# Arnold diffusion in a model of dissipative system

Samuel W. Akingbade  Yeshiva University, Department of Mathematical Sciences,
New York, NY 10016, USA<EMAIL_ADDRESS>,  Marian Gidea  Yeshiva
University, Department of Mathematical Sciences, New York, NY 10016, USA
<EMAIL_ADDRESS>and  Tere M-Seara  Departament de Matemàtiques and IMTECH,
Universitat Politècnica de Catalunya (UPC), Diagonal 647, 08028 Barcelona,
Spain, Centre de Recerca Matemàtica, Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

For a mechanical system consisting of a rotator and a pendulum coupled via a
small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem
asserts the existence of ‘diffusing orbits’ along which the energy of the
rotator grows by an amount independent of the size of the coupling parameter,
for all sufficiently small values of the coupling parameter. There is a vast
literature on establishing Arnold diffusion for such systems. In this work, we
consider the case when an additional, dissipative perturbation is added to the
rotator-pendulum system with coupling. Therefore, the system obtained is not
symplectic but conformally symplectic. We provide explicit conditions on the
dissipation parameter, so that the resulting system still exhibits energy
growth. The fact that Arnold diffusion may play a role in systems with small
dissipation was conjectured by Chirikov. In this work, the coupling is
carefully chosen, however the mechanism we present can be adapted to general
couplings and we will deal with the general case in future work.

S.W.A. and M.G. were partially supported by NSF grant DMS-1814543.

The work of T.M-S. was partially supported by the grants PGC2018-098676-B-100
and PID-2021-122954NB-100 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A
way of making Europe”. This work is also supported by the Spanish State
Research Agency, through the Severo Ochoa and María de Maeztu Program for
Centers and Units of Excellence in R&D (CEX2020-001084-M). T.M-S. was
partially supported by the Catalan Institution for Research and Advanced
Studies via an ICREA Academia Prize 2019.

##  1\. Introduction

The Arnold diffusion problem [Arn64] broadly refers to a universal mechanism
of instability for multi-dimensional Hamiltonian systems that are small
perturbations of integrable ones. Through this mechanism, chaotic transfers of
energy take place between subsystems of a given Hamiltonian system, which, in
particular, can lead to significant growth of energy of one of the subsystems
over time. Chirikov [Chi79] conjectured that Arnold diffusion may play a role
in systems with small dissipation as well.

Studying Hamiltonian systems with small dissipation is important for
applications, as many real-life physical systems experience some energy loss
over time.

A significant class of examples is furnished by Celestial Mechanics, on the
motion of celestial bodies under mutual gravity. As the gravitational force is
conservative, such systems are usually modeled as Hamiltonian systems.
Nevertheless dissipative forces are present in real-world systems, including
tidal forces, Stokes drag, Poynting-Robertson effect, Yarkowski/YORP effects,
atmospheric drag, and their effect may accumulate in the long run. While some
of these effects may be negligible over relatively short time scales, others,
for instance Earth’s atmospheric drag on artificial satellites, can have
significant effects over practical time scales. See, e.g. [MNF87, Cel07,
RR17].

Another class of examples is given by energy harvesting devices. Some of these
devices consist of systems of oscillating beams made of piezoelectric
materials, where on the one hand there is dissipation due to mechanical
friction, and on the other hand there is external forcing, owed to the
movement of the device, that triggers the beams to oscillate. See, e.g. [MH79,
EHI09, Gra17].

Of course, there are many other examples. In this paper we consider a simple
model of a mechanical system, consisting of a rotator and a pendulum with a
small, periodic coupling, subject to a small dissipative perturbation. Coupled
rotator-pendulum systems are fundamental models in the study of Arnold
diffusion in Hamiltonian systems. Adding a dissipative perturbation results in
a system that is non-Hamiltonian. The symplectic structure changes into a
conformally symplectic one [Ban02]. We show that such a system exhibits Arnold
diffusion, in the sense that there exist pseudo-orbits for which the energy of
the rotator subsystem grows by some quantity that is independent of the
smallness parameter. (By a pseudo-orbit here we mean a sequence of orbit
segments of the flow such that the endpoint of each orbit segment is ‘close’
to the starting point of the next orbit segment in the sequence.) We note that
for the unperturbed rotator-pendulum system, the energy of the rotator
subsystem is conserved. The small, periodic coupling added to the system makes
the rotator undergo small oscillations in energy, while the dissipative
perturbation typically yields a loss in energy. The physical significance of
our result is that, despite the dissipation effects, it is possible to overall
gain a significant amount of energy over time.

