show a complex pattern. One sees this clearly in [30] (Figure 4 for
$n=1$, $e=0$). It can be seen that $S_{1}(0)$ fills in a large region about
$P_{2}$, and many smaller regions, whose boundary is $W_{1}(0)$. Many of the
smaller regions are isolated islands. As indicated by other figures in [30],
described in Numerical Results A, as $n$ increases the sets decrease on size,
seen clearly in [30] where $1\leq n\leq 8$. Different values of $e$ show a
similar structure. This is also observed in [13, 17]. These figures suggest
that as $n\rightarrow\infty$, $W^{\prime},\hat{S}$ decrease in size
significantly for $C\in I_{C},\ C\in\hat{I}_{C}$, respectively. For a fixed
$\theta$, $W^{\prime}(\theta)$ is a union of Cantor sets, on each respective
$\Sigma(C)$, that would be hard to discern numerically, as would the set
$\hat{S}(\theta)$, especially if the different $C$ were close in value. See
Figure 5 which is just a rough sketch (not a numerical simulation) for all
$\theta\in[0,2\pi]$.

A numerical investigation of $W^{\prime},\hat{S}$ for large values of $n$ is
beyond the scope of this paper.

A Parametric Plane and an Iteration Plane

The points of $W^{\prime},\ \hat{S}$ can be viewed in the ${\bf
Y}=(Y_{1},Y_{2})$-plane. Each of these sets are defined by first varying a
position in the $(Y_{1},Y_{2})$-plane along $L(\theta)$, for a value of
$\theta$, then using the W-algorithm to see which points exists along
$L(\theta)$ after infinitely many cycles from $W_{n}(\theta)$ and
$S_{n}(\theta)$. Since $\dot{r}=0$, then the other velocity $\dot{\theta}$
component (in polar coordinates) needs to be suitably adjusted satisfying
$E_{2}({\bf Y},\dot{\bf{Y}})<0$ that yields a desired value of $e\in[0,1)$
(see (A.2)).

Each point of $W^{\prime},\ \hat{S}$ in the $(Y_{1},Y_{2})$-plane,
$P_{Y_{1},Y_{2}}$, has associated to it a value of $E_{2}<0$ and satisfies,
$\dot{r}=0$. Thus, the points belong to the projection of $\Lambda$ onto
$P_{Y_{1},Y_{2}}$.
$\Lambda=\\{(Y_{1},Y_{2},\dot{Y}_{1},\dot{Y}_{2})\in\mathbb{R}^{4}\
|\dot{r}({\bf{Y}},\dot{\bf{Y}})=0,E_{2}({\bf{Y}},\dot{\bf{Y}})<0\\}$.

Definition $P_{Y,\Lambda}$ denotes the points of the $(Y_{1},Y_{2})$-plane as
a projection from $\Lambda$.

Thus, $W^{\prime}$, $\hat{S}$ exist on $P_{Y,\Lambda}$. $C\in I$ varies at
different points of $P_{Y,\Lambda}$.

It is noted that the points of $P_{Y,\Lambda}$ can be viewed as the periapsis
points of Keplerian elliptical orbits.

A key observation is that the values of $W^{\prime}$, $\hat{S}$ along
$L(\theta)$ in $P_{Y,\Lambda}$ are used as points to iterate on $S_{\theta}$,
starting on the $r$-axis.

If one chooses $p\in W^{\prime}(\theta)$, then $\Phi^{n}(p)\in\mathcal{C}$ for
all $n\geq 1$, where $\dot{r}$ need not be $0$, which lie on $\Sigma(C)$ for
the corresponding value of $C\in I_{C}$. Similarly, if $p\in\hat{S}(\theta)$
then $\Phi^{n}(p)\in S^{*}(\theta)$ for all $n\geq 1$, where $\dot{r}$ need
not be $0$, which lie on $\Sigma(C)$ for the corresponding value of
$C\in\hat{I}_{C}$.

