# Melnikov method for non-conservative perturbations of the three-body problem

Marian Gidea† Yeshiva University, Department of Mathematical Sciences, New
York, NY 10016, USA<EMAIL_ADDRESS>,  Rafael de la Llave‡ School of
Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
<EMAIL_ADDRESS>and  Maxwell Musser† Yeshiva University,
Department of Mathematical Sciences, New York, NY 10016, USA
<EMAIL_ADDRESS>
###### Abstract.

We consider the planar circular restricted three-body problem (PCRTBP), as a
model for the motion of a spacecraft relative to the Earth-Moon system. We
focus on the Lagrange equilibrium points $L_{1}$ and $L_{2}$. There are
families of Lyapunov periodic orbits around either $L_{1}$ or $L_{2}$, forming
Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov
manifolds around either $L_{1}$ or $L_{2}$, as well as heteroclinic orbits
between the Lyapunov manifold around $L_{1}$ and the one around $L_{2}$. The
motion along the homoclinic/heteroclinic orbits can be described via the
scattering map, which gives the future asymptotic of a homoclinic orbit as a
function of the past asymptotic. In contrast with the more customary Melnikov
theory, we do not need to assume that the asymptotic orbits have a special
nature (periodic, quasi-periodic, etc.).

We add a non-conservative, time-dependent perturbation, as a model for a
thrust applied to the spacecraft for some duration of time, or for some other
effect, such as solar radiation pressure. We compute the first order
approximation of the perturbed scattering map, in terms of fast convergent
integrals of the perturbation along homoclinic/heteroclinic orbits of the
unperturbed system. As a possible application, this result can be used to
determine the trajectory of the spacecraft upon using the thrust.

###### Key words and phrases:

Melnikov method; homoclinic and heteroclinic orbits; three-body problem;
astrodynamics.

###### 2010 Mathematics Subject Classification:

Primary, 37J40; 37D05 37C29; 34C37; 70F07 Secondary, 70H08.

† Research of M.G. was partially supported by NSF grant DMS-1515851, and the
Alfred P. Sloan Foundation grant G-2016-7320.

‡ Research of R.L. was partially supported by NSF grant DMS-1800241, and
H2020-MCA-RISE #734577

† Research of M.M. was partially supported by NSF grant DMS-1515851, and the
Alfred P. Sloan Foundation grant G-2016-7320.

##  1\. Introduction

###  1.1. Motivation

A motivation for this work is the following situation from astrodynamics.
Consider a spacecraft traveling between the Earth and the Moon. Assume that
the spacecraft is coasting along the stable/unstable hyperbolic invariant
manifolds associated to the periodic orbits near one of the center-saddle
equilibrium points, at some fixed energy level. Such an orbit is driven by the
gravitational fields of the Earth and the Moon, and does not require using the
thrusters. The total energy is preserved along the orbit. One can describe the
motion of the spacecraft in terms of some geometrically defined coordinates:
an ‘action’ coordinate describing the size of the periodic orbit associated to
the stable/unstable invariant manifold, an ‘angle’ coordinate describing the
asymptotic phase of the motion, and a pair of ‘hyperbolic’ coordinates
describing the position of the spacecraft relative to the corresponding
stable/unstable manifold.

Suppose now that we want to make a maneuver in order to jump from the
hyperbolic invariant manifold on the given energy level to another hyperbolic
invariant manifold on a different energy level. Mathematically, turning on the
thrusters amounts to adding a small, non-conservative, time-dependent
perturbation to the original system. If such a perturbation is given, we would
like to estimate its effect on the orbit of the spacecraft. More precisely, we
would like to compute the change in the action and angle coordinates
associated to the orbit as a result of applying the perturbation. We want to
obtain such an estimate in terms of the original trajectory of the unperturbed
system, and of the particular perturbation.

Other non-Hamiltonian perturbations, for instance, due to solar radiation
pressure and solar wind, can be considered.

In this paper, we will investigate the general problem of adding a non-
Hamiltonian perturbation to a homoclinic/heteroclinic trajectory and computing
the effect on the homoclinic/heteroclinic orbits.

Note that adding a non-Hamiltonian perturbation (e.g., a small dissipation)
may destroy all the periodic orbits. Nevertheless, the normally hyperbolic
manifold, formed by the collection of periodic orbits, persists (see details
later). In contrast with the most customary versions of the Melnikov theory,
which assume that the asymptotic orbits are periodic or quasiperiodic, and
that they preserve their nature under perturbation, we consider homoclinic
excursions to a normally hyperbolic manifold. The asymptotic orbits could
change their nature under the perturbations. For example, a family of Lyapunov
periodic orbits subject to a small dissipation could get transformed into a
family of orbits that converge to a critical point.

###  1.2. Brief description of the main results and methodology

In the model that we consider, the family of periodic orbits about either
$L_{1}$ or $L_{2}$ forms a normally hyperbolic invariant manifold (NHIM). Each
NHIM has hyperbolic stable and unstable manifolds. We will assume that the
stable and unstable manifolds of the NHIM corresponding to either $L_{1}$ or
$L_{2}$ intersect transversally, and also that the unstable (stable) manifold
of the NHIM corresponding to $L_{1}$ intersects transversally the stable
(unstable) manifold of the NHIM corresponding to $L_{2}$. This assumption can
be verified numerically for a wide range of energy levels and mass parameters
in the PCRTBP (see, e.g., [KLMR00]). Thus, there exist homoclinic orbits to
either one of the NHIM’s, as well as heteroclinic orbits between the two
NHIM’s. There exist scattering maps associated to the transverse homoclinic
intersections, as well as to the transverse heteroclinic intersections.

There exist some neighborhoods of $L_{1}$ and of $L_{2}$ where the Hamiltonian
of the PCRTBP can be written in a normal form, via some suitable symplectic
action-angle and hyperbolic coordinates $(I,\theta,y,x)$. In particular, each
NHIM can be parametrized in terms of the action-angle coordinates
$(I,\theta)$. Therefore, the scattering map can also be described in terms of
these coordinates. In the unperturbed case, the scattering map is particularly
simple: it is a shift in the angle coordinate (a phase shift).

The fact that we use normal form coordinates to express the Hamiltonian, and
we subsequently estimate the scattering map in terms of the action-angle
coordinates, is a matter of practical convenience. Normal forms are often used
to compute numerically, with high precision, the periodic orbits and the
NHIM’s around the equilibrium points, as well as the corresponding stable and
unstable manifolds, e.g., [Jor99, GJMS01].

For applications, it is important to note that the scattering map for the
PCRTBP can be computed numerically with high precision; see [CDMR06, DGR16,
CGdlL16].

In this paper we study the effect of a small, non-Hamiltonian, time-dependent
perturbation that is added to the system. Provided that the perturbation is
small enough, the NHIM’s will persist [Fen72], although periodic orbits inside
the NHIM’s may disappear. Also, the transverse homoclinic/heteroclinic orbits,
hence the scattering map, will survive in the perturbed system.

The main contribution is that we compute the effect of the perturbation on the
action and angle components of the scattering map. More precisely, we use
Melnikov theory to provide explicit estimates – up to first order with respect
to the size of the perturbation – for the difference between the perturbed
scattering map and the unperturbed one, relative to the action and angle
coordinates. The resulting expressions are given in terms of convergent
improper integrals of the perturbation evaluated along segments of
homoclinic/heteroclinic orbits of the unperturbed system. One important aspect
in the computation is that, in the perturbed system, the action is a slow
variable, while the angle is a fast variable.

We stress that, unlike the usual treatments of Melnikov theory, when one
studies orbits homoclinic to hyperbolic fixed points, periodic or quasi-
periodic orbits, here we study orbits homoclinic to NHIM’s. The asymptotic
dynamics inside the NHIM’s may change under the perturbation.

The effect of the perturbation on the action-angle components of the
scattering map can be interpreted, in the context of astrodynamics, as
follows. The difference in the action coordinates between the perturbed and
the unperturbed scattering map can be interpreted as the change in energy due
to the maneuver, or equivalently, the change in the ‘size’ of the periodic
orbit which the homoclinic/heteroclinic orbit is asymptotic to. The difference
in the angle coordinates between the perturbed and the unperturbed scattering
map can be interpreted as the change in asymptotic phase due to the maneuver.

We mention here that there are numerous works on using hyperbolic invariant
manifolds to design low-energy space mission, see, for example [KLMR00,
GLMS01, Bel04, BGT10, PA14], and the references listed there. We hope that our
results can be used to optimize the thrust that needs to be applied in order
to maneuver between hyperbolic invariant manifolds on different energy levels.

###  1.3. Related works

The study of Hamiltonian systems subject to non-conservative perturbations is
of practical interest in physical models, such as in celestial mechanics,
where dissipation leads to migration of satellites and spacecrafts [MNF87,
GEPC17, dlLK19, CCdlL20].

Computation of the scattering map, similar to the one in this paper, have been
done in the case of the pendulum-rotator model subject to Hamiltonian
perturbations, e.g., [DdlLS08, GdlL18]. The rotator-pendulum model is a
product system and is naturally endowed with action-angle and hyperbolic
coordinates. It has two conserved quantities: the action of the rotator and
the total energy of the pendulum. The effect of the perturbation on the action
component of the scattering map is relatively easy to compute directly. On the
other hand, the effect on the angle component of the scattering map is more
complicated to compute, since this is a fast variable. To circumvent this
difficulty, the papers [DdlLS08, GdlL18] use the symplecticity of the
scattering map to estimate indirectly the effect of the perturbation on the
angle component of the scattering map.

The perturbed scattering map has been computed in the case of the pendulum-
rotator model subject to non-conservative perturbations in[GdlLM21]. Since the
perturbations are not Hamiltonian, the symplectic argument used in [DdlLS08]
can no longer be applied. Therefore, to determine the effect of the
perturbation on the angle component of the scattering map, a direct
computation is performed in [GdlLM21].

The PCRTBP model considered in this paper presents some significant
differences from the pendulum-rotator model. First, it has only one conserved
quantity, namely the total energy. Second, the PCRTBP is not a product system,
and does not carry a globally defined system of action-angle and hyperbolic
coordinates. Third, in the unperturbed case the stable and unstable manifolds
associated to a NHIM do not coincide. For these reasons, we construct locally
defined systems of action-angle and hyperbolic coordinates along the unstable
manifold as well as along the stable manifold, respectively. At the
intersection of the unstable and stable manifolds, the two coordinate systems
do not agree in general. So the computation of the perturbed scattering map
has to take into account the ‘mismatch’ between these coordinate systems. The
dynamics in these coordinate systems fails to be of product type, as there is
a coupling between the action-angle and the hyperbolic coordinates, which also
needs to be taken into account in the computation. All of these features make
the computation of the perturbed scattering map for the planar circular
restricted three-body problem more intricate than for the rotator-pendulum
system. Some of the calculations are simplified taking advantage that some of
the variables in the unperturbed system are slow variables, but we can deal
with perturbation theory for fast variables by observing that, near the
NHIM’s, the difference between the variables and their asymptotic values is
slow (a technique already used in [GdlLM21].

###  1.4. Structure of the paper

The main result, Theorem 2.1, is given in Section 2. It provides an expansion
of the perturbed scattering map in terms of the unperturbed scattering map,
where the first order term in the expansion is given explicitly in Section 5,
in Proposition 5.2 and Proposition 5.7. Section 3 describes how to verify the
hypotheses of Theorem 2.1 in the context of the PCRTBP. Section 4 defines some
suitable coordinate systems, which we use to describe the geometric objects of
interest. The proof of the main result is given in Section 4.

##  2\. Set-up and main result

Consider a (real analytic) $\mathcal{C}^{\omega}$, symplectic manifold
$(M,\Omega)$ of dimension $(2m)$. Each point $z\in M$ is described via a
system of local coordinates $z=z(p,q)\in\mathbb{R}^{2m}$ such that $\Omega$
relative to these coordinates is the standard symplectic form

(2.1) $\Omega=dp\wedge dq=\sum_{i=1}^{m}dp_{i}\wedge dq_{i}.$

On $M$ we consider a non-autonomous system of differential equations

(2.2)
$\frac{d}{dt}z=\mathcal{X}^{0}(z)+\varepsilon\mathcal{X}^{1}(z,t;\varepsilon),$

where $\mathcal{X}^{0}:M\to TM$ is a $\mathcal{C}^{\omega}$-smooth vector
field on $M$,
$\mathcal{X}^{1}:M\times\mathbb{R}\times(-\varepsilon_{0},\varepsilon_{0})\to
TM$ is a time-dependent, parameter dependent $\mathcal{C}^{r}$-smooth vector
field on $M$, with $r$ sufficiently large, and $\epsilon\in\mathbb{R}$ is a
‘smallness’ parameter, taking values in the interval
$(-\varepsilon_{0},\varepsilon_{0})$ around $0$. The dependence of
$\mathcal{X}^{1}(z,t;\varepsilon)$ on the time $t$ is assumed to be of a
general type – not necessarily periodic or quasi-periodic.

The flow of (2.2) will be denoted by $\Phi^{t}_{\varepsilon}$.

The assumption that $\mathcal{X}^{0}$ is analytic is motivated by applications
to celestial mechanics. We will consider the case when $\mathcal{X}^{0}$ is
Hamiltonian vector field for the PCRTBP. We will use this assumption only to
be able to quote several normal form theorems. We believe it could be weakened
to finite differentiability at the price of providing some new normal form
theorems.

The assumption that $\mathcal{X}^{1}$ is only $\mathcal{C}^{r}$ and possibly
not Hamiltonian is also motivated by applications, as $\mathcal{X}^{1}$ can
model the thrust applied to a spacecraft for some time. In particular
$\mathcal{X}^{1}$ can have compact support in space and in time. Note that
even if the perturbation were analytic, the NHIM’s which play a role in our
treatment can only be assumed to be finite differentiable. The regularity is
limited by ratios between the tangential and normal contraction rates, as well
as by the regularity of the perturbation. Since the motion on the unperturbed
manifold is integrable, we have that for $\varepsilon$ small enough, the
tangential rates are close to zero, so that the limitations of regularity due
to the rates become irrelevant.

Below, we will require that the vector fields $\mathcal{X}^{0}$,
$\mathcal{X}^{1}$ satisfy additional assumptions.

###  2.1. The unperturbed system

We assume that the vector field $\mathcal{X}^{0}$ represents an autonomous
Hamiltonian vector field, that is, $\mathcal{X}^{0}=J\nabla H_{0}$ for some
$\mathcal{C}^{\omega}$ Hamiltonian function $H_{0}:M\to\mathbb{R}$, where $J$
is an almost complex structure compatible with the standard symplectic form
given by (2.1), and the gradient $\nabla:=\nabla_{z}$ is with respect to the
associated Riemannian metric. The Hamilton equation for the unperturbed system
is:

(2.3) $\frac{d}{dt}z=J\nabla H_{0}(z).$

####  2.1.1. Homoclinic connections

We assume that the Hamiltonian flow associated to $H_{0}$ satisfies the
conditions (A-i), (A-ii) and (A-iii), below. In Section 3, we will see that
these conditions can be verified in the planar circular restricted three-body
problem.

  * (A-i)

_There exists an equilibrium point $L$ of saddle-center type, that is, the
linearized system $DJ\nabla H_{0}$ at $L$ has eigenvalues of the type
$\pm\lambda,\pm i\omega$, with $\lambda,\omega\neq 0$._

Consequently, by the Lyapunov Center Theorem [Mos58], there exists a
$1$-parameter family of periodic orbits $\lambda_{0}(h)$, parametrized by the
energy level $H=h$, for $h\in(H(L),h_{1})$ with $h_{1}$ sufficiently small,
such that $\lambda_{0}(h)$ shrinks to $L$ as $h\to H(L)$. This family of
periodic orbits determines a $2$-dimensional manifold $\Lambda_{0}\simeq
D\times\mathbb{T}\subseteq M$ which is a normally hyperbolic invariant
manifold (NHIM) with boundary for the Hamiltonian flow $\Phi^{t}_{0}$ of
$H_{0}$, where $D$ is closed interval with non-empty interior contained in
$(H(L),h_{1})$. The notion of a NHIM is recalled in Definition A.1, Appendix
A.

The NHIM $\Lambda_{0}$ is symplectic when endowed with the form
$\Omega_{\mid\Lambda_{0}}$, where $\Omega$ is given by (2.1). Moreover,
$\Lambda_{0}$ is foliated by the periodic orbits $\lambda_{0}(h)$, i.e.,
$\Lambda_{0}=\bigcup_{h\in D}\lambda_{0}(h)$. The flow $\Phi^{t}_{0}$ on each
$\lambda_{0}(h)$ is a constant speed flow. In particular, the dynamics
restricted to the NHIM is integrable. Therefore $\Lambda_{0}$ can be
parametrized in terms of symplectic action-angle variables $(I,\theta)$, so
that each periodic orbit $\lambda_{0}(h)$ represents a level set $I_{h}$ of
the action variable. The action $I_{h}$ is uniquely determined by the energy
level $H_{0}=h$. In fact, as we will see in Section 4.1, there exist a system
of action-angle and hyperbolic variables $(I,\theta,y,x)$ in a neighborhood of
$L$ such the Hamiltonian $H_{0}$ can be written in a normal form.

  * (A-ii)

_There exists a relatively compact open set $\mathscr{K}$ in $M$ such that the
unstable manifold $W^{\rm u}_{\mathscr{K}}(\Lambda_{0})$ and the stable
manifold $W^{\rm s}_{\mathscr{K}}(\Lambda_{0})$ inside $\mathscr{K}$ intersect
transversally along a _homoclinic channel_ $\Gamma_{0}\subset\mathscr{K}$._

The definition of a homoclinic channel is given in Appendix B, Definition B.1.

As a consequence of (A-ii), there exist transverse homoclinic orbits to
$\Lambda_{0}$. The unstable and stable manifolds of $\lambda_{0}(h)$ are
contained in the same energy level as $\lambda_{0}(h)$, i.e., $W^{\rm
u}(\lambda_{0}(h)),W^{\rm s}(\lambda_{0}(h))\subseteq M_{h}$, where
$M_{h}=\\{z\in M\,|\,H=h\\}$. Hence, each homoclinic orbit to $\Lambda_{0}$ is
asymptotic, in both forward and backwards time, to the periodic orbit
$\lambda_{0}(h)$ corresponding to its energy level $h$.

In Section 4.1, we will see that the normal form coordinates $(I,\theta,y,x)$
can be extended via the flow $\Phi^{t}_{0}$ along neighborhoods of $W^{\rm
u}(\Lambda_{0})$ and $W^{\rm s}(\Lambda_{0})$, yielding two systems of action-
angle and hyperbolic variables $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm
u})$, $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$, respectively. Relative
to these two systems of coordinates $W^{\rm u}(\Lambda_{0})$ can be described
locally by $y^{\rm u}=0$, and $W^{\rm s}(\Lambda_{0})$ can be described
locally by $x^{\rm s}=0$. The two coordinate systems $(I^{\rm u},\theta^{\rm
u},y^{\rm u},x^{\rm u})$ and $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$
do not agree with one another in a neighborhood of the homoclinic channel
$\Gamma_{0}$.