Specifically, the unperturbed rotator-pendulum system is given by a
Hamiltonian of the form

$H_{0}(p,q,I,\theta)=h_{0}(I)+h_{1}(p,q),$

with
$z=(p,q,I,\theta)\in\mathbb{R}\times\mathbb{T}^{1}\times\mathbb{R}\times\mathbb{T}^{1}$,
where $h_{0}(I)$ represents the Hamiltonian of the rotator, and $h_{1}$
represents the Hamiltonian of the pendulum, and
$\mathbb{T}^{1}=\mathbb{R}/2\pi\mathbb{Z}$. The perturbed system is of the
form

(1.1) $\dot{z}=J\nabla_{z}H_{0}(z)+\varepsilon
J\nabla_{z}H_{1}(z,t)+\mathcal{X}_{\lambda}(z),$

where $H_{1}(z,t)$ is a Hamiltonian that is $2\pi$-periodic in time $t$,
$\varepsilon\geq 0$ is the size of the coupling, $\mathcal{X}_{\lambda}(z)$ is
a dissipative vector field depending on some dissipation parameter
$\lambda=\lambda(\varepsilon)>0$, and

$J=\left(\begin{array}[]{cc}J_{2}&0\\\ 0&J_{2}\\\ \end{array}\right)\textrm{
where }J_{2}=\left(\begin{array}[]{cc}0&-1\\\ 1&0\\\ \end{array}\right).$

Technical conditions on $h_{0},h_{1},H_{1},\mathcal{X}_{\lambda}$ will be
given in Section 3. Under those conditions, the phase space of the perturbed
system has a $3$-dimensional Normally Hyperbolic Invariant Manifold (NHIM from
now on), which contains a $2$-dimensional invariant torus that is an attractor
for the dynamics in the NHIM. This torus creates a ‘barrier’ for the existence
of diffusing orbits by using only the ‘inner dynamics’ (i.e., the dynamics
restricted to the NHIM). The main question is whether there are diffusing
orbits crossing this ‘barrier’ by combining the ‘inner dynamics’ with the
‘outer dynamics’ (i.e., the dynamics along homoclinic orbits to the NHIM).

We show that there exist $C>0$ and $\varepsilon_{0}>0$ such that, for all
$0<|\varepsilon|<\varepsilon_{0}$, there exists a pseudo-orbit $z(t)$,
$t\in[0,T]$, of (1.1), such that

$I(T)-I(0)>C\textrm{ for some }T>0.$

More technical details will be given in Theorem 4.1. In order for the above
result to be of practical interest, the above solution $z(t)$ should be chosen
such that at the beginning $(I(0),\theta(0))$ is below (relative to $I$) the
aforementioned attractor, and at the end $(I(T),\theta(T))$ is above the
aforementioned attractor. Indeed, it is possible to increase $I$ by starting
below the attractor and moving towards the attractor under the effect of the
dissipation alone; obviously, such a solution is not of practical interest.

##  2\. Conservative vs. dissipative systems

Arnold’s conjecture on Hamiltonian instability originated with an example of a
rotator-pendulum system with a small, time-periodic Hamiltonian coupling of
special type [Arn64]. In his example, in the absence of the coupling, the
phase space of the rotator forms a normally hyperbolic invariant manifold
(NHIM) foliated by ‘whiskered’, rotational tori, which have stable and
unstable invariant manifolds that coincide. The coupling in Arnold’s example
was specially chosen so that it vanishes on the family of invariant tori, and
so the tori are preserved. These tori constitute ‘barriers’ for the existence
of diffusing orbits, since orbits in the NHIM always move along these tori and
thus cannot increase their action variable. At the same time the coupling
splits the stable and unstable manifolds, so that the unstable manifold of
each torus intersects transversally the stable manifolds of nearby tori. Thus,
one can form ‘transition chains’ of tori, and show that, by interspersing the
‘outer dynamics’ along the homoclinic orbits to the NHIM with the ‘inner
dynamics’ along the tori, one can obtain ‘diffusing’ orbits along which the
energy of the rotator exhibits a significant growth. Arnold conjectured that
this mechanism of diffusion occurs in close to integrable general systems.