Thus,

(a.) The points of $P_{Y,\Lambda}$ provide the initial points for $\Phi$ to
iterate on $S_{\theta}$ for a given $\theta$. $W^{\prime},\ \hat{S}$ exist on
$P_{Y,\Lambda}$.

(b.) The $(r,\dot{r})$-plane defining $S_{\theta}$ is the plane of iterates
for $\Phi$ that belong to $\mathcal{C},\ S^{*}(\theta)$, respectively (see
Figures 4, 5).

Figure 5. $W^{\prime},\ \hat{S}$ on $P_{Y,\Lambda}$

Definition $P_{Y,\Lambda}$ is referred to as the initial parameter plane for
$\Phi$ and $S_{\theta}$ is referred to as the iteration plane for $\Phi$.

There are points in $P_{Y,\Lambda}$ that don’t belong to $W^{\prime},\
\hat{S}$, and are complementary to these sets on $P_{Y,\Lambda}$.

######  Lemma 5.1.

The complementary points to those of $W^{\prime}(\theta)\cup\hat{S}(\theta)$
relative to $L(\theta)$, on $P_{Y,\Lambda}$, are the unstable points,
$U_{n}(\theta)$, for any finite $n$, that don’t belong to
$W^{\prime}(\theta)$, labeled,
$\bar{U}_{n}(\theta)=\\{p\in(U_{n}(\theta)-W^{\prime}(\theta))\cap(L(\theta)\cap\Lambda),\
n\geq 1\\}$, $C\in I$.

The proof follows by the definition of $W^{\prime}(\theta),\hat{S}(\theta)$.

Geometrically the points of $\bar{U}_{n}=\\{\bar{U}_{n}(\theta),\
\theta\in[0,2\pi]\\}$ represent all the points of $P_{Y,\Lambda}$ in Figure 5
outside the hatched regions defining points of $\hat{S}$.

Trajectories having initial values in $\bar{U}_{n}$ can eventually move out of
$H_{2}$ into $H_{1}$ or $H_{O}$, prior to $n$ cycles of a given trajectory
$\bm{\psi}$ about $P_{2}$ and cycle about $P_{1}$ and not returning to
$H_{2}$, or asymptotically approach $\gamma_{1}$ or $\gamma_{2}$ within
$H_{2}$.

Bounded and unbounded motion are defined as follows,

Definition (bounded motion starting on $P_{Y,\Lambda}$) A point $p\in
P_{Y,\Lambda}\cap H_{2}$ gives bounded motion for a trajectory $\bm{\psi}(t)$
having $p$ as an initial value at $t=t_{0}$, if for $t>t_{0}$, $\bm{\psi}(t)$
does not permanently escape $H_{2}$ for all $t>t_{0}$.

It is noted that this definition implies $\bm{\psi}(t)$ could escape $H_{2}$
at some time $t_{1}$, but it returns to $H_{2}$ at a later time $t_{2}>t_{1}$.

Both $\hat{S},W^{\prime}$ give rise to bounded motion as initial conditions
for trajectories in $H_{2}$.

Definition (unbounded motion starting on $P_{Y,\Lambda}$) A point $p\in
P_{Y,\Lambda}\cap H_{2}$ leads to unbounded motion for a trajectory
$\bm{\psi}(t)$ having $p$ as an initial value at $t=t_{0}$, if for $t>t_{0}$,
$\bm{\psi}(t)$ moves into $H_{1}$ or $H_{O}$ and does not return to $H_{2}$,
or if it asymptotically approaches $\gamma_{1}$ or $\gamma_{2}$.

Unbounded motion could occur for initial points on $\bar{U}_{n}$, but points
in these sets could also give rise to bounded motion.

######  Property 5.2.

Points in $W^{\prime},\hat{S}$ give bounded motion. Points in $\bar{U}_{n}$
may lead to unbounded motion.

The proof follows by definition of bounded and unbounded motion, together with
the dynamics of trajectories with initial values on $W^{\prime},\hat{S}$ and
$\bar{U}_{n}$.