In addition, we require that $H_{0}$ satisfies a non-degeneracy condition
(A-iii), which is formulated in terms of the normal form from Section 4.1.
Condition (A-iii) consists of two parts: (A-iii-a), and (A-iii-b). Condition
(A-iii-a) will be given in Section 4.1, and condition (A-iii-b) will be given
in Section 4.2. These are explicit and verifiable conditions that the
derivatives of certain functions are non-zero.

####  2.1.2. Unperturbed scattering map associated to a homoclinic channel

Let $\Omega^{-}:W^{\rm u}(\Lambda_{0})\to\Lambda_{0}$ be the projection
mapping defined by $\Omega^{-}(z_{0})=z^{-}_{0}$, where
$z_{0}^{-}\in\Lambda_{0}$ is the footpoint of the unstable fiber through
$z_{0}\in W^{\rm u}(\Lambda_{0})$. Similarly, let $\Omega^{+}:W^{\rm
s}(\Lambda_{0})\to\Lambda_{0}$ be the projection mapping defined by
$\Omega^{+}(z_{0})=z^{+}_{0}$, where $z_{0}^{+}\in\Lambda_{0}$ is the
footpoint of the stable fiber through $z_{0}\in W^{\rm s}(\Lambda_{0})$. The
stable and unstable fibers are defined in Appendix A, equation (A.4).

Consider the homoclinic channel $\Gamma_{0}$ from condition (A-ii). By the
definition of a homoclinic channel, $\Omega^{\pm}$ restricted to $\Gamma_{0}$
is a diffeomorphism onto its image. To any homoclinic channel we can associate
a _scattering map_ , which is defined in Appendix B, Definition B.2.
Specifically, the scattering map $\sigma_{0}$ associated to $\Gamma_{0}$ is
given by:

(2.4)
$\begin{split}&\sigma_{0}:\Omega^{-}(\Gamma_{0})\subseteq\Lambda_{0}\to\Omega^{+}(\Gamma_{0})\subseteq\Lambda_{0},\\\
&\sigma_{0}(z_{0}^{-})=z_{0}^{+},\end{split}$

provided that there exists a homoclinic point $z_{0}\in\Gamma_{0}$ such that
$\Omega^{-}(z_{0})=z^{-}_{0}$ and $\Omega^{+}(z_{0})=z^{+}_{0}$. For more
details on the scattering map, see Appendix B.

The energy preservations along the stable and unstable manifolds of periodic
orbits implies that $\sigma_{0}$ leaves each periodic orbit $\lambda_{0}(h)$
invariant, that is, $\sigma_{0}(\lambda_{0}(h))\subseteq\lambda_{0}(h)$.

Fixing a point $z_{0}\in\Gamma$, we have that
$\sigma_{0}(z_{0}^{-})=z_{0}^{+}=z_{0}^{-}+\Delta$, for some $\Delta$
depending on $z_{0}$. The invariance property of the scattering map (B.2), and
the fact that $\Phi_{0}^{t}$ restricted to $\lambda_{0}(h)$ is linear implies
that

$\sigma_{0}(\Phi^{t}_{0}(z_{0}^{-}))=\Phi^{t}(z_{0}^{+})=\Phi^{t}(z_{0}^{-}+\Delta)=\Phi^{t}(z_{0}^{-})+\Delta,$

for all $t$ for which $\Phi^{t}_{0}(z_{0})$ remains in $\Gamma_{0}$. This
implies that, in terms of the action-angle coordinates $(I,\theta)$,
$\sigma_{0}$ is given by a shift in the angle

$\sigma_{0}(I,\theta)=(I,\theta+\Delta(I)),$

for some function $\Delta$ that depends differentiably on $I$. We stress that
$\Delta$ also depends on the choice of the homoclinic channel $\Gamma_{0}$.

###  2.2. The perturbation.

The vector field $\mathcal{X}^{1}$ is a time-dependent, parameter-dependent
vector field on $M$.

  * (A-iv)

_We assume that $\mathcal{X}^{1}=\mathcal{X}^{1}(z,t;\varepsilon)$ is
uniformly $\mathcal{C}^{r}$-differentiable in all variables on
$\mathscr{K}\times\mathbb{R}\times(-\varepsilon_{0},\varepsilon_{0})$, where
the set $\mathscr{K}$ is as in the condition (A-ii)._

Above, we assume that $r$ is suitably large. We will not assume that
$\mathcal{X}^{1}$ is Hamiltonian. Thus, our setting can be used to model
dissipation or forcing applied to a Hamiltonian system. Note that non-
Hamiltonian perturbations are very singular, in the sense that periodic and
homoclinic orbits may disappear. On the other hand, the NHIM’s and their
stable and unstable manifolds persist and can be used as the basis for
perturbative calculations.

As a particular case, we will also write our results for the case when the
perturbation $\mathcal{X}^{1}$ in (2.2) is Hamiltonian, that is, it is given
by

(2.5) $\mathcal{X}^{1}(z,t;\varepsilon)=J\nabla_{z}H_{1}(z,t;\varepsilon),$

where $H_{1}$ is a time-dependent, parameter-dependent
$\mathcal{C}^{r}$-smooth Hamiltonian function on $M$.

When the perturbation is added to the system, as we will see in Section 5.1,
the NHIM for the unperturbed system can be continued to a NHIM for the
perturbed system, and the transverse homoclinic/heteroclinic orbits for the
unperturbed system can be continued to transverse homoclinic/heteroclinic
orbits for the perturbed system, provided that the perturbation is
sufficiently small. Hence there exists an associated scattering map for the
perturbed system.

The goal is to quantify the effect of the perturbation on the corresponding
scattering map.

###  2.3. Extended system

We consider the system (2.2) under the conditions (A-i), (A-ii), and (A-iii).
We associate to it the extended system

(2.6)
$\begin{split}&\frac{d}{d\tau}z=\mathcal{X}^{0}(z)+\varepsilon\mathcal{X}^{1}(z,t;\varepsilon),\\\
&\frac{d}{d\tau}t=1,\\\ \end{split}$

which is defined on the extended phase space
$\widetilde{M}=M\times\mathbb{R}$. We denote
$\tilde{z}=(z,t)\in\widetilde{M}$. The independent variable will be denoted by
$\tau$ from now on. We will denote by $\tilde{\Phi}^{\tau}_{\varepsilon}$ the
extended flow of (2.6). We have

$\tilde{\Phi}^{\tau}_{\varepsilon}(z,t)=({\Phi}^{\tau}_{\varepsilon}(z),t+\tau).$

In the extended phase space we have the following:

${\tilde{\Lambda}}_{0}=\Lambda_{0}\times\mathbb{R}$

is a NHIM with boundary for the extended unperturbed flow
$\tilde{\Phi}^{\tau}_{0}$.

$\tilde{\Gamma}_{0}=\Gamma_{0}\times\mathbb{R}$

is a homoclinic channel for $\tilde{\Phi}^{\tau}_{0}$.

The scattering map associated to $\tilde{\Gamma}_{0}$ is given by

$\tilde{\sigma}_{0}(I,\theta,t)=(I,\theta+\Delta(I),t).$

###  2.4. Main result

######  Theorem 2.1.

Consider the system (2.2).

Assume that the unperturbed system $\mathcal{X}^{0}$ satisfies the conditions
(A-i), (A-ii) and (A-iii), and that the perturbation $\mathcal{X}^{1}$
satisfies the condition (A-iv).

Then, there exists $\varepsilon_{1}$, with
$0<\varepsilon_{1}<\varepsilon_{0}$, such that the following hold true:

  * (i)

For all $\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$, there is a
$\mathcal{C}^{\ell}$-family of NHIMs $\tilde{\Lambda}_{\varepsilon}$ for the
extended flow $\tilde{\Phi}^{t}_{\varepsilon}$, for some $\ell\geq 1$, which
form a continuation of the NHIM $\tilde{\Lambda}_{0}$ of
$\tilde{\Phi}^{t}_{0}$;

  * (ii)

For all $\varepsilon\in(-\varepsilon_{1},\varepsilon_{1})$, the unstable and
stable manifolds of $\tilde{\Lambda}_{\varepsilon}$, $W^{\rm
u}(\tilde{\Lambda}_{\varepsilon})$ and $W^{\rm
s}(\tilde{\Lambda}_{\varepsilon})$, respectively, intersect transversally, in
the extended phase space $\widetilde{M}$, along a homoclinic channel
$\tilde{\Gamma}_{\varepsilon}$;

  * (iii)

The perturbed scattering map $\tilde{\sigma}_{\varepsilon}$ associated to
$\tilde{\Gamma}_{\varepsilon}$ can be written as

(2.7)
$\tilde{\sigma}_{\varepsilon}(I,\theta,t)=\tilde{\sigma}_{0}(I,\theta,t)+\varepsilon\tilde{\mathcal{S}}(I,\theta,t)+O_{\mathcal{C}^{\ell}}(\varepsilon^{2}),$

where
$\tilde{\mathcal{S}}=(\tilde{\mathcal{S}}^{I},\tilde{\mathcal{S}}^{\theta},\tilde{\mathcal{S}}^{t})$
is a mapping from some domain in $D\times\mathbb{T}\times\mathbb{R}$ to its
image in $\mathbb{R}\times\mathbb{T}\times\mathbb{R}$ as follows:

    * (iii-a)

the components $\tilde{\mathcal{S}}^{I}$ and $\tilde{\mathcal{S}}^{\theta}$
are given by (5.17) and (5.40), respectively, and

    * (iii-b)

the component $\tilde{\mathcal{S}}^{t}$ is given by
$\tilde{\mathcal{S}}^{t}(I,\theta,t)=t$.

We recall the notation $O_{\mathcal{C}^{k}}(\cdot)$ used above:
$f=O_{\mathcal{C}^{k}}(g)$ means that $\|f\|_{\mathcal{C}^{k}}\leq
M\|g\|_{\mathcal{C}^{k}}$ for some $M>0$, where $k\geq 0$, and
$\|\cdot\|_{\mathcal{C}^{k}}$ is the ${\mathcal{C}^{k}}$-norm. In the sequel,
to simplify the notation we will write $O(\cdot)$ without the subscript
indicating the function space topology, whenever this can be inferred from the
context.

###  2.5. Heteroclinic connections

Instead of the conditions (A-i), (A-ii), (A-iii) we assume that the
Hamiltonian flow associated to $H_{0}$ satisfies the conditions (A’-i),
(A’-ii), (A’-iii) from below.

Condition (A’-i) has two parts (A’-i-a) and (A’-i-b).

  * (A’-i-a)

_There exists two equilibrium points $L^{1}$, $L^{2}$ of saddle-center type._

We do not assume that the two equilibrium points are on the same energy level,
that is $H(L^{1})\neq H(L^{2})$ in general. Consequently, for each equilibrium
point $L^{1}$, $L^{2}$ there exists a $1$-parameter family of closed orbits
$\lambda^{1}_{0}(h)$, for $h\in D^{1}$, and $\lambda^{2}_{0}(h)$ for $h\in
D^{2}$, where $D^{1}$, $D^{2}$ are some closed intervals contained in some
neighborhoods of $H(L^{1})$, $H(L^{2})$, respectively.

  * (A’-i-b)

_There exists an interval of energies $D\subseteq D^{1}\cap D^{2}$, with non-
empty interior, such that there are periodic orbits $\lambda^{1}_{0}(h)$,
$\lambda^{2}_{0}(h)$ for all $h\in D$. Moreover, there exist normal form
coordinates $(I^{1},\theta^{1},y^{1},x^{1})$ and
$(I^{2},\theta^{2},y^{2},x^{2})$ defined around $\lambda^{1}_{0}(h)$,
$\lambda^{2}_{0}(h)$, respectively, for all $h\in D$. These normal form
coordinates are as in Section 4.1._

This condition implies that each family of periodic orbits determines a
$2$-dimensional NHIM $\Lambda^{1}_{0}=\bigcup_{h\in D}\lambda_{0}(h)$,
$\Lambda^{2}_{0}=\bigcup_{h\in D}\lambda^{2}_{0}(h)$ in $M$. Moreover, there
are neighborhood of $\Lambda^{1}_{0}$ and $\Lambda^{2}_{0}$ where $H_{0}$ can
be written in a normal form as in Section 4.1.

  * (A’-ii)

_There exists a relatively compact open set $\mathscr{K}$ in $M$ such that the
unstable manifold $W^{\rm u}_{\mathscr{K}}(\Lambda^{1}_{0})$ and the stable
manifold $W^{\rm s}_{\mathscr{K}}(\Lambda^{2}_{0})$ inside $\mathscr{K}$
intersect transversally along a _heteroclinic channel_
$\Gamma_{0}\subset\mathscr{K}$._

The definition of a heteroclinic channel is given in Definition B.3, Appendix
B. As a consequence, there exist transverse heteroclinic orbits from
$\Lambda^{1}_{0}$ to $\Lambda^{2}_{0}$. Each such heteroclinic orbit is
asymptotic in backwards time to a periodic orbit $\lambda^{1}_{0}(h)$, and is
asymptotic in forward time to a periodic orbit $\lambda^{2}_{0}(h)$.

As in the case of homoclinic connections, we require some non-degeneracy
condition (A’-iii), formulated in terms of normal forms, which is the analogue
of condition (A-iii) in Section 2. For the sake of brevity, we will not
formulate this condition explicitly.

####  2.5.1. Unperturbed scattering map associated to a heteroclinic channel

As before, we define $\Omega^{-,1}:W^{\rm
u}(\Lambda^{1}_{0})\to\Lambda^{1}_{0}$ by $\Omega^{-,1}(z_{0})=z^{-}_{0}$,
where $z_{0}^{-}\in\Lambda^{1}_{0}$ is the footpoint of the unstable fiber
through $z_{0}\in W^{\rm u}(\Lambda^{1}_{0})$, and $\Omega^{+,2}:W^{\rm
s}(\Lambda^{2}_{0})\to\Lambda^{2}_{0}$ by $\Omega^{+,2}(z_{0})=z^{+}_{0}$,
where $z_{0}^{+}\in\Lambda^{2}_{0}$ is the footpoint of the stable fiber
through $z_{0}\in W^{\rm s}(\Lambda^{2}_{0})$.

Associated to the heteroclinic channel $\Gamma_{0}$ we can define the
scattering map as in Definition B.4 in Appendix B. Specifically,

$\sigma_{0}:\Omega^{-,1}(\Gamma_{0})\subseteq\Lambda^{1}_{0}\to\Omega^{+,2}(\Gamma_{0})\subseteq\Lambda^{2}_{0},$

is given by

$\sigma_{0}(z_{0}^{-})=z_{0}^{+},$

provided that there exists a $z_{0}\in\Gamma_{0}$ such that
$\Omega^{-,1}(z_{0})=z^{-}_{0}$ and $\Omega^{+,2}(z_{0})=z^{+}_{0}$.

Since $H_{0}$ is constant along heteroclinic orbits, we have that
$I(z^{-}_{0})=I(z^{+}_{0})$. Then the scattering map, expressed in terms of
the action-angle coordinates, is given by

$\sigma_{0}(I,\theta)=(I,\theta+\Delta(I)),$

for some function $\Delta$ that depends smoothly on $I$.

In this case, we can obtain a result similar to Theorem 2.1. For brevity, we
will not provide the precise formulas for the components of the corresponding
expansion of the perturbed scattering map, which is analogous to (2.7). Such
formulas are analogues of (5.17) and (5.40).

##  3\. Geometric structures in the planar circular restricted three-body
problem.

In this section we survey the status of the verification of the conditions
(A-i), (A-ii), (A-iii) from Section 2.1, or the conditions (A’-i), (A’-ii),
(A’-iii) from Section 2.5, in the concrete model of the PCRTBP. Some of the
verifications in the literature are rigorous and some of them are numerical.

Of course, the verification of the hypothesis of Theorem 2.1 in a concrete
model does not affect the validity of the rigorous arguments establishing
Theorem 2.1, and the reader interested only in rigorous results may safely
skip this section.

We note that, since our hypothesis are mainly transversality conditions, they
can be verified with finite precision calculations, which seem to be safe
calculations for today’s standard and could well be accessible to _“computer
assisted proofs”_. We hope that this work could stimulate more extensive
verifications.

The PCRTBP is a model describing the motion of an infinitesimal body under the
Newtonian gravity exerted by two heavy bodies moving on circular orbits about
their center of mass, under the assumption that these orbits are not affected
by the gravity of the infinitesimal body.

We can think of the heavy bodies (referred to as primaries) representing the
Earth and Moon, and the infinitesimal mass representing a spaceship.

It is convenient to study the motion of the infinitesimal body relative to a
co-rotating frame which rotates with the primaries around the center of mass,
and to use normalized units. Henceforth, the masses of the heavy bodies are
$1-\mu$ and $\mu$, where $\mu\in(0,1/2]$. Relative to the co-rotating frame,
the heavier mass $1-\mu$ is located at $(\mu,0)$, and the lighter mass $\mu$
is located at $(-1+\mu,0)$. The motion of the infinitesimal body relative to
the co-rotating frame is described via the autonomous Hamiltonian

(3.1)
$H_{0}(p_{1},p_{2},q_{1},q_{2})=\frac{(p_{1}+q_{2})^{2}+(p_{2}-q_{1})^{2}}{2}-V(q_{1},q_{2}),$

where $\left(p,q\right)=\left(p_{1},p_{2},q_{1},q_{2}\right)\in\mathbb{R}^{4}$
represents the momenta and the coordinates of the infinitesimal body with
respect to the co-rotating frame,

(3.2)
$\begin{split}V(q_{1},q_{2})&=\frac{q_{1}^{2}+q_{2}^{2}}{2}+\frac{1-\mu}{r_{1}}+\frac{\mu}{r_{2}},\\\
r_{1}&=\left((q_{1}-\mu)^{2}+q_{2}^{2}\right)^{1/2},\\\
r_{2}&=\left((q_{1}+1-\mu)^{2}+q_{2}^{2}\right)^{1/2}.\end{split}$

Above $V(q_{1},q_{2})$ represents the effective potential, and $r_{1}(t)$,
$r_{2}(t)$ represent the distances from the infinitesimal body to the masses
$1-\mu$ and $\mu$, respectively. The phase space

$M=\\{(p,q)\in\mathbb{R}^{4}\,|\,q\neq(\mu,0),\textrm{ and
}q\neq(-1+\mu,0)\\}$

is endowed with the symplectic form

$\Omega=dp_{1}\wedge dq_{1}+dp_{2}\wedge dq_{2}.$

Note that the phase space $M$ is not compact.