However, in the case of a general coupling not all of the invariant tori in
the NHIM are preserved. The KAM theorem yields a Cantor set of tori that
survive from the unperturbed case, with gaps in between. The splitting of the
stable and unstable manifold makes the unstable manifold of each torus
intersect transversally the stable manifolds of sufficiently close tori,
however the size of the splitting is in general smaller than the size of the
gaps between tori. This is known as the ‘large gap problem’. It was overcome,
for instance, by forming transition chains that, besides rotational tori, also
include ‘secondary’ tori created by the perturbation [DdlLS00, DLS06]. Other
geometric mechanisms use transition chains that include, besides rotational
tori, Aubry-Mather sets [GR13]. Subsequently, [GdlLMS20] described a general
mechanism of diffusion that relies mostly on the outer dynamics, and uses only
the Poincaré recurrence of the inner dynamics (which is automatically
satisfied in Hamiltonian systems over regions of bounded measure).

The references mentioned above encompass geometric ideas that we can adapt to
the dissipative case. However, there are many other geometric mechanisms that
have been used in the Arnold diffusion problem, such as those in [CG94, BT99,
DdlLS00, Tre02, Tre04, DdlLS06a, DdlLS06b, Pif06, GT08, DH09, Tre12, GdlL17,
GT17a, GM22]. A variational program for the Arnold diffusion was formulated in
[Mat04, Mat12] for systems close to integrable. Global variational methods for
diffusion have been used in this setting for convex Hamiltonians [CY09, KZ15,
BKZ16, CX19, KZ20]. A hybrid program combining geometric and variational
methods was started in [BB02, BBB03].

The case of a rotator-pendulum system subject to a non-Hamiltonian
perturbation (consisting of time-periodic Hamiltonian coupling and a
dissipative force) which we consider in this paper, has very different
geometric features from the conservative case. The dissipation added to the
Hamiltonian system is a singular perturbation – the system with positive
dissipation leads to attractors inside the NHIM, which can contain at most one
invariant torus. Poincaré recurrence does not hold for dissipative systems.
The stable and unstable manifolds of the NHIM do not necessarily intersect.
Therefore, the mechanism used for proving diffusion in the Hamiltonian case
does not carry over to the non-Hamiltonian case.

To provide some intuition, we illustrate on a couple of basic examples some
possible effects of dissipation on the geometry of Hamiltonian systems.

######  Example 2.1.

The first example is the standard map.

The (conservative) standard map, which can be viewed as the time-one map of a
non-autonomous Hamiltonian system representing a ‘kicked rotator’, is given by

(2.1) $\begin{split}I^{\prime}=&I+\varepsilon\sin(\theta),\\\
\theta^{\prime}=&\theta+I+\varepsilon\sin(\theta),\end{split}$

where $\varepsilon$ is the perturbative parameter, and $I,\theta$ are defined
$(\textrm{mod}\,2\pi)$. This is a symplectic twist map; the symplecticity
condition being $dI^{\prime}\wedge d\theta^{\prime}=dI\wedge d\theta$ and the
twist condition being $\frac{\partial\theta^{\prime}}{\partial I}\neq 0$. When
$\varepsilon=0$ the resulting map is the time-one map of a rotator, and is
given by

(2.2) $\begin{split}I^{\prime}=&I,\\\ \theta^{\prime}=&\theta+I.\end{split}$

It is an integrable twist map, with all level sets of $I$ being rotational
invariant circles on which the motion is a rigid rotation of frequency
$\omega(I)=I$. For $0<\varepsilon\ll 1$, the KAM theorem asserts that there is
a positive measure set of invariant circles, of Diophantine frequencies, which
survive the perturbation. The measure of the set of the KAM circles tends to
$1$ as $\varepsilon\to 0$. On the other hand, when $\varepsilon>0$ increases,
fewer and fewer invariant circles survive, and eventually only one invariant
circle is left. The last rotational invariant circle for the standard map has
frequency $\omega=\frac{1+\sqrt{5}}{2}$, which is the golden mean [Gre79]. See
Fig. 1a.