Define $M^{*}=W^{\prime}\cup\hat{S}$, $\bar{M}^{*}=\bar{U}_{n}$. This implies,

######  Property 5.3.

If $p\in M^{*}$ then $p$ yields bounded motion as an initial value for a
trajectory. If $p\in\bar{M}^{*}$, then $p$ may lead to unbounded motion.

######  Property 5.4.

$P_{Y,\Lambda}=M^{*}\cup\bar{M}^{*}$. $M^{*}\cap\bar{M}^{*}=\emptyset$.

The definitions of bounded and unbounded motion for trajectories starting on a
point in $P_{Y,\Lambda}\cap H_{2}$ can be similarly made for iterations of
$\Phi(p)$ on $S_{\theta}$ where $p\in S_{\theta}\cap\Lambda$. Let
$\Phi_{r}(p)$ be the projection of $\Phi(p)$ onto the $r$-coordinate.

Let $\rho$ = distance from $P_{2}$ to $L_{2}$. Assume $C\in I$.

$\Phi^{n}(p)$ is unbounded for $n\geq 1$ if $|\Phi_{r}^{n}(p)|>\rho$ for some
$n$, or if there exists an $N$ such that for $n>N$ there are no more iterates
on $S_{\theta}\cap H_{2}$. If $|\Phi_{r}^{n}(p)|<\rho$ for a sequence
$n=n_{j}\geq 1$, $n_{j+1}>n_{j}$, $j=1,2,\ldots$, then $\Phi^{n}(p)$ is
bounded.

######  Summary 5.5.

(Properties of $M^{*}$) Let $p\in M^{*}$, $w\in\bar{M}^{*}$, $C\in I$, then

A.) $\Phi^{n}(p)$ is bounded for all $n\geq 1$ (for each $\theta$)
B.) $\Phi^{n}(w)$ may be unbounded for $n$ sufficiently large,
C.) $\partial M^{*}\equiv W^{\prime}$ is an infinite union of Cantor sets,
over all $\theta\in[0,2\pi],C\in I_{C}$. It is Cantor set (fractal) for each
fixed $\theta,C$.
D.) $M^{*}$ is compact,
E.) $P_{Y,\Lambda}=M^{*}\cup\bar{M}^{*}$, $M^{*}\cap\bar{M}^{*}=\emptyset$.
F.) $M^{*}$ is defined in the initial parameter plane $P_{Y,\Lambda}$ for
$\Phi$ and $\Phi$ is iterated on the iteration plane $S_{\theta}$.

The proof follows by previous results.

In summary,

######  Theorem 5.6.

The boundary of $M^{*}$, is $W^{\prime}$, an infinite union of Cantor sets,
over all $\theta\in[0,2\pi],C\in I_{C}$. It defines the weak stability
boundary, for unstable initial points that give infinite cycling of
trajectories about $P_{2}$. The interior of $M^{*}$ is the set $\hat{S}$ of
initial points for trajectories that cycle about $P_{2}$ infinitely many times
in a stable manner, over all $\theta\in[0,2\pi],C\in\hat{I}_{C}$. $\mu\in
I_{\mu}$ is fixed.

It is interesting to remark that the properties of $M^{*}$ are similar to
those of the classical Mandelbrot set. This is only given since it seems
interesting to note. The definition and properties of a Mandelbrot set, $M$,
are summarized [21, 12]. The following description of this remark can be
skipped since it is not used in the analysis.

$M$ is defined for the complex map, $Q_{c}(z)=z^{2}+c$, $z\in\mathbb{C}$, and
$c\in\mathbb{C}$ a parameter. $M$ is defined in the $c$-plane and iterates in
$z$-plane. The iterates are considered for the critical point $z=0$. That is,
$Q_{c}^{n}(0)$, $n\geq 1$.