The equations of motion of the infinitesimal body are given by the Hamilton
equations corresponding to (3.1), that is

(3.3) $\frac{d}{dt}{z}=J\nabla H_{0}(z),$

where $z=z(p_{1},p_{2},q_{1},q_{2})$ and $J$ represents the standard almost
complex structure.

The Hamiltonian $H_{0}$ is an integral of motion, so the flow $\Phi^{t}_{0}$
of (3.3) leaves invariant each energy hyper-surface

(3.4) $M_{h}:=H_{0}^{-1}(h)=\\{(p_{1},p_{2},q_{1},q_{2})\in
M\,|\,H_{0}(p_{1},p_{2},q_{1},q_{2})=h\\}.$

The system has three equilibrium points $L_{1},L_{2},L_{3}$ located on the
$q_{1}$-axis, and two other equilibrium points $L_{4}$, $L_{5}$, each lying in
the $(q_{1},q_{2})$-plane and forming an equilateral triangle with the
primaries. These are known as the Lagrange points. In our convention, $L_{1}$
is located between the primaries, $L_{2}$ is on the side of the lighter
primary, and $L_{3}$ is on the side of the heavier primary. The linearized
stability of $L_{1},L_{2},L_{3}$ is of center-saddle type. The linearized
stability of $L_{4}$, $L_{5}$ is of center-center type, for $\mu$ less than
some critical value $\mu_{\textrm{cr}}$.

We note that condition (A-i) is satisfied for each of the equilibrium points
$L_{1},L_{2},L_{3}$.

For $i=1,2,3$, for each energy level $h$, with $H(L_{i})<h$ and $h$
sufficiently close to $H(L_{i})$, there exists a unique periodic orbit
$\lambda_{0}(h)$ near the equilibrium point $L_{i}$, which is referred to as a
Lyapunov orbit. The existence of such periodic orbits follows from the
Lyapunov Center Theorem (see, e.g., [Mos58]). Moreover, there exists a
neighborhood of $L_{i}$ in the phase space where the Hamiltonian $H_{0}$ can
be written in a normal form relative to some suitable coordinates
$(I,\theta,y,x)$; see Section 4.1.

Each Lyapunov orbit is hyperbolic in the energy surface, so it has associated
$2$-dimensional unstable and stable manifolds denoted $W^{\rm
u}(\lambda_{0}(h))$ and $W^{\rm s}(\lambda_{0}(h))$.

Numerical evidence shows that these periodic orbits can be continued for
energy levels $h>H(L_{i})$ that are not necessarily close to $H(L_{i})$ (see,
e.g., [Bro68]).

Normally hyperbolic invariant manifold for the unperturbed system. For an
energy range $h\in D$ sufficiently close to the energy of $L_{i}$, the family
of Lyapunov orbits

(3.5) $\Lambda_{0}=\bigcup_{h\in D}\lambda_{0}(h),$

defines a $2$-dimensional NHIM with boundary for the Hamiltonian flow of
(3.3). The NHIM carries the symplectic structure $\Omega_{\mid\Lambda_{0}}$,
and it can be described in terms of the action-angle coordinates $(I,\theta)$.
The action $I=I_{h}$ is uniquely defined by the energy $h$, and $\theta$ is
symplectically conjugate with $I$ with respect to $\Omega_{\mid\Lambda_{0}}$.
The variable $I$ is a first integral along the trajectories of the flow on
$\Lambda_{0}$, and the action level sets are in fact the Lyapunov orbits
$\lambda_{0}(h)$. The motion restricted to each Lyapunov orbit is a rigid
rotation in the variable $\theta$, with the frequency depending on the energy
level.

The NHIM $\Lambda_{0}$ and its unstable and stable manifolds $W^{\rm
u}(\Lambda_{0})$ and $W^{\rm s}(\Lambda_{0})$ have simple descriptions in
terms of the normal form coordinates $(I,\theta,y,x)$ in a neighborhood of
$L_{1}$: $\Lambda_{0}$ corresponds to $x=y=0$, $W^{\rm u}(\Lambda_{0})$
corresponds to $y=0$, and $W^{\rm s}(\Lambda_{0})$ corresponds to $x=0$.

Homoclinic connections. We first focus on the dynamics around the equilibrium
point $L_{1}$. There are analytic arguments (see [LMS85]) showing that, for a
discrete set of values of $\mu$ that are sufficiently small, and for each $h$
sufficiently close to $H(L_{1})$, the branches of $W^{\rm u}(\lambda_{0}(h))$
and $W^{\rm s}(\lambda_{0}(h))$ on the side of the heavier primary do not
collide with the primary and intersect transversally along some homoclinic
orbit $\gamma_{0}(h)$, not necessarily unique. Numerical evidence shows that,
for a large range of values of masses $\mu$ and energies $h$, the branches of
$W^{\rm u}(\lambda_{0}(h))$ and $W^{\rm s}(\lambda_{0}(h))$ on the side of
each primary do not collide with either primary and intersect transversally;
see, e.g. [KLMR00, Cap12].

Each choice of a transverse homoclinic orbit $\gamma_{0}(h)$ can be continued
in energy $h$ to a family of such homoclinic orbits, which forms a homoclinic
manifold $\bigcup_{h\in D}\gamma_{0}(h)$. Moreover, we can ensure that $W^{\rm
u}(\lambda_{0}(h))$ and $W^{\rm s}(\lambda_{0}(h))$ are contained inside some
compact subset $\mathscr{K}$ of the phase space. One can always restrict to a
submanifold

(3.6) $\Gamma_{0}\subseteq\bigcup_{h\in D}\gamma_{0}(h),$

that is a homoclinic channel.

In this way, we can ensure condition (A-ii).

The remaining condition is (A-iii), which consists of (A-iii-a) and (A-iii-b).
These are explicit non-degeneracy conditions that can separately be verified
numerically.

In this way, for the PCRTBP, we can verify the existence of the geometric
structures of interest and the corresponding conditions (A-i), (A-ii), (A-iii)
from Section 2.1.

Heteroclinic connections. We now focus on the dynamics around the equilibrium
points $L_{1}$ and $L_{2}$. They satisfy condition (A’-i). For energy levels
$h$ with $H(L_{1})\lesssim h$, there exists a family $\lambda^{1}_{0}(h)$ of
Lyapunov orbits near $L_{1}$, and for $H(L_{2})\lesssim h$ there exists a
family $\lambda^{2}_{0}(h)$ of Lyapunov orbits near $L_{2}$. Moreover, there
exist normal form coordinates $(I^{1},\theta^{1},y^{1},x^{1})$ and
$(I^{2},\theta^{2},y^{2},x^{2})$ defined around $L_{1}$ and $L_{2}$,
respectively, for some suitable energy ranges. These normal form coordinates
are as in Section 4.1.

Numerical evidence shows that families of periodic orbits near $L_{1}$ and
$L_{2}$ can exist simultaneously, for some energy range. Therefore, we
consider an interval of energies $D$ such that, for $h\in D$ we have the
following: there exists families of periodic orbits $\lambda^{1}_{0}(h)$ near
$L_{1}$, and $\lambda^{2}_{0}(h)$ near $L_{2}$, the following sets

(3.7) $\begin{split}\Lambda^{1}_{0}&=\bigcup_{h\in D}\lambda^{1}_{0}(h),\\\
\Lambda^{2}_{0}&=\bigcup_{h\in D}\lambda^{2}_{0}(h),\end{split}$

are NHIM’s for the Hamiltonian flow of (3.3), and the normal form coordinates
$(I^{1},\theta^{1},y^{1},x^{1})$ and $(I^{2},\theta^{2},y^{2},x^{2})$ are
defined in some neighborhoods of $\Lambda^{1}_{0}$ and $\Lambda^{2}_{0}$,
respectively.

Numerical evidence also shows that there exist transverse heteroclinic
connections determined by $W^{{\rm u}}(\lambda^{1}_{0}(h))\cap W^{{\rm
s}}(\lambda^{2}_{0}(h))$ and $W^{{\rm u}}(\lambda^{2}_{0}(h))\cap W^{{\rm
s}}(\lambda^{1}_{0}(h))$ for certain ranges of energies. See, e.g., [KLMR00,
WZ03, CM06, BGT10]. In either case, we denote the corresponding family of
heteroclinic orbits by $\gamma_{0}(h)$. We consider a range of energies $h\in
D$ for which this additional condition on the existence of transverse
heteroclinic connections is satisfied.

The transverse intersection of the unstable manifold $W^{\rm
u}(\Lambda^{1}_{0})$ with the stable manifold $W^{\rm s}(\Lambda^{2}_{0})$
define a heteroclinic manifold $\bigcup_{h\in D}\gamma_{0}(h)$, which depends
on the choice of the family of heteroclinic orbits $\gamma_{0}(h)$. One can
always restrict to a submanifold

(3.8) $\Gamma_{0}\subseteq\bigcup_{h\in D}\gamma_{0}(h),$

that is a heteroclinic channel.

If that is the case, the condition (A’-ii) is verified.

As in the case of homoclinic connections, the remaining condition (A’-iii)
consists of two explicit non-degeneracy conditions that can be verified
numerically.

Thus, one can verify numerically the conditions (A’-i), (A’-ii), (A’-iii),
from Section 2.5.

It would be of interest to verify if Theorem 2.1 can be applied when the
Lagrange point $L_{3}$ is also considered; some numerical results concerning
the dynamics around $L_{3}$ can be found in [BO06, CCP16].

In summary, in this section we have outlined how the conditions of the Theorem
2.1 can be verified in the PCRTBP. The theoretical results of Theorem 2.1 are
independent on the application to the PCRTBP.

##  4\. Coordinate systems and evolution equations

###  4.1. New coordinate systems for the unperturbed system

We consider the case of homoclinic connections described by conditions (A-i),
(A-ii), (A-iii). Under these assumptions, the manifolds $W^{\rm
u}(\Lambda_{0})$ and $W^{\rm s}(\Lambda_{0})$ intersect transversally along
the homoclinic channel $\Gamma_{0}$.

The next proposition states that, in a neighborhood of each $W^{\rm
u}(\Lambda_{0})$ and $W^{\rm s}(\Lambda_{0})$, there exists a system of
symplectic coordinates such that $\Lambda_{0}$, $W^{\rm u}(\Lambda_{0})$ and
$W^{\rm s}(\Lambda_{0})$ have very simple descriptions, and moreover the
unperturbed Hamiltonian $H_{0}$ can be written in a normal form relative to
the corresponding coordinates. As before, for $z\in W^{{\rm s},{\rm
u}}(\Lambda_{0})$, we denote $z^{\pm}=\Omega^{\pm}(z)$.

######  Proposition 4.1 (Normal Forms).

There exist three systems of symplectic coordinates111coordinates obtained
from $(p,q)$ via a canonical transformation, defined for some range of
energies $h\in D$, as follows:

  * (N)

A coordinate system $(I,\theta,y,x)$ in a neighborhood $\mathscr{N}$ of
$\Lambda_{0}$ such that for $z\in\mathcal{N}$ we have:

    * (N-i)

$z\in\Lambda_{0}$ if and only if $x(z)=y(z)=0$;

    * (N-ii)

$z\in W^{\rm u}(\Lambda_{0})$ if and only if $y=0$, and $z\in W^{\rm
s}(\Lambda_{0})$ if and only if $x=0$;

    * (N-iii)

for $z\in W^{\rm u}(\Lambda_{0})$ we have $I(z)=I(z^{-})$ and
$\theta(z)=\theta(z^{-})$, and for $z\in W^{\rm s}(\Lambda_{0})$ we have
$I(z)=I(z^{+})$ and $\theta(z)=\theta(z^{+})$;

    * (N-iv)

$H_{0}$ restricted to $\mathscr{N}$ can be written in a normal form

(4.1) $\begin{split}H_{0}(p,q)=&H_{0}(I,xy)\\\
=&h_{0}\left(I\right)+(xy)g_{1}\left(I\right)+\left(xy\right)^{2}g_{2}(I,xy)\end{split}$

for some analytic functions $h_{0}=h_{0}(I)$, $g_{1}=g_{1}(I)$, and
$g_{2}=g_{2}(I,xy)$.

  * (U)

A coordinate system $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ in a
neighborhood $\mathscr{N}^{\rm u}$ of $W^{\rm u}(\Lambda_{0})$ such that for
$z\in\mathcal{N}^{\rm u}$ we have:

    * (U-i)

$z\in\Lambda_{0}$ if and only if $x^{\rm u}(z)=y^{\rm u}(z)=0$;

    * (U-ii)

$z\in W^{\rm u}(\Lambda_{0})$ if and only if $y^{{\rm u}}=0$;

    * (U-iii)

for $z\in W^{\rm u}(\Lambda_{0})$ we have $I^{\rm u}(z)=I^{\rm u}(z^{-})$, and
$\theta^{\rm u}(z)=\theta^{\rm u}(z^{-})$;

    * (U-iv)

$H_{0}$ restricted to $\mathscr{N}^{\rm u}$ can be written in a normal form

(4.2) $\begin{split}H_{0}(p,q)=&H_{0}^{\rm u}(I^{\rm u},x^{\rm u}y^{\rm u})\\\
=&h_{0}\left(I^{u}\right)+(x^{u}y^{u})g_{1}\left(I^{{\rm
u}}\right)+\left(x^{{\rm u}}y^{{\rm u}}\right)^{2}g_{2}(I^{\rm u},x^{\rm
u}y^{\rm u})\end{split}$

for some analytic functions $h_{0}=h_{0}(I^{\rm u})$, $g_{1}=g_{1}(I^{\rm
u})$, and $g_{2}=g_{2}(I^{\rm u},x^{\rm u}y^{\rm u})$.

  * (S)

A coordinate system $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$ in a
neighborhood $\mathscr{N}^{\rm s}$ of $W^{\rm s}(\Lambda_{0})$ such that for
$z\in\mathcal{N}^{\rm s}$ we have:

    * (S-i)

$z\in\Lambda_{0}$ if and only if $x^{\rm s}(z)=y^{\rm s}(z)=0$;

    * (S-ii)

$z\in W^{\rm s}(\Lambda_{0})$ if and only if $x^{{\rm s}}=0$;

    * (S-iii)

for $z\in W^{\rm s}(\Lambda_{0})$ we have $I^{\rm s}(z)=I^{\rm s}(z^{+})$, and
$\theta^{\rm s}(z)=\theta^{\rm s}(z^{+})$;

    * (S-iv)

$H_{0}$ restricted to $\mathscr{N}^{\rm s}$ can be written in a normal form

(4.3) $\begin{split}H_{0}(p,q)=&H_{0}^{\rm s}(I^{\rm s},x^{\rm s}y^{\rm s})\\\
=&h_{0}\left(I^{{\rm s}}\right)+(x^{{\rm s}}y^{{\rm s}})g_{1}\left(I^{{\rm
s}}\right)+\left(x^{{\rm s}}y^{{\rm s}}\right)^{2}g_{2}(I^{\rm s},x^{\rm
s}y^{\rm s})\end{split}$

for some analytic functions $h_{0}=h_{0}(I^{\rm s})$, $g_{1}=g_{1}(I^{\rm
s})$, and $g_{2}=g_{2}(I^{\rm s},x^{\rm s}y^{\rm s})$.

The coordinate systems $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ and
$(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$ coincide with
$(I,\theta,y,x)$ on $\mathscr{N}$, i.e.,

(4.4) $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm u})=(I^{\rm s},\theta^{\rm
s},y^{\rm s},x^{\rm s})=(I,\theta,y,x)\textrm{ on }\mathscr{N}.$

The function $h_{0},g_{1},g_{2}$ that appear in (4.1), (4.2) and (4.3) are the
same.

###### Proof.

Part (N) follows from [Gio01], so we will not give a detailed proof.

We only summarize the procedure to obtain the normal form in a neighborhood of
a center-saddle equilibrium point for a $2$-degrees of freedom Hamiltonian
$H_{0}$. First, the Hamiltonian $H_{0}$ is expanded in a Taylor series around
that equilibrium point (shifted to the origin) as

$H_{0}(p,q)=H_{2}(p,q)+H_{3}(p,q)+H_{4}(p,q)+\ldots,$

where $H_{j}(p,q)$ is an homogeneous polynomial of degree $j$ in the variables
$(p_{1},p_{2},q_{1},q_{2})$. Then, by making a linear canonical change of
coordinates $(p,q)\mapsto(x,y)$, with the eigenvectors of the linearized
system given by $J\nabla H_{2}(0,0)$ as the axes of the new system, the
quadratic part $H_{2}$ of $H_{0}$ can be written in the new coordinates
$(x,y)\in\mathbb{R}^{4}$ as

$H_{2}(x,y)=\lambda x_{1}y_{1}+\frac{\omega}{2}(x_{2}^{2}+y_{2}^{2}),$

where $\pm\lambda_{1}=\pm\lambda$ and $\pm\lambda_{2}=\pm i\omega$ are the
eigenvalues of $J\nabla H_{2}(0,0)$. Then, via another linear canonical change
of coordinates

$x_{1}=\xi_{1},\quad y_{1}=\eta_{1},\quad
x_{2}=\frac{\xi_{2}+i\eta_{2}}{\sqrt{2}},\quad\quad
y_{2}=\frac{i\xi_{2}+\eta_{2}}{\sqrt{2}},$

we obtain $H_{2}$ written in complex variables as

$H^{N}_{2}(\xi,\eta)=\lambda\xi_{1}\eta_{1}+i\omega\xi_{2}\eta_{2}=\lambda_{1}\xi_{1}\eta_{1}+\lambda_{2}\xi_{2}\eta_{2}.$

The next step is to apply a sequence of changes of coordinates to kill all
monomials for which the exponent of $\xi_{j}$ is different from the exponent
of $\eta_{j}$. Since the eigenvalue $\lambda_{1}=\lambda$ is real and
$\lambda_{2}=i\omega$ is imaginary, there are no small divisors. The only
possible source of divergence is due to the use of Cauchy’s estimates for the
derivatives required by the normalization procedure. In [Gio01] the
accumulation of derivatives is controlled via a KAM technique. The process can
be continued to any order.

Thus, in the limit $H_{0}$ can be written as an expansion

$H^{N}_{0}(\xi,\eta)=H^{N}_{2}(\xi_{1}\eta_{1},\xi_{2}\eta_{2})+H^{N}_{3}(\xi_{1}\eta_{1},\xi_{2}\eta_{2})+H^{N}_{4}(\xi_{1}\eta_{1},\xi_{2}\eta_{2})+\ldots,$

where $H_{j}$ is an homogeneous polynomial in
$\xi_{1}\eta_{1},\xi_{2}\eta_{2}$ of degree $j$. The series expansion of
$H_{0}^{N}$ is convergent in a neighborhood $\bar{\mathscr{N}}$ of the origin,
and the coordinate change $x=x(\xi,\eta)$, $y=y(\xi,\eta)$ is canonical and
given in terms of convergent series.