The dissipative standard map is defined as

(2.3) $\begin{split}I^{\prime}=&(1-\lambda)I+\mu+\varepsilon\sin(\theta),\\\
\theta^{\prime}=&\theta+(1-\lambda)I+\mu+\varepsilon\sin(\theta),\end{split}$

where $\lambda$ is the dissipative parameter, $0\leq\lambda<1$, and $\mu$ is
the drift parameter; $\lambda=0$ corresponds to no dissipation. The map is no
longer symplectic, but conformally symplectic, that is $dI^{\prime}\wedge
d\theta^{\prime}=(1-\lambda)dI\wedge d\theta$, and still satisfies a twist
condition.

When $\varepsilon=0$, the resulting map

(2.4) $\begin{split}I^{\prime}=&(1-\lambda)I+\mu,\\\
\theta^{\prime}=&\theta+(1-\lambda)I+\mu,\end{split}$

has a single rotational invariant circle $I=\frac{\mu}{\lambda}$ of frequency
$\omega_{*}:=\frac{\mu}{\lambda}$. The KAM theorem for conformally symplectic
systems asserts that for each $0<\varepsilon\ll 1$ there is one rotational
invariant circle, of Diophantine frequency, that survives the perturbation,
and that circle is a local attractor for the system (see, e.g. [CC09,
CCdlL13b, CCDlL13a, CCdlL20]). In order for the surviving circle to be of
Diophantine frequency $\omega_{*}$, we need to properly adjust the drift
parameter $\mu$. See Fig. 1b and Fig. 1c.

We can rewrite the dissipative standard map in terms of a frequency parameter
$\omega_{*}$ rather than in terms of the drift parameter
$\mu=\lambda\omega_{*}$ obtaining:

(2.5)
$\begin{split}I^{\prime}=&I-\lambda(I-\omega_{*})+\varepsilon\sin(\theta),\\\
\theta^{\prime}=&\theta+I-\lambda(I-\omega_{*})+\varepsilon\sin(\theta).\end{split}$

In this case, by the persistence of normal hyperbolicity of the torus given by
$I=\omega_{*}$, as $0<\varepsilon\ll 1$ is varied, there exists an invariant
torus of frequency $\omega=\omega(\varepsilon)$ close to $\omega_{*}$; not all
frequencies $\omega$ yield KAM circles but only those $\omega(\varepsilon)$
which are Diophantine.

(a)  Conservative standard map

(b)  Dissipative standard map with un-adjusted drift

(c) Dissipative standard map with adjusted drift (d) The basins of attraction
of the attractors appearing in (c), using a colour scale based on rotation
numbers (Credit Matteo Manzi)

Figure 1. The conservative and dissipative standard map

######  Example 2.2.

The second example is the pendulum, given by the Hamiltonian

$h_{1}(p,q)=\frac{p^{2}}{2}+(\cos(q)-1).$

As it is well known, the pendulum has a hyperbolic fixed point whose stable
and unstable manifolds coincide (see Fig. 2a).

When dissipation is added to the pendulum

$\begin{split}\dot{p}=&-\lambda p+\sin(q),\\\ \dot{q}=&p,\end{split}$

the origin is again a hyperbolic fix point with eigenvalues
$\pm\sqrt{1+(\frac{\lambda}{2})^{2}}-\frac{\lambda}{2}$. Nevertheless, its
stable and unstable manifolds cease to intersect, for dissipative coefficient
$\lambda>0$ (see Fig. 2b).