More precisely, $M$ is defined by those values of $c$ such that the iterates
$Q_{c}^{n}(0)$ in the $z$-plane are bounded for $n\geq 1$. $M$ has the
properties,

######  Summary 5.7.

(Properties of $M$) Let $c\in M$, $\hat{c}\in\bar{M}$,

A.) $Q_{c}^{n}(0)$ are bounded for all $n\geq 1$,
B.) $Q_{\hat{c}}^{n}(0)$ is unbounded for $n\rightarrow\infty$,
C.) $\partial M$ is a connected fractal set,
D.) $M$ is compact,
E.) $\mathbb{C}=M\cup\bar{M}$, $M\cap\bar{M}=\emptyset$,
F.) $M$ is defined in the parameter plane $c\in\mathbb{C}$ for $Q_{c}(0)$ and
$Q_{c}(0)$ is iterated on the complex $z$-plane.

######  Summary 5.8.

(Differences between $M^{*},\ M$)

i.) The iterates $Q^{n}_{c}(0)$ on the $z$-plane are parameterized for
different $c\in\mathbb{C}$; The iterates $\Phi^{n}(c)$ on the
$(r,\dot{r})$-plane are parameterized different $c\in
P_{Y,\Lambda}\subset\mathbb{R}^{2}$

ii.) $\partial M$ is a connected fractal curve; $\partial M^{*}=W^{\prime}$ is
fractal for each fixed $\theta,C$, and totally disconnected.

iii.) For points in $\bar{M}$, the iterates are unbounded, and for the points
in $\bar{M}^{*}$, the iterates may be unbounded.

$M^{*}$ and $M$ have some similarities. They are both defined in a parameter
plane ($M^{*}$ in $P_{Y,\Lambda}$ and $M$ in the $c$-plane), and their
iterates in another plane. They both have fractal boundaries, with the main
difference that $\partial M^{*}$ is totally disconnected for each $\theta,C$,
whereas $\partial M$ is connected. $M$ is defined for a complex analytic map,
whereas, $M^{*}$ is defined for a real analytic map. This is summarized in
Table 1.

$M^{*}$ | $M$
---|---
$\Phi^{n}(c),n\geq 1,$ bounded, $c\in P_{Y,\Lambda}\subset\mathbb{R}^{2}$ |  $Q_{c}^{n}(0)$, $n\geq 1$, bounded $c\in\mathbb{C}$
$\partial M^{*}$ fractal, totally disconnected for each $\theta,C$ |  $\partial M$ fractal, connected
$\Phi^{n}(c)\in S_{\theta}\subset\mathbb{R}^{2}$ | $Q_{c}^{n}(0)\in\mathbb{C}$
Table 1. Comparison between $M^{*}$ and $M$.

##  6\. Non-existence of KAM tori

KAM tori exist for $C$ sufficiently large as follows from [19]. See Theorem
3.5 for $C\in C_{L}$. When $C\notin C_{L}$ then the general existence of KAM
tori in $H_{2}$ about $P_{2}$ is an open problem to theoretically understand.
When $C$ is not large and the radius of $H_{2}$ increases, then the
gravitational perturbative force from $P_{1}$ can be strong enough relative to
the gravitational force of $P_{2}$, for $\mu$ sufficiently small, so that KAM
tori may not exist on $\Sigma(C)$.

The range of $C$ considered is $C\in I$, and $\mu\in I_{\mu}$ is fixed. The
techniques in [19] are not applicable for $C\in I$.

######  Theorem 6.1.

Consider a fixed energy surface $\Sigma(C),\ C\in I_{C}$, $\mu\in I_{\mu}$,
and a section $S_{\theta}$ on this energy surface for any given
$\theta\in[0,2\pi]$. The iterates of an initial point $p_{0}\in
P_{Y,\Lambda}$, $\Phi^{k}(p_{0})$ on $S_{\theta}$, cannot lie on a KAM torus
if $p_{0}\in\partial M^{*}$. If $p_{0}\in\bar{M}^{*}$ the iterates may not lie
on a KAM torus. If $p_{0}\in\text{the interior of}\ M^{*}$, then it is
possible the iterates could lie on a KAM torus.