There exist periodic orbits $\lambda_{0}(h)$ around the equilibrium point for
all energy levels $h$ sufficiently close to the energy level of the
equilibrium point. This implies that the NHIM
$\bar{\Lambda}_{0}=\bigcup_{h\in\bar{D}}\lambda_{0}(h)$ is contained in
$\bar{\mathscr{N}}$, for some suitable energy range $h\in\bar{D}$.

To express $H_{0}^{N}$ in action-angle coordinates one applies the canonical
transformation

$\xi_{2}=\sqrt{\bar{I}}\exp(i\bar{\theta}),\quad\eta_{2}=-i\sqrt{\bar{I}}\exp(-i\bar{\theta}).$

Finally, denote $\bar{x}=\xi_{1}$, $\bar{y}=\eta_{1}$. We obtain the normal
form

(4.5)
$H_{0}^{N}(\bar{I},\bar{x}\bar{y})=\lambda\bar{x}\bar{y}+{\omega}\bar{I}+H_{3}(\bar{I},\bar{x}\bar{y})+H_{4}(\bar{I},\bar{x}\bar{y})+\ldots.$

Moreover, in these coordinates the following hold:

  * (i)

The normally hyperbolic invariant manifold $\bar{\Lambda}_{0}$ is given by
$\bar{x}=\bar{y}=0$, and each periodic orbit in $\bar{\Lambda}_{0}$
corresponds to a level set of $\bar{I}$;

  * (ii)

The local unstable invariant manifold $W^{\rm
u}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$ is given by $\bar{y}=0$;

  * (iii)

The local stable invariant manifold $W^{\rm
s}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$ is given by $\bar{x}=0$.

The equations of motion are

(4.6) $\begin{split}\frac{d}{dt}\bar{I}=&0,\\\
\frac{d}{dt}\bar{\theta}=&\omega+\frac{\partial
H_{3}^{N}}{\partial\bar{I}}+\frac{\partial
H_{4}^{N}}{\partial\bar{I}}+\ldots,\\\
\frac{d}{dt}\bar{y}=&-\lambda\bar{y}-\frac{\partial
H_{3}^{N}}{\partial\bar{x}}-\frac{\partial
H_{4}^{N}}{\partial\bar{x}}+\ldots,\\\
\frac{d}{dt}\bar{x}=&\lambda\bar{x}+\frac{\partial
H_{3}^{N}}{\partial\bar{y}}+\frac{\partial
H_{4}^{N}}{\partial\bar{y}}+\ldots.\end{split}$

Note that Hamiltonian $H_{0}^{N}$ on $\bar{\mathscr{N}}$ has two first
integrals $\bar{I}$ and $\bar{x}\bar{y}$, which are independent and in
involution.

This implies that, if $z\in W^{\rm u}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$
(resp. $z\in W^{\rm s}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$), since
$\bar{x}\bar{y}=0$, we have $\bar{I}(z)=\bar{I}(z^{-})$ (resp.
$\bar{I}(z)=\bar{I}(z^{+})$).

The Hamiltonian $H_{0}^{N}$ restricted to $\bar{\Lambda}_{0}$ is given by

$h_{0}(\bar{I}):={\omega}\bar{I}+H_{3}^{N}(\bar{I})+H_{4}^{N}(\bar{I})+\ldots,$

where we denote $H^{N}_{0}(\bar{I},0)=h_{0}(\bar{I})$, and
$H^{N}_{j}(\bar{I},0)=H^{N}_{j}(\bar{I})$ for all $j$. Each Lyapunov orbit
$\lambda_{h}$ in $\bar{\Lambda}_{0}$ corresponds to a unique level set of
$\bar{I}$, so we can write $\lambda_{0}(h)=\lambda_{0}(\bar{I})$.

By truncating the expansion (4.5) at the first order we can write

(4.7)
$H_{0}^{N}(\bar{I},\bar{\theta},\bar{y},\bar{x})=h_{0}(\bar{I})+(\bar{x}\bar{y})g_{1}(\bar{I})+(\bar{x}\bar{y})^{2}g_{2}(\bar{I},\bar{x}\bar{y}),$

for some analytic functions $h_{0}=h_{0}(\bar{I})$,
$g_{1}=g_{1}(\bar{I},\bar{x}\bar{y})$, and
$g_{2}=g_{2}(\bar{I},\bar{x}\bar{y})$.

For points $z\in W^{\rm u}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$ or $z\in
W^{\rm s}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$, since $\bar{x}\bar{y}=0$,
we have

$\frac{d}{dt}\theta(z)=\frac{\partial h_{0}}{\partial\bar{I}}(\bar{I}(z)).$

This implies that if $z\in W^{\rm u}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$
(resp. $z\in W^{\rm s}_{\bar{\mathscr{N}}}(\bar{\Lambda}_{0})$), we have
$\bar{\theta}(z)=\bar{\theta}(z^{-})$ (resp.
$\bar{\theta}(z)=\bar{\theta}(z^{+})$).

The coordinate system $(\bar{I},\bar{\theta},\bar{y},\bar{x})$ constructed
above is not yet the coordinate system from part (N). The desired coordinate
system $(I,\theta,y,x)$ will be constructed below by flowing in time the
coordinate system $(\bar{I},\bar{\theta},\bar{y},\bar{x})$.

Now we construct the coordinate system claimed in part (U).

We extend the coordinate system $(\bar{I},\bar{\theta},\bar{y},\bar{x})$ along
the flow to a neighborhood $\mathscr{N}^{\rm u}$ of $W^{\rm
u}(\bar{\Lambda}_{0})$, up to a neighborhood of the homoclinic manifold, as
follows. Let $T>0$ be a time such that
$\Phi^{T}_{0}(\bar{\mathscr{N}})\supseteq\Gamma$. Let $\mathscr{N}^{\rm
u}:=\Phi^{T}_{0}(\bar{\mathscr{N}})$. Each point $z\in\mathscr{N}^{\rm u}$ is
of the form $z=\Phi^{T}_{0}(\zeta)$ with $\zeta\in\bar{\mathscr{N}}$. We
define the coordinates $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ of $z$
to be equal to the coordinates $(\bar{I},\bar{\theta},\bar{y},\bar{x})$ of
$\zeta$, or equivalently

(4.8) $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm
u})(z)=(\bar{I},\bar{\theta},\bar{y},\bar{x})(\Phi^{-T}_{0}(z)).$

The restriction of the coordinates $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm
u})$ to $\bar{\mathscr{N}}\cap\mathscr{N}^{\rm u}$ is given by
$(\bar{I},\bar{\theta},,\bar{y}\bar{x})\circ\Phi^{-T}_{0}$.

Since the coordinates $(I^{\rm u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ of a
point $z$ are the coordinates $(\bar{I},\bar{\theta},\bar{y},\bar{x})$ of
$\Phi^{-T}_{0}(z)$, then they are symplectic, and they yield the same normal
form expansion of $H_{0}$ as (4.7). More precisely,

$\begin{split}H(I^{\rm u},\theta^{\rm u},&y^{\rm u},x^{\rm u})\\\
=&H(\bar{I}\circ\Phi^{-T}_{0},\bar{\theta}\circ\Phi^{-T}_{0},\bar{y}\circ\Phi^{-T}_{0},\bar{x}\circ\Phi^{-T}_{0})\\\
=&h_{0}(\bar{I}\circ\Phi^{-T}_{0})\\\
&+\left((\bar{x}\circ\Phi^{-T}_{0})\cdot(\bar{y}\circ\Phi^{-T}_{0})\right)g_{1}(\bar{I}\circ\Phi^{-T}_{0})\\\
&+\left((\bar{x}\circ\Phi^{-T}_{0})\cdot(\bar{y}\circ\Phi^{-T}_{0})\right)^{2}g_{2}\left((\bar{I}\circ\Phi^{-T}_{0},\bar{x}\circ\Phi^{-T}_{0})\cdot(\bar{y}\circ\Phi^{-T}_{0})\right)\\\
=&h_{0}(I^{\rm u})+(x^{\rm u}y^{\rm u})g_{1}(I)+(x^{\rm u}y^{\rm
u})^{2}g_{2}(I^{\rm u},x^{\rm u}y^{\rm u}).\end{split}$

Now we construct the coordinate system claimed in part (S).

We extend the coordinate system $(I,\theta,y,x)$ along the flow to a
neighborhood $\mathscr{N}^{\rm s}$ of $W^{\rm s}(\bar{\Lambda}_{0})$, up to
the homoclinic manifold, as follows. Start with the coordinates
$(\bar{I},\bar{\theta},\bar{y},\bar{x})\circ\Phi^{-2T}_{0}$ defined in the
neighborhood $\Phi^{-2T}_{0}(\bar{\mathscr{N}})$ of $\bar{\Lambda}_{0}$. Let
$\mathscr{N}^{\rm s}:=\Phi^{-3T}_{0}(\bar{\mathscr{N}})$. Each point
$z\in\mathscr{N}^{\rm s}$ is of the form $z=\Phi^{-T}_{0}(\zeta)$ with
$\zeta\in\Phi^{-2T}_{0}(\bar{\mathscr{N}})$. We define the coordinates
$(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$ of $z$ to be equal to the
coordinates $(\bar{I},\bar{\theta},\bar{y},\bar{x})\circ\Phi^{-2T}_{0}$ of
$\zeta$, or equivalently

(4.9) $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm
s})(z)=(\bar{I},\bar{\theta},\bar{y},\bar{x})(\Phi^{-2T}_{0}(\Phi^{T}_{0}(z)))=(\bar{I},\bar{\theta},\bar{y},\bar{x})(\Phi^{-T}_{0}(z)).$

Since the coordinates $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$ of a
point $z$ are the coordinates
$(\bar{I},\bar{\theta},\bar{y},\bar{x})\circ\Phi^{-2T}_{0}$ of
$\Phi^{T}_{0}(z)$, then they are symplectic, and they yield the same normal
for expansion of $H_{0}$ as in (4.7), that is

$H^{N}_{\mathscr{N}^{\rm s}}(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm
s})=h_{0}(I^{\rm s})+(x^{\rm s}y^{\rm s})g_{1}(I)+(x^{\rm s}y^{\rm
s})^{2}g_{2}(I^{\rm s},x^{\rm s}y^{\rm s}).$

The restriction of the coordinates $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm
s})$ to $\bar{\mathscr{N}}\cap\mathscr{N}^{\rm s}$ is given by
$(\bar{I},\bar{\theta},\bar{y},\bar{x})\circ\Phi^{-T}_{0}$.

Now we define the coordinate system claimed in part (N).

We construct the coordinate system $(I,\theta,y,x)$ in a neighborhood
$\mathscr{N}:=\bar{\mathscr{N}}\cap\mathscr{N}^{\rm u}\cap\mathscr{N}^{\rm s}$
by

(4.10)
$(I,\theta,y,x)(z)=(\bar{I},\bar{\theta},\bar{y},\bar{x})(\Phi^{-T}_{0}(z)).$

In terms of the coordinates $(I,\theta,y,x)$, the Hamiltonian has the same
normal form expansion as in (4.7), that is

$H^{N}_{\mathscr{N}}(I,\theta,y,x)=h_{0}(I)+(xy)g_{1}(I)+(xy)^{2}g_{2}(I,xy).$

We note that $\mathscr{N}$ is a small neighborhood of the equilibrium point in
the phase space. Thus we restrict the Hamiltonian to an energy range $h\in D$
such that $\Lambda_{0}=\bigcup_{h\in D}\lambda_{h}\subseteq\mathscr{N}$.
Moreover, we choose $D$ such that $W^{\rm u}(\Lambda_{0})$, $W^{\rm
s}(\Lambda_{0})$ are contained in $\mathscr{K}$, where the set $\mathscr{K}$
is as in condition (A-ii).

By the above constructions, the coordinates $(I,\theta,y,x)$, $(I^{\rm
u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ and $(I^{\rm s},\theta^{\rm s},y^{\rm
s},x^{\rm s})$ satisfy the properties listed in Proposition 4.1. ∎

We now formulate some non-degeneracy condition, in terms of the above normal
form coordinates, that $H_{0}$ must satisfy for our results.

  * (A-iii-a)

_The Hamiltonian $H_{0}$, written in the normal form given by Proposition 4.1
satisfies:_

(4.11) $\begin{split}\frac{\partial h_{0}}{\partial I}(I_{0})\neq&0\textrm{
for all }I_{0}=I_{0}(h)\textrm{ with }h\in D,\\\ g_{1}(I_{0})\neq&0\textrm{
for all }I_{0}=I_{0}(h)\textrm{ with }h\in D.\end{split}$

######  Remark 4.2.

It is important to note that the coordinates $(I^{\rm u},\theta^{\rm u},x^{\rm
u},y^{\rm u})$ and $(I^{\rm s},\theta^{\rm s},x^{\rm s},y^{\rm s})$ do not
generally agree at homoclinic points away from the Lyapunov orbit, where both
coordinate systems are well defined. Nevertheless, for any homoclinic point
$z\in M_{h}\cap W^{\rm u}(\Lambda_{0})\cap W^{\rm s}(\Lambda_{0})$, we have
that $I^{\rm u}(z)=I^{\rm s}(z)=I_{h}$.

######  Remark 4.3.

The above result on the existence of a convergent normal form in a
neighborhood of a center-saddle point, obtained via a convergent canonical
coordinate transformation, is valid for 2-degrees of freedom Hamiltonian
systems. For higher degree of freedom Hamiltonian systems, the same result is
true under some additional non-resonance conditions (see [Gio01]). A related
approach to the normal form that we use here can be found in [Mos58]. A
numerical methodology for the effective computations of normal forms is
developed in [Jor99].

######  Remark 4.4.

Now we discuss the case when we have heteroclinic connections between two
NHIMs $\Lambda^{1}_{0}$ around $L^{1}$ and $\Lambda^{2}_{0}$ around $L^{2}$,
as in Section 2.5. The manifolds $W^{\rm u}(\Lambda^{1}_{0})$ and $W^{\rm
s}(\Lambda^{2}_{0})$ are assumed to intersect transversally.

The construction of the normal form coordinates from Proposition 4.1 only
works in a small neighborhood of the equilibrium point. In the case of
heteroclinic connections, since $L^{1}$ and $L^{2}$ are on different energy
level, the theory does not guarantee the simultaneous existence two normal
form coordinate system around $L^{1}$ and $L^{2}$ respectively, for some
common energy range.

In this case, we need to make a separate assumption that there exist two
systems of normal form coordinates around $L^{1}$ and $L^{2}$, for some common
energy range. Indeed, this assumption is already made in (A’-i-b).

Based on this assumption, we can construct, as in the proof of Proposition
4.1, two systems of coordinates

  * •

$(I^{1,{\rm u}},\theta^{1,{\rm u}},y^{1,{\rm u}},x^{1,{\rm u}})$ in a
neighborhood $\mathscr{N}^{1,{\rm u}}\subseteq\mathscr{K}$ of $W^{\rm
u}(\Lambda^{1}_{0})$,

  * •

$(I^{2,{\rm s}},\theta^{2,{\rm s}},y^{2,{\rm s}},x^{2,{\rm s}})$ in a
neighborhood $\mathscr{N}^{2,{\rm s}}\subseteq\mathscr{K}$ of $W^{\rm
s}(\Lambda^{2}_{0})$,

so that they satisfy properties similar to those in the case of homoclinic
connections.

###  4.2. The scattering map for the unperturbed system

Consider the scattering map ${\sigma}_{0}$ associated to ${\Gamma}_{0}$. We
will express the scattering map in terms of the coordinates $(I^{\rm
u},\theta^{\rm u},y^{\rm u},x^{\rm u})$ and $(I^{\rm s},\theta^{\rm s},y^{\rm
s},x^{\rm s})$.

Consider a homoclinic point $z_{0}\in W^{\rm u}(\lambda_{h})\cap W^{\rm
s}(\lambda_{h})$. Both coordinate systems $(I^{\rm u},\theta^{\rm u},y^{\rm
u},x^{\rm u})$ and $(I^{\rm s},\theta^{\rm s},y^{\rm s},x^{\rm s})$ are
defined in the neighborhood of $z_{0}$.

By Proposition 4.1 the action coordinate of $z_{0}\in{\Gamma}_{0}$ is the same
as the action of the unstable and stable foot-points
$z_{0}^{-},z_{0}^{+}\in{\Lambda}_{0}$, that is $I(z_{0}^{-})=I^{\rm
u}(z_{0}^{-})=I^{\rm u}(z_{0})=I^{\rm s}(z_{0})=I^{\rm
s}(z_{0}^{+})=I(z_{0}^{+})$. Therefore the scattering map ${\sigma}_{0}$
preserves the $I$-coordinate. Hence ${\sigma}_{0}$ is a phase shift on
$I$-level-sets in ${\Lambda}_{0}$ wherever it is defined:

(4.12) ${\sigma}_{0}(I,\theta)=(I,\theta+\Delta(I)).$

In general, $\theta^{\rm u}(z_{0})\neq\theta^{\rm s}(z_{0})$. It is easy to
see that the phase-shift determined by the unperturbed scattering map is given
by the ‘mismatch’ between the two angle coordinates.

######  Proposition 4.5.

Let $h$ be a fixed energy level and let $z_{0}\in\Gamma_{0}\cap M_{h}$.

(i) The angle mismatch $\theta^{\rm s}(z_{0})-\theta^{\rm u}(z_{0})$ is a
constant that depends only on $h$, so we write it as $\theta^{\rm
s}(h)-\theta^{\rm u}(h)$.

(ii) The scattering map $\sigma_{0}$ is given by
$(I,\theta)\mapsto\sigma_{0}(I,\theta)=(I,\theta+\Delta(I))$, where
$\Delta(I)=\theta^{\rm s}(h)-\theta^{\rm u}(h)$ for $I=I_{h}$.

###### Proof.

(i) If $z_{0}$ is a point in $\Gamma_{0}\cap M_{h}$, then $\Gamma_{0}\cap
M_{h}$ consists of points of the form $\Phi^{t}_{0}(z)$, $t\in[t_{1},t_{2}]$,
for some interval $[t_{1},t_{2}]$ containing $0$.

From $\Omega^{+}(z_{0})=z_{0}^{+}$ and $\Omega^{-}(z_{0})=z_{0}^{-}$, by the
equivariance property (A.5) we have
$\Omega^{+}(\Phi^{t}_{0}(z_{0}))=\Phi^{t}_{0}(z_{0}^{+})$ and
$\Omega^{-}(\Phi^{t}_{0}(z_{0}))=\Phi^{t}_{0}(z_{0}^{-})$ for
$t\in[t_{1},t_{2}]$. By By Proposition 4.1 (U-iii) and (S-iii), and by the
fact that the dynamics restricted to $\lambda_{h}$ is a rigid rotation in
$\theta$, we have

$\begin{split}\theta^{\rm s}(\Phi^{t}_{0}(z_{0}))-\theta^{\rm
u}(\Phi^{t}_{0}(z_{0}))=&\theta(\Phi^{t}_{0}(z_{0}^{+}))-\theta(\Phi^{t}_{0}(z_{0}^{-}))\\\
=&\theta(z_{0}^{+})-\theta(z_{0}^{-}),\end{split}$

which depends only on the energy level $h$.