However, when both dissipation and periodic forcing are added to the pendulum,

$\begin{split}\dot{p}=&-\lambda p+\sin(q)+\varepsilon\sin(t),\\\
\dot{q}=&p,\end{split}$

for certain parameter values $\lambda>0$ and $\varepsilon>0$, the time-$2\pi$
map exhibits chaotic attractors in the Poincaré section (see Fig. 2c).

These simple examples illustrate that adding dissipation to a Hamiltonian
system typically destroys – sometimes dramatically – some of the geometric
structures – KAM tori, homoclinic connections – that are relevant in Arnold’s
mechanism of diffusion, and creates new geometric structures – attractors –
that act as barriers for diffusion. On the other hand, the addition of forcing
can compensate the effects of dissipation.

(a) Phase space of the conservative pendulum

(b) Phase space of the dissipative pendulum

(c) Poincaré section for the pendulum subject to dissipation and periodic
forcing

Figure 2. The pendulum

##  3\. Model

The model that we consider is described by an integrable Hamiltonian system
subject to a time-dependent, Hamiltonian perturbation (or coupling), and to a
second, non-Hamiltonian, perturbation that is dissipative.

The unperturbed Hamiltonian corresponds to an uncoupled rotator-pendulum
system and is given by

(3.1) $H_{0}(p,q,I,\theta)=\frac{I^{2}}{2}+\frac{p^{2}}{2}+(\cos q-1),$

where
$(p,q,I,\theta)\in\mathbb{R}\times\mathbb{T}^{1}\times\mathbb{R}\times\mathbb{T}^{1}$,
which is endowed with the standard symplectic structure $\omega=dp\wedge
dq+dI\wedge d\theta$.

For the rotator part of the Hamiltonian, given by $h_{0}(I)=\frac{I^{2}}{2}$,
each level set $I=\textrm{constant}$ is invariant under the flow of $h_{0}$,
and the corresponding dynamics is a rigid rotation of frequency

(3.2) $\omega(I):=\frac{\partial h_{0}}{\partial I}=I.$

The pendulum part of the Hamiltonian, is given by

(3.3) $h_{1}(p,q)=\frac{p^{2}}{2}+(\cos q-1),$

it has a hyperbolic fixed point at $(p,q)=(0,0)$ and an elliptic fixed point
at $(p,q)=(0,\pi)$. The stable and unstable manifolds of the hyperbolic fixed
point $(0,0)$ coincide, and can be parametrized as

(3.4) $(p_{0}(t),q_{0}(t))=\left(\pm\frac{2}{\cosh t},4\arctan e^{\pm
t}\right).$

Since the system $H_{0}$ is uncoupled, $I$ is a conserved quantity and so each
hypersurface $\\{I=\textrm{const.}\\}$ constitutes a barrier for the dynamics
of $H_{0}$: there are no trajectories along which the variable $I$ can change.

When we add the time-dependent, Hamiltonian perturbation, we have

(3.5) $H_{\varepsilon}(p,q,I,\theta,t)=H_{0}(p,q,I,\theta)+\varepsilon
H_{1}(p,q,I,\theta,t),$

where $t\in\mathbb{T}^{1}$, meaning that the perturbation $H_{1}$ is
$2\pi$-periodic in time.

We will assume that $H_{1}$ is of the form

(3.6) $H_{1}(p,q,I,\theta,t)=f(q)\cdot g(\theta,t).$

The dissipative perturbation is given by a vector field
$\mathcal{X}_{\lambda}$ that is added to the Hamiltonian vector field $J\nabla
H_{\varepsilon}$ of $H_{\varepsilon}$, where

(3.7) $\mathcal{X}_{\lambda}(p,q,I,\theta)=(-\lambda
p,0,-\lambda(I-\omega_{*}),0),$

where $\lambda$ is the dissipation coefficient, and $\omega_{*}$ is a fixed
Diophantine frequency. For the moment, we will treat $\lambda$ as an
independent parameter, but for most of the paper we will consider $\lambda$ of
the form $\lambda=\varepsilon\rho$, with $\rho$ being a sufficiently small
independent parameter. In our main result Theorem 4.1 we will use
$\lambda=\varepsilon\rho(\varepsilon)$ where
$0<\rho(\varepsilon)=\frac{\bar{\rho}}{\log{\frac{1}{\varepsilon}}}\ll 1$,
where $\bar{\rho}$ is a positive constant.