Proof -

Consider a trajectory ${\bm{\psi}}(t,\mu)$ cycling about $P_{2}$ in ${\bf
Y}$-coordinates under these assumptions for $t>0$, with an inital value at
$t=t_{0}$ on $M^{*}\subset P_{Y,\Lambda}$.

The points of intersection of ${\bm{\psi}}(t,\mu)$ with
$S_{\theta}\cap\Sigma(C)$ occur at a sequence of times $t_{k},k\geq 1$,
$t_{k+1}>t_{k}$. These generate the iterates of $\Phi$, labeled
$\Phi^{k}(p_{0})$, where $p_{0}$ is the initial point of ${\bm{\psi}}(t,\mu)$
on $S_{\theta}$ at $t=t_{0}$.

$\Phi^{k}(p_{0})$, $k=1,2,\ldots$ lie on a Cantor set $\mathcal{C}_{C}$ on
$S_{\theta}$ by assumptions.

The $\Phi^{k}(p_{0})$, $k=1,2,\ldots$ cannot lie on a KAM torus. This is shown
as follows: Assume the iterates did lie on such a KAM torus, $T^{*}$, two-
dimensional. It intersects $S_{\theta}$ in a topological circle,
$\tilde{S}^{1}$. Since the iterates belong to $\mathcal{C}_{C}$, they are
nowhere dense. This contradicts the assumption they lie on $\tilde{S}^{1}$,
since then the iterates would have to be dense on $\tilde{S}^{1}$ by the Moser
Twist Theorem for a monotone twist map, $M$ [23, 28]. This follows since by
KAM theory, $M$ takes the form,
$\phi\rightarrow\phi+2\pi\lambda(\rho)+\mathcal{O}(\mu),\
\rho\rightarrow\rho+\mathcal{O}(\mu)$, $\phi\in[0,2\pi],\rho>0$ are polar
coordinates for points on $\tilde{S}^{1}$, where the angular frequency for a
given value of $\rho$, $\lambda$, satisfies, $d\lambda/d\rho>0$, and satisfies
diophantine conditions. Thus the iterates cannot lie on $T^{*}$.

These iterates, however, could belong to points in a resonance gap between KAM
tori, if they existed, on $\Sigma(C)$. If that were the case, then how wide
could this gap be on $\Sigma(C)$? In the complementory region
$\bar{M}^{*}\cap\Sigma(C)$, initial values for $\Phi$ may not yield iterates
that lie on the intersection of KAM tori with $S_{\theta}\cap\Sigma(C)$. This
is because the region $\bar{M}^{*}$ consists of points in $\bar{U}_{n}$ for
finite $n\geq 1$. They may be unbounded by iteration with $\Phi$ on
$S_{\theta}$. Thus, this indicates that initial points in $\bar{M}^{*}$ to use
for iteration of $\Phi$ on $S_{\theta}\cap\Sigma(C)$ may not lie on KAM tori.

What happens for $p_{0}\in\text{the interior of}\ M^{*}$? These points belong
to $\hat{S}_{C}(\theta)$, cycling about $P_{2}$ for all time, and could lie on
KAM tori. This is not known.

This proves Theroem 6.1

This theorem only pertains to the fate of the iterations of points on
$P_{Y,\Lambda}$.

##  7\. Discussion of Results

One of the main results of this paper is that the weak stability boundary,
$W^{\prime}$, about $P_{2}$ for infinitely many cycles of $P$ about $P_{2}$,
is a union of infinitely many Cantor sets of hyperbolic points. This answers
the question on the fractal structure of the weak stability boundary, at least
in the case of infinite cycling.

Another result is that KAM tori cannot exist on $W^{\prime}$, but they may
exist on the stable set $\hat{S}$ that it bounds for the range of $C$. Results
indicate that that KAM tori may not exist beyond $W^{\prime}$, so this set may
yield a kind of boundary for KAM tori about $P_{2}$.

A curious result is that $M^{*}=W^{\prime}\cup\hat{S}$ has similar properties
to a classical Mandelbrot set, even though they are defined completely
differently. The main difference between $M^{*}$ and $M$ is that although the
boundaries are both fractal sets, the boundary of $M$ is continuous whereas
the boundary of $M^{*}$ is totally discontinuous.