(ii) In action-angle coordinates, if $z_{0}^{-}=(I,\theta)$, then
$\sigma_{0}(z_{0}^{-})=z_{0}^{+}=(I,\theta+\Delta(I))$. From (i) we
immediately deduce that $\Delta(I)=\Delta(I(h))=\theta^{\rm s}(h)-\theta^{\rm
u}(h)$. ∎

Turning now the attention to the unperturbed homoclinic channel
$\tilde{\Gamma}_{0}$ in the extended phase space, this can be parametrized in
terms of the coordinates $(I^{\rm u},\theta^{\rm u},t)$, as well as in terms
of the coordinates $(I^{\rm s},\theta^{\rm s},t)$:

(4.13) $\begin{split}\tilde{\Gamma}_{0}=&\\{(I^{\rm u},\theta^{\rm u},y^{\rm
u},x^{\rm u},t)\,|\,y^{\rm u}=0,\,x^{\rm u}=x^{\rm u}_{0}(I^{\rm
u},\theta^{\rm u},t)\\},\\\ =&\\{(I^{\rm s},\theta^{\rm s},x^{\rm s},y^{\rm
s},t)\,|\,y^{\rm s}_{0}=y^{\rm s}(I^{\rm s},\theta^{\rm s},t),\,x^{\rm
s}=0\\}.\end{split}$

That is, $\tilde{\Gamma}_{0}$ is a graph over the $(I^{\rm u},\theta^{\rm
u},t)$-variables, as well as a graph over the $(I^{\rm s},\theta^{\rm
s},t)$-variables. Each homoclinic point $\tilde{z}_{0}\in\tilde{\Gamma}_{0}$
is associated to unique $I^{\rm u}=I^{\rm s}=I_{0}$, $\theta^{\rm u}$,
$\theta^{\rm s}$ with $\theta^{\rm s}(\tilde{z}_{0})-\theta^{\rm
u}(\tilde{z}_{0})=\Delta(I_{0})$. Moreover, for the points on the homoclinic
orbit $\tilde{\Phi}^{\tau}(\tilde{z}_{0})$, we also have that $\theta^{\rm
s}(\tilde{\Phi}^{\tau}(\tilde{z}_{0}))-\theta^{\rm
u}(\tilde{\Phi}^{\tau}(\tilde{z}_{0}))=\Delta(I_{0})$.

The corresponding scattering map is given by

$\tilde{\sigma}_{0}(I,\theta,t)=(I,\theta+\Delta(I),t).$

Define

(4.14) $\begin{split}\mathscr{D}_{I_{0}}=&\\{\tilde{z}_{0}\,|\,\theta^{\rm
s}(\tilde{z}_{0})-\theta^{\rm u}(\tilde{z}_{0})=\Delta(I_{0}),\,I^{\rm
s}(\tilde{z}_{0})-I^{\rm u}(\tilde{z}_{0})=0\\}.\end{split}$

We require the following condition

  * (A-iii-b)

_For all $I_{0}=I_{0}(h)$ with $h\in D$, $(\Delta(I_{0}),0)$ is a regular
value of the function_

$\tilde{z}_{0}\mapsto(\theta^{\rm s}(\tilde{z}_{0})-\theta^{\rm
u}(\tilde{z}_{0}),I^{\rm s}(\tilde{z}_{0})-I^{\rm u}(\tilde{z}_{0})).$

Condition (A-iii-b) implies that $\mathscr{D}$ is a codimension-$2$ surface in
$\widetilde{M}$, which intersects transversally $\tilde{\Gamma}_{0}$ along a
$1$-dimensional curve.

At this point, we have completed the formulation of condition (A-iii) in
Theorem 2.1, consisting of (A-iii-a), given in Section 4.1 and (A-iii-b),
given above.

###  4.3. Perturbed evolution equations

In the sequel, we will identify the vector fields $\mathcal{X}^{0}$ and
$\mathcal{X}^{1}$ with derivative operators acting on functions. In general,
given a smooth vector field $\mathcal{X}$ and a smooth function $f$ on a
manifold $M$, and $(z_{j})_{j\in\\{1,\ldots,\dim(M)\\}}$ a system of local
coordinates, then

(4.15) $(\mathcal{X}f)(z)=\sum_{j}(\mathcal{X})_{j}(z)(\partial_{z_{j}}f)(z).$

Consider one of the coordinate systems defined in Section 4.1. To simplify
notation, we will denote such a coordinate system by $(I,\theta,x,y)$. Below
we provide evolution equations of these coordinates, expressing the time-
derivative of each coordinate along a solution of the perturbed system. We
include the expression for a general perturbation, as well as for the case
when the perturbation is Hamiltonian:

(4.16)
$\begin{split}\frac{d}{dt}I&=(\mathcal{X}^{0}+\varepsilon\mathcal{X}^{1})(I)=-\frac{\partial{H_{0}}}{\partial\theta}+\varepsilon\mathcal{X}^{1}(I)\\\
&=-\frac{\partial{H_{0}}}{\partial\theta}-\varepsilon\frac{\partial{H_{1}}}{\partial\theta}.\end{split}$
(4.17)
$\begin{split}\frac{d}{dt}\theta&=(\mathcal{X}^{0}+\varepsilon\mathcal{X}^{1})(\theta)=\frac{\partial{H_{0}}}{\partial
I}+\varepsilon\mathcal{X}^{1}(\theta)\\\ &=\frac{\partial{H_{0}}}{\partial
I}+\varepsilon\frac{\partial{H_{1}}}{\partial I}.\end{split}$ (4.18)
$\begin{split}\frac{d}{dt}y&=\mathcal{X}^{0}(y)+\varepsilon\mathcal{X}^{1}(y)=-\frac{\partial{H_{0}}}{\partial
x}+\varepsilon\mathcal{X}^{1}(y)\\\ &=-\frac{\partial{H_{0}}}{\partial
x}-\varepsilon\frac{\partial{H_{1}}}{\partial x}.\end{split}$ (4.19)
$\begin{split}\frac{d}{dt}x&=\mathcal{X}^{0}(x)+\varepsilon\mathcal{X}^{1}(x)=\frac{\partial{H_{0}}}{\partial
y}+\varepsilon\mathcal{X}^{1}(x)\\\ &=\frac{\partial{H_{0}}}{\partial
y}+\varepsilon\frac{\partial{H_{1}}}{\partial y}.\end{split}$

##  5\. Proof of the main result

In this section we prove Theorem 2.1.

###  5.1. Perturbed normally hyperbolic invariant manifolds

In this section we prove the assertions (i) and (ii) of Theorem 2.1.

We only give the details in the case when $H_{0}$ satisfies the conditions
(A-i), (A-ii), (A-iii). The case when $H_{0}$ satisfies (A’-i), (A’-ii),
(A’-iii) follows similarly.

####  5.1.1. Persistence of the normally hyperbolic invariant manifold under
perturbation

We have that $\Lambda_{0}$ is a NHIM for the flow $\Phi^{t}_{0}$ of
$\mathcal{X}^{0}$. Then $D\Phi^{t}_{0}(z)$ satisfies expansion/contraction
rates as in Appendix A, for all $z\in\Lambda_{0}$, where we denote the
constant and the expansion and contraction rates by $C$, $\lambda_{-}$,
$\lambda_{+}$, $\lambda_{c}$, $\mu_{c}$, $\mu_{-}$, $\mu_{+}$, respectively.

It is immediate that ${\tilde{\Lambda}}_{0}=\Lambda_{0}\times\mathbb{R}$ is a
NHIM for the flow $\tilde{\Phi}^{\tau}_{0}$ of the extended system (2.6).

Under the assumptions of Theorem 2.1,
$\mathcal{X}^{1}=\mathcal{X}^{1}(z,t;\varepsilon)$ is uniformly differentiable
in all variables. The theory of normally hyperbolic invariant manifolds,
[Fen72, HPS77, Pes04] (a handy summary of the results of the theory is
[DdlLS06]), asserts that there exists $\varepsilon_{1}$ such that the manifold
${\tilde{\Lambda}}_{0}$ persists as a normally hyperbolic manifold
${\tilde{\Lambda}}_{\varepsilon}$, for all $|\varepsilon|<\varepsilon_{1}$,
which is locally invariant under the flow $\tilde{\Phi}^{\tau}_{\varepsilon}$.
The persistent NHIM ${\tilde{\Lambda}}_{\varepsilon}$ is $O(\varepsilon)$
close in the $\mathcal{C}^{\ell}$-topology to ${\tilde{\Lambda}}_{0}$, where
$\ell$ is as in A.3. The locally invariant manifolds are in fact invariant
manifolds for an extended system, and they depend on the extension. Hence,
they do not need to be unique. Nevertheless, we point out that, given a family
of systems, it is possible to choose the invariant manifolds in such a way
that the invariant manifods depend smoothly on parameters, as well as the
stable and unstable bundles and the stable and unstable manifolds.

For the perturbed NHIM ${\tilde{\Lambda}}_{\varepsilon}$,
$|\varepsilon|<\varepsilon_{1}$, there exists an invariant splitting of the
tangent bundle $T{\tilde{\Lambda}}_{\varepsilon}$, similar to that in (A.1),
so that $D\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z})$ satisfies
expansion/contraction relations similar to those in (A.2) for all
$\tilde{z}\in\tilde{\Lambda}_{\varepsilon}$, for some constants $\tilde{C}$,
$\tilde{\lambda}_{-}$, $\tilde{\lambda}_{+}$, $\tilde{\lambda}_{c}$,
$\tilde{\mu}_{c}$, $\tilde{\mu}_{-}$, $\tilde{\mu}_{+}$. These constants are
independent of $\varepsilon$, and can be chosen as close as desired to the
unperturbed ones, that is, to $C$, $\lambda_{-}$, $\lambda_{+}$,
$\lambda_{c}$, $\mu_{c}$, $\mu_{-}$, $\mu_{+}$, respectively, by choosing
$\varepsilon_{1}$ suitably small.

There exist unstable and stable manifolds $W^{\rm
u}({\tilde{\Lambda}}_{\varepsilon})$, $W^{\rm
s}({\tilde{\Lambda}}_{\varepsilon})$ associated to
${\tilde{\Lambda}}_{\varepsilon}$, and there exist corresponding projection
maps $\Omega^{-}:W^{\rm
u}({\tilde{\Lambda}}_{\varepsilon})\to{\tilde{\Lambda}}_{\varepsilon}$, and
$\Omega^{+}:W^{\rm
s}({\tilde{\Lambda}}_{\varepsilon})\to{\tilde{\Lambda}}_{\varepsilon}$.

For $\tilde{z}^{+}=\Omega^{+}(\tilde{z})$, with $\tilde{z}\in W^{\rm
s}({\tilde{\Lambda}}_{\varepsilon})$ we have

(5.1)
$d(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}),\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}))\leq
C_{\tilde{z}}e^{\tau\tilde{\lambda}_{+}},\quad\textrm{ for all }\tau\geq 0,$

and for $\tilde{z}^{-}=\Omega^{-}(\tilde{z})$, with $\tilde{z}\in W^{\rm
u}({\tilde{\Lambda}}_{\varepsilon})$ we have

(5.2)
$d(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}),\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{-})\leq
C_{\tilde{z}}e^{\tau\tilde{\mu}_{-}},\quad\textrm{ for all }\tau\leq 0,$

for some ${C}_{\tilde{z}}>0$. The constant $\tilde{C}_{\tilde{z}}$ can be
chosen uniformly bounded, provided that we restrict to $z$ to the compact
neighborhood $\mathscr{K}$ given by (A-ii), and we use the fact that
$\mathcal{X}^{1}=\mathcal{X}^{1}(z,t;\varepsilon)$ is uniformly differentiable
in all variables.

To simplify notation, from now on we will drop the symbol $\tilde{}$ from
$\tilde{C}$, $\tilde{C}_{\tilde{z}}$ $\tilde{\lambda}_{-}$,
$\tilde{\lambda}_{+}$, $\tilde{\mu}_{-}$, $\tilde{\mu}_{+}$,
$\tilde{\lambda}_{c}$, $\tilde{\mu}_{c}$.

In the sequel, we will fix a choice ${\tilde{\Lambda}}_{\varepsilon}$, and all
computations will be performed relative to that choice. Nevertheless the
estimate for the perturbed scattering map $\tilde{\sigma}_{\varepsilon}$ are
independent of the choice of the locally invariant manifold
${\tilde{\Lambda}}_{\varepsilon}$.

####  5.1.2. Persistence of the transverse intersection of the hyperbolic
invariant manifolds under perturbation

For the unperturbed system the unstable and stable manifolds $W^{\rm
u}({\tilde{\Lambda}}_{0})$, $W^{\rm s}({\tilde{\Lambda}}_{0})$ intersect
transversally along the $3$-dimensional homoclinic channel
$\tilde{\Gamma}_{0}$. By the persistence of transversality under small
perturbations, it follows that $W^{\rm u}({\tilde{\Lambda}}_{\varepsilon})$,
$W^{\rm s}({\tilde{\Lambda}}_{\varepsilon})$ intersect transversally along
$\tilde{\Gamma}_{\varepsilon}$, for all $|\varepsilon|<\varepsilon_{1}$,
provided $\varepsilon_{1}$ is chosen small enough. The condition (B.1) in the
definition of a homoclinic/heteroclinic channel is also a transversality-type
condition, so it is also persistent under small perturbations. We conclude
that (i) and (ii) from Theorem 2.1 hold true for all
$|\varepsilon|<\varepsilon_{1}$, provided $\varepsilon_{1}$ is chosen small
enough.

Recall from Section 4.2 that the unperturbed homoclinic channel
$\tilde{\Gamma}_{0}$ can be described as a graph over the $(I^{\rm
u},\theta^{\rm u},t)$-variables, as well as a graph over the $(I^{\rm
s},\theta^{\rm s},t)$-variables. Therefore, the perturbed homoclinic channel
$\tilde{\Gamma}_{\varepsilon}$, for $|\varepsilon|<\varepsilon_{1}$, can also
be described as a graph over the $(I^{\rm u},\theta^{\rm u},t)$-variables, as
well as a graph over the $(I^{\rm s},\theta^{\rm s},t)$-variables. Therefore,
each homoclinic point $\tilde{z}_{\varepsilon}\in\tilde{\Gamma}_{\varepsilon}$
is associated to unique coordinate triples $(I^{\rm u},\theta^{\rm u},t)$, and
$(I^{\rm s},\theta^{\rm s},t)$. More precisely, we have

(5.3) $\begin{split}\tilde{\Gamma}_{\varepsilon}=&\\{(I^{\rm u},\theta^{\rm
u},y^{\rm u},x^{\rm u},t)\,|\,y^{\rm u}=y^{\rm u}_{\varepsilon}(I^{\rm
u},\theta^{\rm u},t),\,x^{\rm u}=x^{\rm u}_{\varepsilon}(I^{\rm u},\theta^{\rm
u},t)\\},\\\ \tilde{\Gamma}_{\varepsilon}=&\\{(I^{\rm s},\theta^{\rm s},y^{\rm
s},x^{\rm s},t)\,|\,y^{\rm s}=y^{\rm s}_{\varepsilon}(I^{\rm s},\theta^{\rm
s},t),\,x^{\rm s}=x^{\rm s}_{\varepsilon}(I^{\rm s},\theta^{\rm
s},t)\\},\end{split}$

with $y^{\rm u}_{\varepsilon}(I^{\rm u},\theta^{\rm u},t)=O(\varepsilon)$ and
$x^{\rm s}_{\varepsilon}(I^{\rm s},\theta^{\rm s},t)=O(\varepsilon)$.

By condition (A-iii-b), since $\mathscr{D}_{I_{0}}$ intersects
$\tilde{\Gamma}_{0}$ transversally, it follows that $\mathscr{D}_{I_{0}}$
intersects $\tilde{\Gamma}_{\varepsilon}$ for $\varepsilon$ sufficiently
small. Therefore, given $\tilde{z}_{0}$ in $\tilde{\Gamma}_{0}$ we will
associate to it a homoclinic point
$\tilde{z}_{\varepsilon}\in\mathscr{D}_{I_{0}}$, that is, satisfying the
conditions

(5.4) $\begin{split}\theta^{\rm s}(\tilde{z}_{\varepsilon})-\theta^{\rm
u}(\tilde{z}_{\varepsilon})=&\theta^{\rm s}(\tilde{z}_{0})-\theta^{\rm
u}(\tilde{z}_{0}),\\\ I^{\rm s}(\tilde{z}_{\varepsilon})-I^{\rm
u}(\tilde{z}_{\varepsilon})=&0.\end{split}$

Note that a homoclinic point $\tilde{z}_{\varepsilon}$ satisfying these
conditions is not uniquely defined. We can impose on $\tilde{z}_{\varepsilon}$
an additional condition, for instance
$t(\tilde{z}_{\varepsilon})=t(\tilde{z}_{0})$, or $I^{\rm
s}(\tilde{z}_{\varepsilon})=I^{\rm u}(\tilde{z}_{\varepsilon})=I_{0}$. Such an
extra condition can be useful for applications.

In the sequel we will compare the scattering map associated to $\tilde{z}_{0}$
with the scattering map associated to $\tilde{z}_{\varepsilon}$ satisfying
(5.4).

###  5.2. Perturbed scattering map

In this section we prove the assertion (iii) of Theorem 2.1.

We start with the unperturbed system (2.3). We recall that for a given
homoclinic channel $\Gamma_{0}$, the corresponding scattering map
$\sigma_{0}$, is a phase-shift of the form

$\sigma_{0}(I,\theta)=(I,\theta+\Delta(I)).$

We choose and fix an energy level $h$ of $H_{0}$, and a point
$z_{0}\in\Gamma_{0}\cap M_{h}$. In the $(I^{{\rm u},{\rm s}},\theta^{{\rm
u},{\rm s}},y^{{\rm u},{\rm s}},x^{{\rm u},{\rm s}})$-coordinates, $z_{0}$ is
given by

(5.5) $z_{0}=(I^{\rm s}_{0},\theta^{\rm s}_{0},y^{\rm s}_{0},0)=(I^{\rm
u}_{0},\theta^{\rm u}_{0},0,x^{\rm u}_{0}),$

where $I^{\rm s}_{0}=I^{\rm u}_{0}=I_{0}$. The effect of the flow
$\Phi^{\tau}_{0}$ on $z_{0}$ in these coordinates is given by

(5.6) $\Phi^{\tau}_{0}(z_{0})=(I^{\rm s}_{0},\theta^{\rm
s}_{0}+\omega(I_{0})\tau,y^{\rm s}(\tau),0)=(I^{\rm u}_{0},\theta^{\rm
u}_{0}+\omega(I_{0})\tau,0,x^{\rm u}(\tau)),$

where $y^{\rm s}(\tau)$ and $x^{\rm u}(\tau)$ are the $y^{\rm s}$-component
and the $x^{\rm u}$-component, respectively, of $\Phi^{\tau}_{0}(z_{0})$
evaluated at time $\tau$, and $\omega(I_{0})=\frac{\partial h_{0}}{\partial
I}$.