The system of interest is

(3.8) $\dot{z}=J\nabla_{z}H_{0}(z)+\varepsilon
J\nabla_{z}H_{1}(z,t)+\mathcal{X}_{\lambda}(z),\quad z=(p,q,I,\theta).$

We obtain the following equations:

(3.9) $\begin{cases}\dot{p}=\sin(q)-\varepsilon f^{\prime}(q)\cdot
g(\theta,t)-\lambda p,\\\ \dot{q}=p,\\\
\dot{I}=-\lambda(I-\omega_{*})-\varepsilon f(q)\cdot\frac{\partial
g}{\partial\theta}(\theta,t),\\\ \dot{\theta}=I.\end{cases}$

As we shall see, the dissipative perturbation yields the existence of
attractors for the dynamics restricted to the NHIM, that is, the dynamics in
the $(I,\theta)$-variables. In particular, we can have attractors that act as
barriers on the NHIM, in the sense that they separate the NHIM into
topologically non-trivial connected components. As all trajectories within the
basin of attractors move towards the attractors, the action $I$ will increase
along some trajectories and will decrease along some other trajectories,
however there are no trajectories within the NHIM that start on one side of
the attractor and end on the other side.

Below, we will consider two concrete examples of Hamiltonian perturbations:

Vanishing perturbation:

$H_{1}$ vanishes at $(p,q)=(0,0)$

(3.10) $\begin{split}f(q)=\cos(q)-1,\\\
g(\theta,t)=a_{00}+a_{10}\cos\theta+a_{01}\cos t.\end{split}$

Non-vanishing perturbation:

$H_{1}$ does not vanish at $(p,q)=(0,0)$

(3.11) $\begin{split}f(q)=\cos(q),\\\
g(\theta,t)=a_{00}+a_{10}\cos\theta+a_{01}\cos t.\end{split}$

Above, $a_{00}$, $a_{10}$, and $a_{01}$ are real numbers with
$a_{10}a_{01}\neq 0$.

######  Remark 3.1.

The choice of the coupling of the form $H_{1}(p,q,I,\theta,t)=f(q)g(\theta,t)$
has been made in order to deal with a simple model. The fact that the function
$f$ satisfies $f^{\prime}(0)=0$ implies that the normally hyperbolic invariant
manifold, which is exhibited by the unperturbed system, is not affected by the
perturbation; see Section 5.4. We do not need to invoke the theory of
persistence of normally hyperbolic invariant manifolds under perturbation. The
function $g(\theta,t)$ can be viewed as a truncation to the first two
harmonics of the Fourier expansion of an analytic function. We will deal with
the general case with infinitely many harmonics, as well as with perturbations
that do not preserve the NHIM, in future work.

######  Remark 3.2.

We note that instead of (3.7) we can consider more general perturbations of
the form

$\mathcal{X}_{\lambda}(p,q,I,\theta)=(-\lambda_{1}p,0,-\lambda_{2}(I-\omega_{*}),0),$

for $\lambda_{1}=\varepsilon\bar{\rho}_{1}$ and
$\lambda_{2}=\varepsilon\frac{\bar{\rho}_{2}}{\log{\frac{1}{\varepsilon}}}$
with $\bar{\rho}_{1},\bar{\rho}_{2}>0$. One will be able to see that the
arguments below also apply to this case and therefore, the main result, stated
below, remains valid.