This paper uses previous numerical results to motivate the analysis. This is
done for relatively small values of $n$. It would be interesting to
numerically explore $M^{*}$ for large values of $n$, which is beyond the scope
of this paper.

A potential application of these results pertains to the low energy permanent
weak capture of $P$ about $P_{2}$ from the Hill’s region about $P_{1}$. $P$
would move through the channel, $\mathcal{C}$, and into the Hill’s region
about $P_{2}$ to the boundary region, $W^{\prime}$ about $P_{2}$ and become
permanetly captured taking infinitely many cycles. The location of
$W^{\prime}$ relative to $P_{2}$ would give regions where permanent capture
can occur. Permanent capture using $W^{\prime}$ could be used to design
spacecraft trajectories that never need orbit maintenance maneuvers while
orbiting $P_{2}$.

The results of this paper may have an interesting interpretation on the nature
of the interaction of the gravitational fields between two bodies on a
particle of negligible mass. Defining this boundary over infinite cycles
yields a boundary consisting of the union of Cantor sets. Thus, in general, no
matter how small one magnifies this boundary, the self similarity yields the
same structure. This could just be a curious mathematical property or it also
may say something more about interacting gravitational fields and their scale
properties.

##  Appendix A Supporting Calculations

###  A.1. Kepler Energy, $\bm{E_{2}}$

In a $P_{2}$-centered inertial coordinate system, ${\bf{X}}=(X_{1},X_{2})$,
the Kepler energy of $P$ relative to $P_{2}$ is

(A.1) $E_{2}=(1/2)|\dot{\bf{X}}|^{2}-\mu|{\bf X}|^{-1},$

where $X_{1}=x_{1}-1,X_{2}=x_{2}$. ${\bf{x}}=(x_{1},x_{2})$ are
$P_{1}$-centered inertial coordinates. .

In a $P_{2}$-centered rotating coordinates, ${\bf{Y}}=(Y_{1},Y_{2})$, obtained
by setting $Y_{1}=y_{1}-1,Y_{2}=y_{2}$, where ${\bf{y}}=(y_{1},y_{2})$ are
$P_{1}$-centered rotating coordinates defined in (2.1);

(A.2) $E_{2}({\bf Y},\dot{\bf{Y}})=(1/2)\dot{Y}^{2}-\frac{\mu}{Y}-L({\bf
Y},\dot{\bf{Y}})+(1/2)Y^{2},$

where $L({\bf Y},{\bm{\dot{Y}}})=\dot{Y}_{1}Y_{2}-\dot{Y}_{2}Y_{1}$,
$Y=|{\bf{Y}}|,\dot{Y}=|\dot{\bf{Y}}|$.

###  A.2. $\bm{E_{2}\geq 0}$ When $\bm{P}$ Cycles About $\bm{P_{1}}$

It is assumed $C_{2}<C\lessapprox C_{1}$. $P$ has an initial conditon at
$t=t_{0}$ on $L(\theta)$ with eccentricity $e\in[0,1)$, and with initial
velocity (3.1), at a distance $r$ from $S$. This implies at the initial point,
$E_{2}<0$. $P$ cycles around $S$ $n\geq 1$ times and escapes $S$ through the
channel $\mathcal{C}_{1}$. Since this an unstable motion, then this implies
the trajectory for $P$ will make at least one cycle about $P_{1}$. In
particular, $P$ will cross the negative $Y_{1}$-axis, to the left of $P_{1}$,
where $r>1$. It will do this while moving near the zero velocity curve about
$P_{1}$ (see (Belbruno2004, Figure 3.10), (McGehee1969)). Thus, there exists a
time $t_{1}>t_{0}$, where $Y_{2}(t_{1})=0,Y_{1}(t_{1})\gtrapprox-2$. Also, for
the velocities, $|\dot{Y}_{2}(t_{1})|>0$ since $P$ will cross the $Y_{1}$-axis
moving upward or downward, and $\dot{Y}_{1}(t_{1})\approx 0$ since $P$ crosses
the $Y_{2}$-axis near the zero velocity curve. Thus, at the time of crossing,
$L(t_{1})\approx 0$. This implies
$E_{2}(t_{1})\approx(1/2)\dot{Y}^{2}+(1/2)Y^{2}-\frac{\mu}{Y}$ which is
positive since $\mu$ is small. Thus, $E_{2}(t_{1})>0$.