There exist uniquely defined points $z^{-}_{0}$, $z^{+}_{0}$ in
$\lambda_{0}(h)$ such that $W^{\rm u}(z^{-}_{0})\cap(\Gamma_{0}\cap
M_{h})=W^{\rm s}(z^{+}_{0})\cap(\Gamma_{0}\cap M_{h})=\\{z_{0}\\}$. In the
$(I,\theta,y,x)$-coordinates, the foot-points $z_{0}^{\pm}$ are given by

(5.7) $\begin{split}z^{-}_{0}=(I_{0},\theta^{-}_{0},0,0),\quad
z^{+}_{0}=(I^{\rm s}_{0},\theta^{+}_{0},0,0),\end{split}$

where $\theta^{-}_{0}=\theta^{\rm u}_{0}$ and $\theta^{+}_{0}=\theta^{\rm
s}_{0}$. The effect of the flow $\Phi^{\tau}_{0}$ on $z_{0}^{\pm}$ in these
coordinates is given by

(5.8)
$\begin{split}\Phi^{\tau}_{0}(z^{-}_{0})=(I_{0},\theta^{-}_{0}+\omega(I_{0})\tau,0,0),\\\
\Phi^{\tau}_{0}(z^{+}_{0})=(I_{0},\theta^{+}_{0}+\omega(I_{0})\tau,0,0).\end{split}$

The scattering map $\sigma_{0}$ takes $z^{-}_{0}\in\lambda_{0}(h)$ into
$z^{+}_{0}\in\lambda_{0}(h)$.

In the extended system (2.6), the corresponding homoclinic point is
$\tilde{z}_{0}=(z_{0},t_{0})$ for some $t_{0}\in\mathbb{R}$. The scattering
map $\tilde{\sigma}_{0}$ takes $\tilde{z}^{-}_{0}=(z^{-}_{0},t_{0})$ into
$\tilde{z}^{+}_{0}=(z^{+}_{0},t_{0})$.

We will compute the effect of the perturbation on the scattering map
$\tilde{\sigma}_{0}$.

When we add the perturbation there exists a homoclinic point
$\tilde{z}_{\varepsilon}\in\tilde{\Gamma}_{\varepsilon}$ corresponding to
$\tilde{z}_{0}=(z_{0},t_{0})$ from the unperturbed case, such that
$\tilde{z}_{\varepsilon}$ satisfies the condition (A-iii-b). Associated to
$\tilde{z}_{\varepsilon}\in\tilde{\Gamma}_{\varepsilon}$ we have the points
$\tilde{z}^{-}_{\varepsilon}$, $\tilde{z}^{+}_{\varepsilon}$ in
$\tilde{\Lambda}_{\varepsilon}$ such that $W^{\rm
u}(\tilde{z}^{-}_{\varepsilon})\cap\tilde{\Gamma}_{\varepsilon}=\\{\tilde{z}_{\varepsilon}\\}$,
and $W^{\rm
s}(\tilde{z}^{+}_{\varepsilon})\cap\tilde{\Gamma}_{\varepsilon}=\\{\tilde{z}_{\varepsilon}\\}$.
The scattering map $\tilde{\sigma}_{\varepsilon}$ takes
$\tilde{z}^{-}_{\varepsilon}\in{\tilde{\Lambda}}_{\varepsilon}$ into
$\tilde{z}^{+}_{\varepsilon}\in{\tilde{\Lambda}}_{\varepsilon}$.

In the sequel we will make a quantitative comparison between

$\tilde{z}^{-}_{0}\mapsto\tilde{\sigma}_{0}({z}^{-}_{0}):=\tilde{z}^{+}_{0},$

and

$\tilde{z}^{-}_{\varepsilon}\mapsto\tilde{\sigma}_{\varepsilon}(\tilde{z}^{-}_{\varepsilon}):=\tilde{z}^{+}_{\varepsilon}.$

####  5.2.1. Estimates

Below we will refer to the notation in (5.5), (5.6), (5.7), and (5.8). To
simplify notation, we denote $I^{{\rm s}}_{\varepsilon}=I^{\rm
s}(\tilde{z}_{\varepsilon})$, $I^{{\rm u}}_{\varepsilon}=I^{\rm
u}(\tilde{z}_{\varepsilon})$, $I^{{\rm s}+}_{\varepsilon}=I^{\rm
s}(\tilde{z}^{+}_{\varepsilon})$, $I^{{\rm u}-}_{\varepsilon}=I^{\rm
u}(\tilde{z}^{-}_{\varepsilon})$, $\xi^{{\rm s}}_{\varepsilon}=(x^{\rm
s}y^{\rm s})(\tilde{z}_{\varepsilon})$, $\xi^{{\rm s}+}=(x^{\rm s}y^{\rm
s})(\tilde{z}^{+}_{\varepsilon})$, $\xi^{{\rm u}}_{\varepsilon}=(x^{\rm
u}y^{\rm u})(\tilde{z}_{\varepsilon})$, $\xi^{{\rm u}-}=(x^{\rm u}y^{\rm
u})(\tilde{z}^{-}_{\varepsilon})$.

Note that in the following, the coordinates of the scattering map can be
considered as functions of the point. Hence, the symbols $O(\varepsilon)$ can
be interpreted as relative to the $C^{r}$ norm.

######  Lemma 5.1.

  * (i)

Estimates on $I$:

(5.9) $\begin{split}I^{{\rm s}+}_{\varepsilon}-I^{{\rm
s}}_{\varepsilon}=O(\varepsilon),\,I^{{\rm u}-}_{\varepsilon}-I^{{\rm
u}}_{\varepsilon}=O(\varepsilon),\,I^{{\rm s}+}_{\varepsilon}-I^{{\rm
u}-}_{\varepsilon}=O(\varepsilon).\end{split}$

  * (ii)

Estimates on $h_{0}$:

(5.10) $\begin{split}h_{0}(I^{{\rm s}+}_{\varepsilon})-h_{0}(I^{{\rm
s}}_{\varepsilon})=&(I^{{\rm s}+}_{\varepsilon}-I^{{\rm
s}}_{\varepsilon})\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)+O(\varepsilon^{2}),\\\ h_{0}(I^{{\rm
u}-}_{\varepsilon})-h_{0}(I^{{\rm u}}_{\varepsilon})=&(I^{{\rm
u}-}_{\varepsilon}-I^{{\rm u}}_{\varepsilon})\left(\frac{\partial
h_{0}}{\partial I}(I_{0})\right)+O(\varepsilon^{2}).\end{split}$

  * (iii)

Estimates on $\xi$:

(5.11) $\xi^{{\rm s}}_{\varepsilon}=O(\varepsilon),\,\xi^{{\rm
u}}_{\varepsilon}=O(\varepsilon),\,\xi^{{\rm
s}+}_{\varepsilon}=O(\varepsilon^{2}),\,\xi^{{\rm
u}-}_{\varepsilon}=O(\varepsilon^{2}).$

  * (iv)

Estimates on $g_{1}$:

(5.12) $\begin{split}g_{1}(I^{{\rm
s}+}_{\varepsilon})=&g_{1}(I_{0})+\frac{\partial g_{1}}{\partial
I}(I_{0})(I^{{\rm s}+}_{\varepsilon}-I_{0})+O(\varepsilon^{2}),\\\
g_{1}(I^{{\rm u}-}_{\varepsilon})=&g_{1}(I_{0})+\frac{\partial g_{1}}{\partial
I}(I_{0})(I^{{\rm u}-}_{\varepsilon}-I_{0})+O(\varepsilon^{2}),\end{split}$

  * (v)

Estimates on $\frac{\partial g_{1}}{\partial I}$:

(5.13) $\begin{split}\frac{\partial g_{1}}{\partial I}(I^{{\rm
s}+}_{\varepsilon})=&\frac{\partial g_{1}}{\partial
I}(I_{0})+\frac{\partial^{2}g_{1}}{\partial I^{2}}(I_{0})(I^{{\rm
s}+}_{\varepsilon}-I_{0})+O(\varepsilon^{2}),\\\ \frac{\partial
g_{1}}{\partial I}(I^{{\rm u}-}_{\varepsilon})=&\frac{\partial g_{1}}{\partial
I}(I_{0})+\frac{\partial^{2}g_{1}}{\partial I^{2}}(I_{0})(I^{{\rm
u}-}_{\varepsilon}-I_{0})+O(\varepsilon^{2}),\end{split}$

###### Proof.

(i) Since $\tilde{z}_{\varepsilon}=\tilde{z}_{0}+O(\varepsilon)$ and
$\tilde{z}^{\pm}_{\varepsilon}=\tilde{z}^{\pm}_{0}+O(\varepsilon)$, we have
$I^{{\rm s},{\rm u}}(\tilde{z}_{\varepsilon})=I^{{\rm s},{\rm
u}}(\tilde{z}_{0})+O(\varepsilon)$, and $I^{{\rm s},{\rm
u}}(\tilde{z}^{\pm}_{\varepsilon})=I^{{\rm s},{\rm
u}}(\tilde{z}^{\pm}_{0})+O(\varepsilon)$. The fact that $I^{{\rm s},{\rm
u}}(\tilde{z}_{0})=I^{{\rm s},{\rm u}}(\tilde{z}^{\pm}_{0})=I_{0}$ yields the
estimates in (i).

(ii) We estimate the term $h_{0}(I^{{\rm s}+}_{\varepsilon})-h_{0}(I^{{\rm
s}}_{\varepsilon})$. Applying the integral form of the Mean Value Theorem we
have

(5.14) $\begin{split}h_{0}(I^{{\rm s}+}_{\varepsilon})-h_{0}(I^{{\rm
s}}_{\varepsilon})=&(I^{{\rm s}+}_{\varepsilon}-I^{{\rm
s}}_{\varepsilon})\int_{0}^{1}\frac{\partial h_{0}}{\partial I}(tI^{{\rm
s}+}_{\varepsilon}+(1-t)I^{{\rm s}}_{\varepsilon})dt.\end{split}$

We write the integrand of (5.14) as a Taylor expansion

(5.15) $\begin{split}\frac{\partial h_{0}}{\partial I}(tI^{{\rm
s}+}_{\varepsilon}+(1-t)I^{{\rm s}}_{\varepsilon})=&\frac{\partial
h_{0}}{\partial I}(I_{0})\\\ &+\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})(t(I^{{\rm s}+}_{\varepsilon}-I^{{\rm s}+}_{0})+(1-t)(I^{{\rm
s}}_{\varepsilon}-I^{{\rm s}}_{0}))\\\ &+O(\varepsilon^{2})\end{split}$

where we used

$\frac{\partial h_{0}}{\partial I}(tI^{{\rm s}+}_{0}+(1-t)I^{{\rm
s}}_{0})=\frac{\partial h_{0}}{\partial I}(I_{0})\textrm{ and
}\frac{\partial^{2}h_{0}}{\partial I^{2}}(tI^{{\rm s}+}_{0}+(1-t)I^{{\rm
s}}_{0})=\frac{\partial^{2}h_{0}}{\partial I^{2}}(I_{0}).$

Since $I^{{\rm s}+}_{\varepsilon}-I^{{\rm s}+}_{0}=O(\varepsilon)$ and
$I^{{\rm s}}_{\varepsilon}-I^{{\rm s}}_{0}=O(\varepsilon)$, we have

(5.16) $\frac{\partial h_{0}}{\partial I}(tI^{{\rm s}+}_{0}+(1-t)I^{{\rm
s}}_{0})=\frac{\partial h_{0}}{\partial I}(I_{0})+O(\varepsilon).$

Since $I^{{\rm s}+}_{\varepsilon}-I^{{\rm s}}_{\varepsilon}=O(\varepsilon)$
from (5.14) we obtain the first estimate in (5.10).

The other estimate follows similarly.

(iii) The fact that $\xi^{{\rm s},{\rm u}}=O(\varepsilon)$ follows from
$\tilde{z}_{\varepsilon}=\tilde{z}_{0}+O(\varepsilon)$, and $\xi^{{\rm s},{\rm
u}}(\tilde{z}_{0})=0$, hence $\xi^{{\rm s},{\rm
u}}(\tilde{z}_{\varepsilon})=\xi^{{\rm s},{\rm
u}}(\tilde{z}_{0})+O(\varepsilon)=O(\varepsilon)$.

In the same way we obtain $\xi^{{\rm s}+}=\xi^{{\rm u}-}=O(\varepsilon)$. To
prove that in fact $\xi^{{\rm s}+}=\xi^{{\rm u}-}=O(\varepsilon^{2})$, we
proceed as follows. For $\xi^{{\rm s}+}_{\varepsilon}$, we use the Taylor
expansion:

$\begin{split}\xi^{{\rm s}+}_{\varepsilon}=&\xi^{{\rm s}+}_{0}+D\xi^{{\rm
s}+}_{0}\cdot(\xi^{{\rm s}+}_{\varepsilon}-\xi^{{\rm
s}+}_{0})+O(\varepsilon^{2}),\\\ \end{split}$

where $\cdot$ denotes the dot product, and we used that $\xi^{{\rm
s}+}_{\varepsilon}-\xi^{{\rm s}+}_{0}=O(\varepsilon)$. We have

$\xi^{{\rm s}+}_{0}=0$

and

$D\xi^{{\rm s}+}_{0}\cdot(\xi^{{\rm s}+}_{\varepsilon}-\xi^{{\rm
s}+}_{0})=x^{\rm s}(\tilde{z}^{+}_{0})(y^{{\rm s}+}_{\varepsilon}-y^{{\rm
s}+}_{0})+y^{\rm s}(\tilde{z}^{+}_{0})(x^{{\rm s}+}_{\varepsilon}-x^{{\rm
s}+}_{0})=0,$

since $y^{\rm s}(\tilde{z}^{+}_{0})=x^{\rm s}(\tilde{z}^{+}_{0})=0$ on
${\tilde{\Lambda}}_{0}$. Therefore (5.2.1) implies

$\begin{split}\xi^{{\rm s}+}_{\varepsilon}=&O(\varepsilon^{2}).\end{split}$

Similarly we obtain

$\begin{split}\xi^{{\rm u}-}_{\varepsilon}=&O(\varepsilon^{2}).\end{split}$

(iv) We write $g_{1}(I^{{\rm s}+}_{\varepsilon})$ as a Taylor expansion, using
that $I^{{\rm s}+}_{\varepsilon}-I^{{\rm s}+}_{0}=O(\varepsilon)$, obtaining

$g_{1}(I^{{\rm s}+}_{\varepsilon})=g_{1}(I^{{\rm s}+}_{0})+\frac{\partial
g_{1}}{\partial I}(I^{{\rm s}+}_{0})(I^{{\rm s}+}_{\varepsilon}-I^{{\rm
s}+}_{0})+O(\varepsilon^{2}).$

Since $I^{{\rm s}+}_{0}=I_{0}$, the first equation in (5.12) follows.

Similarly

$g_{1}(I^{{\rm u}-}_{\varepsilon})=g_{1}(I^{{\rm u}-}_{0})+\frac{\partial
g_{1}}{\partial I}(I^{{\rm u}-}_{0})(I^{{\rm u}-}_{\varepsilon}-I^{{\rm
u}-}_{0})+O(\varepsilon^{2}),$

and $I^{{\rm u}-}_{0}=I_{0}$ yield the second equation in (5.12).

(v) The proof is similar to that of (iv). ∎

####  5.2.2. Change in action by the scattering map

We now give the expression of the action-component $\tilde{\mathcal{S}}^{I}$
of the mapping $\tilde{\mathcal{S}}$ in (2.7). Below we will refer to the
notation in (5.5), (5.6), (5.7), and (5.8).

######  Proposition 5.2.

The change in $I$ by the scattering map $\tilde{\sigma}_{\varepsilon}$ is
given by the following formula

(5.17)
$\begin{split}I\left(\tilde{z}^{+}_{\varepsilon}\right)&-I\left(\tilde{z}^{-}_{\varepsilon}\right)\\\
=&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})({\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0})-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

When the perturbation is Hamiltonian $\mathcal{X}^{1}=J\nabla H_{1}$, in
(5.17) we have $\mathcal{X}^{1}H_{0}=\\{H_{0},H_{1}\\}$, where
$\\{\cdot,\cdot\\}$ denotes the Poisson bracket.

Proposition 5.2 implies that

$\tilde{\sigma}^{I}_{\varepsilon}(I,\theta,t)=\tilde{\sigma}^{I}_{0}(I,\theta,t)+\varepsilon\tilde{\mathcal{S}}^{I}(I,\theta,t)+O(\varepsilon^{2}),$

where $\varepsilon\tilde{\mathcal{S}}^{I}$ is given by the first two terms on
the right hand-side of (5.17). The expression of
$\varepsilon\tilde{\mathcal{S}}^{I}$ is particularly simple since $I$ is a
slow variable.