##  4\. Main result

######  Theorem 4.1.

Consider the Hamiltonian system (3.1) subject to the time-periodic Hamiltonian
perturbation (3.10) or (3.11), and to the dissipative perturbation (3.7), with
dissipation coefficient
$\lambda=\varepsilon\rho(\varepsilon)=\varepsilon\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$
with $\bar{\rho}>0$ suitably small.
Then there exist $0<I_{1}<I_{2}$ and $\varepsilon_{0}>0$ such that, for every
$\omega_{*}$ Diophantine number with $0<I_{1}<\omega_{*}<I_{2}$, and every
$0<\varepsilon<\varepsilon_{0}$, there exist pseudo-orbits $z(t)$,
$t\in[0,T]$, such that

$I(z(0))<I_{1}\textrm{ and }I(z(T))>I_{2}.$

Here by a pseudo-orbit $z(t)$ we mean a finite collection of trajectories
$z^{i}(t)$, ${t\in[t_{i},t_{i+1}]}$ of (3.8), for some times
$0=t_{0}<t_{1}<\ldots<t_{m}=T$, where $m>0$, such that

$\begin{split}I(z^{0}(0))<I_{1}\textrm{ and }I(z^{m}(T))>I_{2},\\\
d(z^{i}(t_{i+1}),z^{i+1}(t_{i+1}))<\delta(\varepsilon),\textrm{ for
}i=0,\ldots,m-1,\end{split}$

for some $\delta(\varepsilon)=O(\varepsilon)>0$.

The diffusion time along the pseudo-orbit $z(t)$, ${t\in[0,T]}$, is
$T=O\left(\frac{1}{\varepsilon}\log(\frac{1}{\varepsilon})\right)$.

Above, we used the notation $f=O(g)=O_{\mathcal{C}^{k}}(g)$ for a pair of
functions $f$, $g$ satisfying $\|f\|_{\mathcal{C}^{k}}\leq
M\|g\|_{\mathcal{C}^{k}}$ for some $M>0$, where $\|\cdot\|_{\mathcal{C}^{k}}$
is the ${\mathcal{C}^{k}}$-norm for some suitable $k\geq 0$.

We illustrate the phenomenon described by Theorem 4.1 in Fig. 3a, Fig. 3b,
Fig. 3c, Fig. 3d. In Fig. 3a, using the inner dynamics alone, orbits with
$I(0)<\omega_{*}$ cannot pass beyond the attractor shown in Fig. 3b. However,
using both the inner and outer dynamics, there are orbits with
$I(0)<\omega_{*}$ that end up with $I(T)>\omega_{*}$, as shown in Fig. 3d.
These orbits move close to the separatix of the pendulum $h_{1}(q,p)=0$, as
shown in Fig. 3c.

(a) Inner dynamics

(b) Attractor for the inner dynamics

(c) Outer dynamics

(d) Combined inner and outer dynamics

Figure 3. Orbits of the rotator-pendulum system

Theorem 4.1 gives us diffusing pseudo-orbits. Applying a Shadowing Lemma type
of results similar to those in [Zgl09, GT17b, GdlLMS20, CG], we will be able
to show that there exist true orbits $z(t)$ of (1.1) such that $I(z(0))<I_{1}$
and $I(z(T))>I_{2}$. We leave the technical details for a future work.

The above pseudo-orbits are such that the end-point of one is
$\delta(\varepsilon)$-close to the starting-point of the next one, where
$\delta(\varepsilon)=O(\varepsilon)$. We remark here that we can also obtain
pseudo-orbits with $\delta(\varepsilon)=O(\varepsilon^{p})$, for any $p\geq
1$, with the same diffusion time order
$T=O\left(\frac{1}{\varepsilon}\log(\frac{1}{\varepsilon})\right)$.

In practical applications one can pass from pseudo-orbits to true orbits by
applying small controls; for example, in the case of artificial satellites
perturbed by atmospheric drag, the small controls can be satellite maneuvers.

######  Remark 4.2.

In Theorem 4.1, the action levels $I_{1}$ and $I_{2}$ can be chosen
explicitly, depending on the Hamiltonian perturbation (3.10) or (3.11) that is
considered. See Section 7.7.

The condition on choosing $\omega_{*}$ a Diophantine number between $I_{1}$
and $I_{2}$ is not necessary for the proof of the theorem; see Section 8.2.
The reason for requiring this condition is to be able to apply the KAM theorem
for conformally symplectic systems [CCdlL20], which implies the existence of a
KAM torus that is an attractor for the inner dynamics, and hence represents a
barrier for the inner dynamics. In other words, we want to show that diffusing
pseudo-orbits exist even if there is a barrier inside the NHIM.