###  A.3. Definition of $\bm{S_{\theta_{0}}}$

In a $P_{2}$-centered rotating system, $(Y_{1},Y_{2})$, the Jacobi integral in
the $S$-centered coordinates is

(A.3)
$\tilde{J}=-(|\dot{\bf{Y}}|^{2})+2[(1-\mu)/r_{1}+\mu/r]+[(Y_{1}+1-\mu)^{2}+Y_{2}^{2}]+\mu(1-\mu),$

where $r_{1}^{2}=(Y_{1}-1)^{2}+Y_{2}^{2}$, $r=|{\bf{Y}}|$. $\tilde{J}=C$
defines the energy surface, $\Sigma\Sigma(C)$.

In polar coordinates, $(r,\theta)$, where
$Y_{1}=r\cos\theta,Y_{2}=r\sin\theta$, it is verified that the Jacobi
integral, $\tilde{J}$ becomes

(A.4) $\hat{J}=-(\dot{r}^{2}+r^{2}\dot{\theta}^{2})+f(r,\theta),$

where $f$ is well defined for $r>0,r_{1}>0$, and does not depend on
$\dot{r},\dot{\theta}$. Thus, $\hat{J}=C$, implies that
$\dot{\theta}^{2}=\dot{\theta}^{2}(r,\dot{r},\theta)=r^{-2}[\dot{r}^{2}+f(r,\theta)]$.
Choosing the positive root, $\dot{\theta}=\dot{\theta}>0$, as required for
$S_{\theta_{0}}$ on $\hat{J}=C$,
$S_{\theta_{0}}=\\{r,\dot{r}|\theta=\theta_{0},\\}$, where
$\dot{\theta}=\dot{\theta}(r,\dot{r},\theta)$.

###  A.4. Trajectory Properties

The trajectories satisfy topological conditions. The range of $C$ is
$C_{a}<C<C_{2}$ and $C_{2}\leq C<C_{1}$, These conditions are:

Hypothesis $A$

(i) All the trajectories on $W^{s}(\gamma_{1})$ in $H_{2}$ make at least $n$
consecutive cycles about $P_{2}$, $n=1,2,\ldots$,

(ii) All the trajectories on $W^{u}(\gamma_{1})$ in $H_{1}$ must make at least
$1$ cycle about $P_{1}$,

(iii) All the trajectories on $W^{s}(\gamma_{2})$ in $H_{2}$ must make at
least $n$ consecutive cycles about $P_{2}$,

(iv) All the trajectories on $W^{u}(\gamma_{2})$ in $H_{O}$ must make at least
$1$ cycle about $P_{1}$.

In Hypothesis $A$, cases occur where these assumptions are not satisfied. For
this and other details, see[7].

Acknowledgements

Many thanks to Marian Gidea for helpful comments. I would like to thank
Princeton University, Dept. of Astrophysical Sciences. Research partially
funded by NSF grant DMS-1814543.