We note that we can express the right-hand side of (5.17) in terms of the
$(I^{{\rm u},{\rm s}},\theta^{{\rm u},{\rm s}},y^{{\rm u},{\rm s}},x^{{\rm
u},{\rm s}})$ coordinates, by making the following substitutions:

$\begin{split}\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0})=&\left(I^{\rm
s}_{0},\theta^{\rm s}_{0}+\omega(I_{0})\tau,0,0,t_{0}+\tau\right),\\\
\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0})=&\left(I^{\rm s}_{0},\theta^{\rm
u}_{0}+\omega(I_{0})\tau,0,0,t_{0}+\tau\right),\\\
\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0})=&\left(I^{\rm s}_{0},\theta^{\rm
s}_{0}+\omega(I_{0})\tau,y^{\rm s}(\tau),0,t_{0}+\tau\right)\\\ =&\left(I^{\rm
u}_{0},\theta^{\rm u}_{0}+\omega(I_{0})\tau,0,x^{\rm
u}(\tau),t_{0}+\tau\right).\end{split}$

To prove Proposition 5.2 we will use the following:

######  Lemma 5.3.

The change in $H_{0}$ by the scattering map $\tilde{\sigma}_{\varepsilon}$ is
given by the following equation

(5.18)
$\begin{split}H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}^{-}_{\varepsilon})=&-\varepsilon\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&-\varepsilon\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

###### Proof of Lemma 5.3.

First, note that

$(\mathcal{X}^{0}+\varepsilon\mathcal{X}^{1})H_{0}=\mathcal{X}^{0}H_{0}+\varepsilon\mathcal{X}^{1}H_{0}=\\{H_{0},H_{0}\\}+\varepsilon\mathcal{X}^{1}H_{0}=\varepsilon\mathcal{X}^{1}H_{0}.$

Second, applying Lemma D.1 and Lemma D.2 from Appendix D, for
$\mathbf{F}=H_{0}$, we have

(5.19)
$\begin{split}H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}_{\varepsilon})=&-\varepsilon\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{1+\varrho}\right)\end{split}$ (5.20)
$\begin{split}H_{0}(\tilde{z}^{-}_{\varepsilon})-H_{0}(\tilde{z}_{\varepsilon})=&\varepsilon\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{1+\varrho}\right).\end{split}$

Subtracting the two equations from above, after cancelling out the common term
$H_{0}(\tilde{z}_{\varepsilon})$ representing the value of $H_{0}$ at the
homoclinic point $\tilde{z}_{0}$, we obtain (5.18) with an error term of order
$O(\varepsilon^{1+\varrho})$. Since the function
$H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}^{-}_{\varepsilon})$ can be
expanded as a Taylor series in $\varepsilon$, by matching the corresponding
terms of this Taylor expansion with the terms in (5.19) minus (5.20), it
follows that the error term $O(\varepsilon^{1+\varrho})$ must equal
$O(\varepsilon^{2})$. ∎

###### Proof of Proposition 5.2.

By Proposition 4.1 we have

(5.21) $\begin{split}H_{0}(\tilde{z}^{+}_{\varepsilon})=&h_{0}(I^{{\rm
s}+}_{\varepsilon})+(\xi^{{\rm s}+}_{\varepsilon})g_{1}(I^{{\rm
s}+}_{\varepsilon})+(\xi^{{\rm s}+}_{\varepsilon})^{2}g_{2}(I^{{\rm
s}+}_{\varepsilon},\xi^{{\rm s}+}_{\varepsilon}),\end{split}$ (5.22)
$\begin{split}H_{0}(\tilde{z}^{-}_{\varepsilon})=&h_{0}(I^{{\rm
u}-}_{\varepsilon})+(\xi^{{\rm u}-}_{\varepsilon})g_{1}(I^{{\rm
u}-}_{\varepsilon})+(\xi^{{\rm u}-}_{\varepsilon})^{2}g_{2}(I^{{\rm
u}-}_{\varepsilon},\xi^{{\rm u}-}_{\varepsilon}).\end{split}$

Subtracting we obtain

(5.23)
$\begin{split}H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}^{-}_{\varepsilon})=&(h_{0}(I^{{\rm
s}+}_{\varepsilon})-h_{0}(I^{{\rm u}-}_{\varepsilon}))\\\ &+((\xi^{{\rm
s}+}_{\varepsilon})g_{1}(I^{{\rm s}+}_{\varepsilon})-(\xi^{{\rm
u}-}_{\varepsilon})g_{1}(I^{{\rm u}-}_{\varepsilon}))\\\
&+O(\varepsilon^{2})\end{split}$

where the error term $O(\varepsilon^{2})$ in the above is due to Lemma 5.1
equation (5.11).

The term $h_{0}(I^{{\rm s}+}_{\varepsilon})-h_{0}(I^{{\rm u}-}_{\varepsilon})$
in (5.23) is given, by Lemma 5.1 equation (5.10), as

(5.24) $h_{0}(I^{{\rm s}+}_{\varepsilon})-h_{0}(I^{{\rm
u}-}_{\varepsilon})=(I^{{\rm s}+}_{\varepsilon}-I^{{\rm
u}-}_{\varepsilon})\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)+O(\varepsilon^{2}).$

Since, by Lemma 5.1 equation (5.11), we have

(5.25) $\begin{split}\xi^{{\rm
s}+}_{\varepsilon}=O(\varepsilon^{2}),\,\xi^{{\rm
u}-}_{\varepsilon}=O(\varepsilon^{2}).\end{split}$

we obtain

(5.26) $(\xi^{{\rm s}+}_{\varepsilon})g_{1}(I^{{\rm
s}+}_{\varepsilon})-(\xi^{{\rm u}-}_{\varepsilon})g_{1}(I^{{\rm
u}-}_{\varepsilon})=O(\varepsilon^{2}).$

Thus, from (5.23) and using (5.24) we have

(5.27)
$H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}^{-}_{\varepsilon})=(I^{{\rm
s}+}_{\varepsilon}-I^{{\rm u}-}_{\varepsilon})\left(\frac{\partial
h_{0}}{\partial I}(I)\right)+O(\varepsilon^{2}).$

As the left-hand side of (5.27) is given by (5.18), since $\frac{\partial
h_{0}}{\partial I}\neq 0$ by condition (A-iii-a), solving for $I^{{\rm
s}+}_{\varepsilon}-I^{{\rm u}-}_{\varepsilon}$ yields

(5.28) $\begin{split}I^{{\rm s}+}_{\varepsilon}-I^{{\rm
u}-}_{\varepsilon}=&\left(\frac{\partial h_{0}}{\partial
I}(I)\right)^{-1}\left(H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}^{-}_{\varepsilon})\right)\\\
=&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I)\right)^{-1}\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}^{+}_{0}\right)\right)-(\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\tau\\\
&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I)\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}^{-}_{0}\right)\right)-(\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\tau\\\
&+O\left(\varepsilon^{1+\varrho}\right).\end{split}$

By the same argument as in the proof of Lemma 5.3, the error term
$O(\varepsilon^{1+\varrho})$ in (5.28) must equal $O(\varepsilon^{2})$. This
shows (5.17). ∎

######  Proposition 5.4.

(5.29) $\begin{split}I^{{\rm s}+}_{\varepsilon}-&I^{{\rm
s}}_{\varepsilon}=\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int^{+\infty}_{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int^{+\infty}_{0}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

(5.30) $\begin{split}I^{{\rm u}-}_{\varepsilon}-&I^{{\rm
u}}_{\varepsilon}=\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{-\infty}^{0}\left((\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

###### Proof.

The formula for
$H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}_{\varepsilon})$ is given
by (5.19). The normal form expansion of $H_{0}$ at a homoclinic point
$\tilde{z}_{\varepsilon}$ can be written with respect to the two sets of
coordinates as

(5.31) $H_{0}(\tilde{z}_{\varepsilon})=h_{0}(I^{{\rm
s}}_{\varepsilon})+(\xi^{{\rm s}}_{\varepsilon})g_{1}(I^{{\rm
s}}_{\varepsilon})+(\xi^{{\rm s}}_{\varepsilon})^{2}g_{2}(I^{{\rm
s}}_{\varepsilon},\xi^{{\rm s}}_{\varepsilon})$ (5.32)
$H_{0}(\tilde{z}_{\varepsilon})=h_{0}(I^{{\rm u}}_{\varepsilon})+(\xi^{{\rm
u}}_{\varepsilon})g_{1}(I^{{\rm u}}_{\varepsilon})+(\xi^{{\rm
u}}_{\varepsilon})^{2}g_{2}(I^{{\rm u}}_{\varepsilon},\xi^{{\rm
u}}_{\varepsilon})$

Subtracting (5.31) from (5.21) we obtain

(5.33)
$\begin{split}H_{0}(\tilde{z}^{+}_{\varepsilon})-H_{0}(\tilde{z}_{\varepsilon})=&h_{0}(I^{{\rm
s}+}_{\varepsilon})-h_{0}(I^{{\rm s}}_{\varepsilon})+(\xi^{{\rm
s}+}_{\varepsilon})g_{1}(I^{{\rm s}+}_{\varepsilon})-(\xi^{{\rm
s}}_{\varepsilon})g_{1}(I^{{\rm s}}_{\varepsilon})+O(\varepsilon^{2})\\\
=&\left(\frac{\partial h_{0}}{\partial I}(I_{0})\right)(I^{{\rm
s}+}_{\varepsilon}-I^{{\rm s}}_{\varepsilon})\\\ &+(\xi^{{\rm
s}+}_{\varepsilon})(g_{1}(I^{{\rm s}+}_{\varepsilon})-g_{1}(I^{{\rm
s}}_{\varepsilon}))\\\ &+(\xi^{{\rm s}+}_{\varepsilon}-\xi^{{\rm
s}}_{\varepsilon})g_{1}(I^{{\rm s}}_{\varepsilon})+O(\varepsilon^{2})\\\
=&\left(\frac{\partial h_{0}}{\partial I}(I_{0})\right)(I^{{\rm
s}+}_{\varepsilon}-I^{{\rm s}}_{\varepsilon})\\\ &+(\xi^{{\rm
s}+}_{\varepsilon}-\xi^{{\rm
s}}_{\varepsilon})g_{1}(I_{0})+O(\varepsilon^{2})\end{split}$

In the above we have used Lemma 5.1, equations (5.10), (5.11), and (5.12).

Applying Lemma D.1 and Lemma D.2 from Appendix D, for $\mathbf{F}=\xi^{\rm
s}$, we have

(5.34) $\begin{split}\xi^{{\rm s}+}_{\varepsilon}-\xi^{{\rm
s}}_{\varepsilon}=&-\varepsilon\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}^{+}_{0}\right)\right)-(\mathcal{X}^{1}\xi^{\rm
s})\left(\tilde{\Phi}^{\tau}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\tau\\\
&+O\left(\varepsilon^{1+\varrho}\right).\end{split}$

Thus, using (5.33), (5.19), and (5.34) we obtain (5.29).

Equation (5.30) follows similarly.

The argument that the error term $O(\varepsilon^{1+\varrho})$ can be replaced
by $O(\varepsilon^{2})$ follows in the same way as in the proof of Lemma 5.3.
∎

Let $\tau\in\mathbb{R}$ be some value of the time variable. Applying the
formula (5.29) to
$\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon})$ and
$\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon})$ instead
$\tilde{z}^{+}_{\varepsilon}$ and $\tilde{z}_{\varepsilon}$, respectively, and
the formula (5.30) to
$\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{-}_{\varepsilon})$ and
$\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon})$ instead
$\tilde{z}^{-}_{\varepsilon}$ and $\tilde{z}_{\varepsilon}$, respectively, we
obtain:

######  Corollary 5.5.

For any time $\tau\in\mathbb{R}$ we have

(5.35) $\begin{split}I^{{\rm
s}}_{\varepsilon}(\tilde{\Phi}^{\tau}(\tilde{z}_{\varepsilon}^{+}))-&I^{{\rm
s}}_{\varepsilon}(\tilde{\Phi}^{\tau}(\tilde{z}_{\varepsilon}))\\\
=&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}^{+}_{0}\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\varsigma\\\
&+\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}^{+}_{0}\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
s})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\varsigma\\\
&+O\left(\varepsilon^{2}\right).\end{split}$ (5.36) $\begin{split}I^{{\rm
u}}_{\varepsilon}(\tilde{\Phi}^{\tau}(\tilde{z}_{\varepsilon}^{-}))-&I^{{\rm
u}}_{\varepsilon}(\tilde{\Phi}^{\tau}(\tilde{z}_{\varepsilon}))\\\
=&\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}^{-}_{0}\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\varsigma\\\
&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{-\infty}^{0}\left((\mathcal{X}^{1}\xi^{\rm
u})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}^{-}_{0}\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
u})\left(\tilde{\Phi}^{\tau+\varsigma}_{0}\left(\tilde{z}_{0}\right)\right)\right)d\varsigma\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

######  Corollary 5.6.

(5.37) $\begin{split}&\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau\\\
&\qquad-\int_{-\infty}^{0}\left((\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau=0.\end{split}$

###### Proof.

Let us denote

(5.38) $\begin{split}J^{+}=&\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau,\\\
J^{-}=&-\int_{-\infty}^{0}\left((\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)d\tau.\end{split}$

Recall that, by condition (A-iii-b) and (5.4), we have $I^{{\rm
u}}_{\varepsilon}=I^{{\rm s}}_{\varepsilon}$. Subtracting (5.36) from (5.35),
and comparing with (5.17), we should have

$\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})(J^{+}+J^{-})=0.$

By condition (A-iii-a), we have that $\frac{\partial h_{0}}{\partial
I}(I_{0})\neq 0$ and $g_{1}(I_{0})\neq 0$, therefore

(5.39) $J^{+}+J^{-}=0.$

∎

####  5.2.3. Change in angle by the scattering map

We now give the expression of the angle-component
$\tilde{\mathcal{S}}^{\theta}$ of the mapping $\tilde{\mathcal{S}}$ in (2.7).

######  Proposition 5.7.

The change in $\theta$ by the scattering map $\tilde{\sigma}_{\varepsilon}$ is
given by the following equation

(5.40) $\begin{split}\theta^{\rm
s}\left(\tilde{z}^{+}_{\varepsilon}\right)-&\theta^{\rm
u}\left(\tilde{z}_{\varepsilon}^{-}\right)\\\
=&\Delta(I_{0})-\varepsilon\int^{+\infty}_{0}\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))d\tau\\\
&\qquad\,\,-\varepsilon\int_{-\infty}^{0}\mathcal{X}^{1}\theta^{\rm
u}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-\mathcal{X}^{1}\theta^{\rm
u}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))d\tau\\\
&+\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int^{+\infty}_{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau
d\tau\\\ &+\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau
d\tau\\\ &+O(\varepsilon^{2}),\end{split}$

where $\Delta(I_{0})$ is the phase-shift on the action level set $I_{0}$ that
defines the unperturbed scattering map $\tilde{\sigma}_{0}$ (see Proposition
4.5).

Proposition 5.7 implies that

$\tilde{\sigma}^{\theta}_{\varepsilon}(I,\theta,t)=\tilde{\sigma}^{\theta}_{0}(I,\theta,t)+\Delta(I_{0})+\varepsilon\tilde{\mathcal{S}}^{\theta}(I,\theta,t)+O(\varepsilon^{2}),$

where $\varepsilon\tilde{\mathcal{S}}^{\theta}$ is given by the first four
terms on the right hand-side of (5.40). The expression of
$\varepsilon\tilde{\mathcal{S}}^{\theta}$ is more complicated than the one for
$\varepsilon\tilde{\mathcal{S}}^{I}$ since $\theta$ is a fast variable.

###### Proof.

We will begin by computing the difference of the $\theta^{{\rm s}}$ evaluated
at a homoclinic point and at the footpoint of the stable fiber through the
homoclinic point. By Lemma D.2 from Appendix D, we have

$\theta^{{\rm s}}\left(\tilde{z}^{+}_{\varepsilon}\right)-\theta^{{\rm
s}}\left(\tilde{z}_{\varepsilon}\right)=-\int^{+\infty}_{0}\frac{d}{d\tau}\left[\theta^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-\theta^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))\right]d\tau$

Now by equation (4.17) we have:

$\frac{d\theta^{{\rm s}}}{d\tau}=\frac{\partial H_{0}}{\partial I^{{\rm
s}}}+\varepsilon\mathcal{X}^{1}\theta^{{\rm s}}.$

We can break the integral up into two parts:

(5.41) $A=-\int^{+\infty}_{0}\frac{\partial H_{0}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-\frac{\partial
H_{0}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau$

and

(5.42) $B=-\varepsilon\int^{+\infty}_{0}\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau.$

As for the integral $B$, $\varepsilon\mathcal{X}^{1}\theta^{\rm s}$ is
$O(\varepsilon)$. So, by Lemma D.4 from Appendix D, we can express the
integral in terms of the unperturbed system plus an error term:

$B=-\varepsilon\int^{+\infty}_{0}\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-\mathcal{X}^{1}\theta^{\rm
s}(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))d\tau+O(\varepsilon^{1+\varrho}).$

Returning to the integral (5.41), we now use the normal form (4.3) of $H_{0}$
given by Proposition 4.1, yielding

(5.43) $\frac{\partial H_{0}}{\partial I^{{\rm s}}}=\frac{\partial
h_{0}}{\partial I^{{\rm s}}}(I^{{\rm s}})+(\xi^{\rm s})\frac{\partial
g_{1}}{\partial I^{{\rm s}}}(I^{{\rm s}})+(\xi^{\rm s})^{2}\frac{\partial
g_{2}}{\partial I^{{\rm s}}}(I^{{\rm s}},\xi^{\rm s}),$

where $\xi^{\rm s}=x^{\rm s}y^{\rm s}$.

Thus, the integral $A$ given by (5.41) breaks into three parts

(5.44) $\displaystyle A_{1}=$
$\displaystyle\displaystyle-\int^{+\infty}_{0}\frac{\partial h_{0}}{\partial
I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-\frac{\partial
h_{0}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau,$ (5.45)
$\displaystyle A_{2}=$ $\displaystyle\displaystyle-\int^{+\infty}_{0}\xi^{{\rm
s}}\frac{\partial g_{1}^{\rm s}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-\xi^{{\rm
s}}\frac{\partial g_{1}^{\rm s}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau,$ (5.46)
$\displaystyle A_{3}=$
$\displaystyle\displaystyle-\int^{+\infty}_{0}(\xi^{{\rm
s}})^{2}\frac{\partial g_{2}^{\rm s}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-(\xi^{{\rm
s}})^{2}\frac{\partial g_{2}^{\rm s}}{\partial I^{{\rm
s}}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau.$

From Lemma 5.1 equation (5.11) we have $\xi^{\rm
s}(\tilde{\Phi}(\tilde{z}^{+}_{\varepsilon}))=O(\varepsilon^{2})$ and
$\xi^{\rm s}(\tilde{\Phi}(\tilde{z}_{\varepsilon}))=O(\varepsilon)$ in (5.46).
Thus, we can immediately obtain that $A_{3}$ is $O(\varepsilon^{2})$.

We use the integral form of the Mean Value Theorem to rewrite the integral
$A_{1}$. Recall that

$\frac{\partial F}{\partial x}(b)-\frac{\partial F}{\partial
x}(a)=(b-a)\int^{1}_{0}\frac{\partial^{2}F}{\partial
x^{2}}\left(a+t(b-a)\right)dt$

Applying this result to $A_{1}$ for $b=I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))$,
$a=I^{{\rm s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))$,
and $F=\frac{\partial h_{0}}{\partial I^{{\rm s}}}$, yields

(5.47) $A_{1}=-\int^{+\infty}_{0}(I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon})))C_{\varepsilon}(\tau)\,d\tau,$

where $C_{\varepsilon}$ stands for the integral

(5.48)
$C_{\varepsilon}(\tau)=\int^{1}_{0}\frac{\partial^{2}h_{0}}{\partial(I^{{\rm
s}})^{2}}\left[I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))+t\left(I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))\right)\right]dt.$

We evaluate the expression $I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-I^{{\rm
s}}(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))$ in (5.47) by
invoking Corollary 5.5, obtaining

(5.49) $\begin{split}&-\varepsilon\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}_{0}))\right)d\varsigma\\\
&+\varepsilon\left(\frac{\partial h_{0}}{\partial I^{\rm
s}}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}_{0}))\right)d\varsigma\\\
&+O\left(\varepsilon^{2}\right).\end{split}$

The next part of the integrand is

$C_{\varepsilon}(\tau)=\int^{1}_{0}\frac{\partial^{2}h_{0}}{\partial(I^{{\rm
s}})^{2}}\left[I^{{\rm
s}}\left(\tilde{\Phi}^{\tau}_{\varepsilon}\left(\tilde{z}_{\varepsilon}\right)\right)+t\left(I^{{\rm
s}}\left(\tilde{\Phi}^{\tau}_{\varepsilon}\left(\tilde{z}^{+}_{\varepsilon}\right)\right)-I^{{\rm
s}}\left(\tilde{\Phi}^{\tau}_{\varepsilon}\left(\tilde{z}_{\varepsilon}\right)\right)\right)\right]dt.$

Using Gronwall’s inequality – Lemma (C.1) –, we can write

$C_{\varepsilon}(\tau)=C_{0}(\tau)+O(\varepsilon^{\varrho}),$

where $0<\varrho<1$. However, when $\varepsilon=0$, $I^{{\rm s}}$ along the
flow of the footpoint $\tilde{z}^{+}_{0}$ is equal to $I^{{\rm s}}$ along the
flow of the homoclinic point $\tilde{z}_{0}$. Since $I^{{\rm
s}}\left(\tilde{z}_{0}\right)=I_{0}$ we obtain

$C_{0}(\tau)=\frac{\partial^{2}h_{0}}{\partial I^{2}}(I_{0}),$

which is a constant.