## References

  * [1] Arnold, V. I.: Mathematical Methods of Classical Mechanics. GTM 60 , Springer Verlag, Berlin (1989)
  * [2] Belbruno, E.: Lunar capture orbits, a method of constructing Earth-Moon trajectories and the Lunar GAS mission, with Ballistic Capture and the Lunar GAS Mission. in Proceedings of the 19th AIAA/DGLR/JSASS Inter. Elec. Propl. Conf., Paper no. 87-1054, Colorado Springs (1987) (arc.aiaa.org/doi/10.2514/6.1987-1054).
  * [3] Belbruno, E., Miller, J.: Sun-perturbed Earth-to-Moon transfers with ballistic capture. J. Guidance, Control, Dynamics 16, 770-775 (1993)
  * [4] Belbruno, E., Marsden, B.: Resonance hopping in comets. AJ 113, 1433-1444 (1997)
  * [5] Belbruno, E.: Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, Princeton (2004)
  * [6] Belbruno, E.: Fly Me to the Moon: An Insiders Guide to the New Science of Space Travel. Princeton University Press, Princeton (2007)
  * [7] Belbruno, E., Gidea, M., Topputo, F. Weak stability boundary and invariant manifolds. SIAM J Appl. Dyn. Sys., 9, 1061-1089 (2010)
  * [8] Celletti, A., Chierchia, L. KAM stability and celestial mechanics. Memoirs of the American Mathematical Society 187 (2007)
  * [9] Circi, C., P. Teofilato, P.: On the dynamics of weak stability boundary lunar transfers. Cel. Mech. Dyn. Astr. 79, 41-72 (2001)
  * [10] Conley, C.: Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732-746 (1968)
  * [11] Conley, C.: On the ultimate behavior of orbits with respect to an unstable critical point. J. Diff Equ, 5, 136-158 (1969)
  * [12] Devaney, R.: A First Course in Chaotic Dynamical Systems. CRC Press, Boca Raton, Florida (2022)
  * [13] Garcia, F., Gomez, G.: A Note on Weak Stability Boundaries. Cel. Mech. Dyn. Astr. 97, 87-100 (2007)
  * [14] Gidea, M., Masdemont, J.: Geometry of homoclinic connections in the planar circular restricted three-body problem. Inter. J. Bifur. Chaos. Sci. Engrg. 17, 1151-1169 (2007)
  * [15] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42 Springer Verlag, Berlin (1983)
  * [16] Jehn, R., et. al.: Navigating BepiColombo during the weak-stability capture at Mercury. Advances in Space Research 42, 1364-1369 (2008)
  * [17] Jian, L., Yi-Sui, S.: A survey of weak stability boundaries in the Sun-Mars system. RAA 15 376 (2015)
  * [18] Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10 , 427-469 (2000)
  * [19] Kummer, M.: On the stability of Hill’s solutions of the plane restricted three-body problem. Am. J. Math. 101, 1333-135 (1979)
  * [20] Llibre, J., Martinez, R., Simó, C.: Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near $L_{2}$ in the Restricted three-body problem. J. Diff Equ 58 104-156 (1985)
  * [21] Mandelbrot, B.: Fractal aspects of the iteration of $z\mapsto\lambda z(1-z)$ for complex $\lambda,z^{n}$, Annals of the New York Academy of Sciences 357, 249-259 (1980)
  * [22] Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematics Studies, 160, third edition, Princeton University Press, Princeton (2006)
  * [23] Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachrichten der Akademie der Wissenschaften Gottingen, Math.-Phys. Kl. 2, 1-20 (1962)
  * [24] Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton University Press, Princeton (1973)
  * [25] Poincaré, H.: Les Methodes Nouvelles de la Méchanique Celeste, Vols 1-3. Gauthier Villars, Paris (1899)
  * [26] Romagnoli, D., Circi, C.: Earth-Moon weak stability boundaries in the restricted three and four body problem. Cel. Mech. Dyn. Astr. 10, 79-103 (2009)
  * [27] Schoenmaekers, J., Horas, D., Pulido, D.: SMART-1: With solar electric propulsion to the Moon. in Proceedings of the 16th International Symposium on Space Flight Dynamics, Pasadena, CA, 114-130 (2001)
  * [28] Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Grundlehren Series 187, Springer-Verlag, Berlin (1971)
  * [29] Simó, C., Stuchi, T.: Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140, 1-32. (2000)
  * [30] Topputo, F., Belbruno, E.: (2009), Computation of weak stability boundaries: Sun-Jupiter. Cel. Mech. Dyn. Astr., 105, 3-17 (2009)