Putting these expressions together, we can write $A_{1}$ as

(5.50) $\begin{split}&\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}\left(I_{0}\right)\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}\int^{+\infty}_{0}\int_{0}^{+\infty}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}_{0}))\right)d\varsigma
d\tau\\\ &-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{0}^{+\infty}\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}_{0}))\right)d\varsigma
d\tau\\\ &+O\left(\varepsilon^{1+\varrho}\right).\end{split}$

We will now write the double integrals in (5.50) in a simpler form. We show
the details of the computation from the first double integral that appears in
(5.50), as the second double integral can be treated in a similar fashion.
Denote by $\mathscr{I}^{\rm s}$ the following improper integral

(5.51) $\begin{split}\mathscr{I}^{\rm
s}(\tau)=&-\int_{\tau}^{+\infty}((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}_{0})))d\upsilon.\end{split}$

Since
$(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}_{0})))$
approaches $0$ exponentially as $\upsilon\to+\infty$, the above integral is
convergent and moreover

$\frac{d}{d\tau}(\mathscr{I}^{\rm
s}(\tau))=(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0})).$

That is, $\mathscr{I}^{\rm s}(\tau)$ is the antiderivative of
$\tau\mapsto(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))$
satisfying the condition that it equals $0$ at $+\infty$.

Making the change of variable $\upsilon=\sigma+\tau$ with $d\upsilon=d\sigma$
the double integral in (5.50) becomes

(5.52)
$\begin{split}\int_{0}^{+\infty}\int_{\tau}^{+\infty}((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\upsilon}_{0}(\tilde{z}_{0})))d\upsilon
d\tau=-\int_{0}^{+\infty}\mathscr{I}^{\rm s}(\tau)d\tau.\end{split}$

Using Integration by Parts we obtain

(5.53) $\begin{split}-\int_{0}^{+\infty}\mathscr{I}^{\rm
s}(\tau)d\tau=&-\tau\mathscr{I}^{\rm s}(\tau)\biggr{|}_{0}^{+\infty}\\\
&+\int_{0}^{+\infty}((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0})))\tau
d\tau\\\
=&\int_{0}^{+\infty}((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0})))\tau
d\tau.\end{split}$

In the above, the quantity $\tau\mathscr{I}^{\rm s}(\tau)$ obviously equals
$0$ at $\tau=0$, and equals $0$ when $\tau\to+\infty$ since, by l’Hospital
Rule

$\lim_{\tau\to+\infty}\frac{\mathscr{I}^{\rm
s}(\tau)}{\tau^{-1}}=\lim_{\tau\to+\infty}-\frac{(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))}{\tau^{-2}}=0,$

since
$(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))$
approaches $0$ at exponential rate as $\tau\to+\infty$.

A similar computation can be done to write the second double integral that
appears in (5.50) as a single integral.

Thus, we obtain the following expression of $A_{1}$:

(5.54) $\begin{split}&\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int^{+\infty}_{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau
d\tau\\\ &-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&+O\left(\varepsilon^{1+\varrho}\right).\end{split}$

Finally, we turn to the integral $A_{2}$ given by (5.45). Using Lemma 5.1
equations (5.13) and (5.11), (5.34), as well as integration by parts similarly
to above, we express $A_{2}$ as

(5.55) $\begin{split}-&\int_{0}^{+\infty}(\xi^{{\rm s}}\frac{\partial
g_{1}}{\partial
I})(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon}))-(\xi^{{\rm
s}}\frac{\partial g_{1}}{\partial
I})(\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))d\tau\\\
=&-\left(\frac{\partial g_{1}}{\partial
I}(I_{0})\right)\int_{0}^{+\infty}(\xi^{\rm
s}((\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}^{+}_{\varepsilon})))-\xi^{\rm
s}((\tilde{\Phi}^{\tau}_{\varepsilon}(\tilde{z}_{\varepsilon}))))d\tau\\\
&+O(\varepsilon^{1+\varrho})\\\ =&\varepsilon\left(\frac{\partial
g_{1}}{\partial
I}(I_{0})\right)\int_{0}^{+\infty}\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau+\varsigma}_{0}(\tilde{z}_{0}))\right)d\varsigma
d\tau\\\ &+O(\varepsilon^{1+\varrho})\\\ =&\varepsilon\left(\frac{\partial
g_{1}}{\partial
I}(I_{0})\right)\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&+O(\varepsilon^{1+\varrho})\end{split}$

In the above, we have used that

$(\mathcal{X}^{0}+\varepsilon\mathcal{X}^{1})\xi^{s}=\mathcal{X}^{0}\xi^{s}+\varepsilon\mathcal{X}^{1}\xi^{s}=\\{\xi^{s},H_{0}\\}+\varepsilon\mathcal{X}^{1}\xi^{s}=\varepsilon\mathcal{X}^{1}\xi^{s}+O(\varepsilon^{2}).$

Combining (5.54) and (5.55) we obtain

(5.56) $\begin{split}\theta^{\rm
s}\left(\tilde{z}^{+}_{\varepsilon}\right)-&\theta^{\rm
s}\left(\tilde{z}_{\varepsilon}\right)\\\
=&\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int^{+\infty}_{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau
d\tau\\\ &-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&+\varepsilon\left(\frac{\partial g_{1}}{\partial
I}(I_{0})\right)\int_{0}^{+\infty}\left((\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{+}_{0}))-(\mathcal{X}^{1}\xi^{\rm
s})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&+O(\varepsilon^{2})\end{split}$

In the above, we have also replaced the error term
$O(\varepsilon^{1+\varrho})$ by $O(\varepsilon^{2})$, by calling the same
argument as in the proof of Proposition 5.2.

A similar computation yields:

(5.57) $\begin{split}\theta^{\rm
u}\left(\tilde{z}^{-}_{\varepsilon}\right)-&\theta^{\rm
u}\left(\tilde{z}_{\varepsilon}\right)\\\
=&-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0})\right)^{-1}\int_{-\infty}^{0}\left((\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}H_{0})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau
d\tau\\\ &+\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})\int^{0}_{-\infty}\left((\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-(\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&-\varepsilon\left(\frac{\partial g_{1}}{\partial
I}(I_{0})\right)\int^{0}_{-\infty}\left((\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}^{-}_{0}))-(\mathcal{X}^{1}\xi^{\rm
u})(\tilde{\Phi}^{\tau}_{0}(\tilde{z}_{0}))\right)\tau d\tau\\\
&+O(\varepsilon^{2})\end{split}$

Subtracting (5.57) from (5.56) yields $\theta^{\rm
s}\left(\tilde{z}^{+}_{\varepsilon}\right)-\theta^{\rm
u}\left(\tilde{z}^{-}_{\varepsilon}\right)=\theta^{\rm
s}\left(\tilde{z}_{\varepsilon}\right)-\theta^{\rm
u}\left(\tilde{z}_{\varepsilon}\right)$, plus an epsilon order term consisting
of the sum of six integrals, plus an error term of order $O(\varepsilon^{2})$.
Using the notation (5.38), four of these integrals are

(5.58) $\begin{split}&-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})J^{+},\\\
&-\varepsilon\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\left(\frac{\partial h_{0}}{\partial
I}(I_{0}))\right)^{-1}g_{1}(I_{0})J^{-},\\\ &\varepsilon\left(\frac{\partial
g_{1}}{\partial I}(I_{0})\right)J^{+},\\\ &\varepsilon\left(\frac{\partial
g_{1}}{\partial I}(I_{0})\right)J^{-}.\end{split}$

By Corollary 5.6, since $J^{+}+J^{-}=0$, the sum of the first two expressions
in (5.58) equals $0$, and the sum of the last two expressions in (5.58) equals
$0$.

Also, we recall from Section 5.1.2, that for a given unperturbed homoclinic
point $\tilde{z}_{0}$ we selected a perturbed homoclinic point
$\tilde{z}_{\varepsilon}$ satisfying condition (A-iii-b) (5.4), that is
$\theta^{\rm s}(\tilde{z}_{\varepsilon})-\theta^{\rm
u}(\tilde{z}_{\varepsilon})=\theta^{\rm s}(\tilde{z}_{0})-\theta^{\rm
u}(\tilde{z}_{0})=\Delta(I(\tilde{z}_{0}))$.

Combining these results we obtain (5.40). ∎

## Acknowledgements

We are grateful to Rodney Anderson and Angel Jorba for discussions and
comments.

##  Appendix A Normally hyperbolic invariant manifolds

We briefly recall the notion of a normally hyperbolic invariant manifold.

######  Definition A.1.

Let $M$ be a $\mathcal{C}^{r}$-smooth manifold, $\Phi^{t}$ a
$\mathcal{C}^{r}$-flow on $M$. A submanifold (with or without boundary)
$\Lambda$ of $M$ is a normally hyperbolic invariant manifold (NHIM) for
$\Phi^{t}$ if it is invariant under $\Phi^{t}$, and there exists a splitting
of the tangent bundle of $TM$ into sub-bundles over $\Lambda$

(A.1) $T_{z}M=E^{{\rm u}}_{z}\oplus E^{{\rm s}}_{z}\oplus
T_{z}\Lambda,\quad\forall z\in\Lambda$

that are invariant under $D\Phi^{t}$ for all $t\in\mathbb{R}$, and there exist
rates

$\lambda_{-}\leq\lambda_{+}<\lambda_{c}<0<\mu_{c}<\mu_{-}\leq\mu_{+}$

and a constant ${C}>0$, such that for all $x\in\Lambda$ we have

(A.2)
$\begin{split}{C}e^{t\lambda_{-}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{t\lambda_{+}}\|v\|\textrm{
for all }t\geq 0,&\textrm{ if and only if }v\in E^{{\rm s}}_{z},\\\
{C}e^{t\mu_{+}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{t\mu_{-}}\|v\|\textrm{
for all }t\leq 0,&\textrm{ if and only if }v\in E^{{\rm u}}_{z},\\\
{C}e^{|t|\lambda_{c}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{|t|\mu_{c}}\|v\|\textrm{
for all }t\in\mathbb{R},&\textrm{ if and only if }v\in
T_{z}\Lambda.\end{split}$

In the case when $\Phi^{t}$ is a Hamiltonian flow, the rates can be chosen so
that

$\lambda_{-}=-\mu_{+},\,\lambda_{+}=-\mu_{-},\,\textrm{ and
}\lambda_{c}=-\mu_{c}.$

The regularity of the manifold $\Lambda$ depends on the rates $\lambda^{-}$,
$\lambda^{+}$, $\mu^{-}$, $\mu^{+}$, $\lambda_{c}$, and $\mu_{c}$. More
precisely, $\Lambda$ is $\mathcal{C}^{\ell}$-differentiable, with $\ell\leq
r-1$, provided that

(A.3) $\begin{split}&\ell{\mu}_{c}+{\lambda}_{+}<0,\\\
&\ell{\lambda}_{c}+{\mu}_{-}>0.\end{split}$

The manifold $\Lambda$ has associated unstable and stable manifolds of
$\Lambda$, denoted $W^{{\rm u}}(\Lambda)$ and $W^{{\rm s}}(\Lambda)$,
respectively, which are $\mathcal{C}^{\ell-1}$-differentiable. They are
foliated by $1$-dimensional unstable and stable manifolds (fibers) of points,
$W^{{\rm u}}(z)$, $W^{{\rm s}}(z)$, $z\in\Lambda$, respectively, which are as
smooth as the flow.

These manifolds are defined by:

(A.4) $\begin{split}W^{\rm
s}(\Lambda)=&\\{y\,|\,d({\Phi}^{t}_{\varepsilon}(y),\Lambda)\rightarrow
0\textrm{ as {$t\to+\infty$} }\\}\\\
=&\\{y\,|\,d({\Phi}^{t}_{\varepsilon}(y),\Lambda)\leq
C_{y}e^{t\lambda_{+}},t\geq 0\\},\\\ W^{\rm
u}(\Lambda)=&\\{y\,|\,d({\Phi}^{t}_{\varepsilon}(y),\Lambda)\rightarrow
0\textrm{ as {$t\to-\infty$} }\\}\\\
=&\\{y\,|\,d({\Phi}^{t}_{\varepsilon}(y),\Lambda)\leq C_{y}e^{t\mu_{-}},t\geq
0\\},\\\ W^{{\rm
s}}(x)=&\\{y\,|\,d(\Phi^{t}(y),\Phi^{t}(x))<C_{y}e^{t\lambda_{+}},\,t\geq
0\\},\\\ W^{{\rm
u}}(x)=&\\{y\,|\,d(\Phi^{t}(y),\Phi^{t}(x))<C_{y}e^{t\mu_{-}},\,t\leq
0\\}.\end{split}$

The fibers $W^{{\rm u}}(x)$, $W^{{\rm s}}(x)$ are not invariant by the flow,
but _equivariant_ in the sense that

(A.5) $\begin{split}\Phi^{t}(W^{\rm u}(z))&=W^{\rm u}(\Phi^{t}(z)),\\\
\Phi^{t}(W^{\rm s}(z))&=W^{\rm s}(\Phi^{t}(z)).\end{split}$

Since $W^{{\rm s},{\rm u}}(\Lambda)=\bigcup_{z\in\Lambda}W^{s,u}(z)$, we can
define the projections along the fibers

(A.6) $\begin{split}\Omega^{+}:W^{{\rm
s}}(\Lambda)\to\Lambda,\quad&\Omega^{+}(z)=z^{+}\textrm{ iff }z\in W^{{\rm
s}}(z^{+}),\\\ \Omega^{-}:W^{{\rm
u}}(\Lambda)\to\Lambda,\quad&\Omega^{-}(z)=z^{-}\textrm{ iff }z\in W^{{\rm
u}}(z^{-}).\end{split}$

The point $z^{+}\in\Lambda$ is characterized by

(A.7) $d({\Phi}^{t}(z),{\Phi}^{t}(z^{+}))\leq
C_{z}e^{t\lambda_{+}},\quad\textrm{ for all }t\geq 0.$

and the point $z^{-}\in\Lambda$ by

(A.8) $d({\Phi}^{t}(z),{\Phi}^{t}(z^{-}))\leq C_{z}e^{t\mu_{-}},\quad\textrm{
for all }t\leq 0,$

for some $C_{z}>0$.

For our applications, the most important result about NHIM’s is that they
persist when we perturb the flow. This is the fundamental result of [Fen72,
HPS77, Pes04]. In the case when the manifold has a boundary, the persistence
result requires a step of extending the flow. This makes that the persistent
manifold is not invariant but only locally invariant and not unique (it
depends on the extension).

When we are given a family of flows, it is possible to choose the extensions
depending smoothly on parameters and obtain that the manifolds depend smoothly
on parameters.

The precise meaning of the smooth dependence is that the we can find
diffeomorphisms $k_{\varepsilon}:\Lambda_{0}\rightarrow\Lambda_{\varepsilon}$.
The maps $k_{\varepsilon}(x)$ are jointly $C^{r}$ as functions of
$x,\varepsilon$. The proof of this well known result is not very difficult. It
suffices to consider an extended flow
$\tilde{\Phi}^{t}(x,\varepsilon)=(\Phi_{\varepsilon}^{t}(x),\varepsilon)$,
which is a small perturbation of
$\tilde{\Phi}^{t}_{0}(x,\varepsilon)=(\Phi_{0}^{t}(x),\varepsilon)$. The
regularity of the NHIM of $\tilde{\Phi}^{t}$ gives the claimed regularity of
the NHIM of $\Phi^{t}$ with respect to parameters.

From the same proof (using the invariant objects of the extended flow), it
easily follows the regularity with respect to parameters of the stable and
unstable bundles and the stable and unstable manifolds.

##  Appendix B Scattering map

Assume that $W^{\rm u}(\Lambda)$, $W^{\rm s}(\Lambda)$ have a transverse
intersection along a manifold $\Gamma$ satisfying:

(B.1) $\begin{split}&T_{z}\Gamma=T_{z}W^{\rm s}(\Lambda)\cap T_{z}W^{\rm
u}(\Lambda),\textrm{ for all }z\in\Gamma,\\\ &T_{z}M=T_{z}\Gamma\oplus
T_{z}W^{\rm u}(z^{-})\oplus T_{z}W^{\rm s}(z^{+}),\textrm{ for all
}z\in\Gamma.\end{split}$

Under these conditions the projection mappings $\Omega^{\pm}$ restricted to
$\Gamma$ are local diffeomorphisms. We can restrict $\Gamma$ if necessary so
that $\Omega^{\pm}$ are diffeomorphisms from $\Gamma$ onto open subsets
$U^{\pm}$ in $\Lambda$.

######  Definition B.1.

A homoclinic channel is a homoclinic manifold $\Gamma$ satisfying the strong
transversality condition B.1, and such that

$\Omega^{\pm}_{\mid\Gamma}:\Gamma\to U^{\pm}:=\Omega^{\pm}(\Gamma)$

is a $\mathcal{C}^{\ell-1}$-diffeomorphism.

######  Definition B.2.

Given a homoclinic channel $\Gamma$, the scattering map associated to $\Gamma$
is defined as

$\begin{split}\sigma:=&\sigma^{\Gamma}:U^{-}\subseteq\Lambda\to
U^{+}\subseteq\Lambda,\\\
\sigma=&\Omega^{+}\circ(\Omega^{-})^{-1}.\end{split}$

Equivalently, $\sigma(z^{-})=z^{+}$, provided that $W^{\rm u}(z^{-})$
intersects $W^{\rm s}(z^{+})$ at a unique point $z\in\Gamma$.

The meaning of the scattering map is that, given a homoclinic excursion, it
has two orbits in the manifold is asymptotic to. It is asymptotic to an orbit
in the past and to another orbit in the future. The scattering map considers
the future asymptotic orbit as a function of the asymptotic in the past. When
we consider all the homoclinic orbits in a homoclinic channel we obtain a