The choice of the dissipation coefficient
$\lambda=\varepsilon\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$ is related
to the time $T_{h}=O(\log(\frac{1}{\varepsilon}))$ required for a point
starting in an $\varepsilon$-neighborhood of the NHIM to travel along a
homoclinic orbit and arrive in an $\varepsilon$-neighborhood of the NHIM. This
choice of $\lambda$ implies that $\lambda\cdot
T_{h}=O(\bar{\rho}\varepsilon)$, while the order of the change in action by
the scattering map is $O(\varepsilon)$. By choosing a suitable small enough
constant $\bar{\rho}$, we can ensure that, when the growth in $I$ by the
scattering map competes with the decay in $I$ by the dissipation, which is of
order $O(\lambda T_{h})=O(\bar{\rho}\varepsilon)$, we will make the former to
win against the latter.

If we do not impose that the homoclinic orbits get $\varepsilon$-close to the
NHIM, then we can choose a shorter time $T_{h}$ along the homoclinic orbits
and implicitly a larger $\lambda$, as long as $\lambda\cdot T_{h}$ is
$O(\varepsilon)$; for example, we can choose $T_{h}=O(1)$ and
$\lambda=\bar{\rho}\varepsilon$ for some $\bar{\rho}>0$ suitably small.

##  5\. Preliminaries

###  5.1. Extended system

Since the perturbation $H_{1}$ is time-dependent, it is convenient to consider
time as an independent variable $t$ and to work in the extended phase space
$\tilde{z}=(p,q,I,\theta,s)\in\mathbb{R}\times\mathbb{T}^{1}\times\mathbb{R}\times\mathbb{T}^{1}\times\mathbb{T}^{1}$,
adding the equation $\dot{s}=1$ to system (3.9) to obtain:

(5.1) $\begin{array}[]{rcl}\dot{z}&=&J\nabla_{z}H_{0}(z)+\varepsilon
J\nabla_{z}H_{1}(z,s)+\mathcal{X}_{\lambda}(z),\quad z=(p,q,I,\theta)\\\
\dot{s}&=&1\end{array}$

We denote by $\tilde{\Phi}^{t}_{0}$ the unperturbed extended flow, and by
$\tilde{\Phi}^{t}_{\varepsilon}$ the perturbed extended flow.

###  5.2. Normally hyperbolic invariant manifolds

We briefly recall the notion of a normally hyperbolic invariant manifold
(NHIM) [Fen74, HPS77].

Let $M$ be a $\mathscr{C}^{r}$-smooth manifold, $\Phi^{t}$ a
$\mathscr{C}^{r}$-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 $\Lambda$

(5.2) $T_{z}M=E^{\textrm{u}}_{z}\oplus E^{\textrm{s}}_{z}\oplus
T_{z}\Lambda,\quad\forall z\in\Lambda$

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

(5.3)
$\begin{split}{C}e^{t\lambda_{-}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{t\lambda_{+}}\|v\|\textrm{
for all }t\geq 0,&\textrm{ if and only if }v\in E^{\textrm{s}}_{z},\\\
{C}e^{t\mu_{+}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{t\mu_{-}}\|v\|\textrm{
for all }t\leq 0,&\textrm{ if and only if }v\in E^{\textrm{u}}_{z},\\\
{C}e^{|t|\lambda_{c}}\|v\|\leq\|D\Phi^{t}(z)(v)\|\leq{C}e^{|t|\mu_{c}}\|v\|\textrm{
for all }t\in\mathbb{R},&\textrm{ if and only if }v\in
T_{z}\Lambda.\end{split}$

It is known that $\Lambda$ is $\mathscr{C}^{\ell}$-differentiable, with
$\ell\leq r-1$, provided that

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

The manifold $\Lambda$ has associated unstable and stable manifolds, denoted
$W^{\textrm{u}}(\Lambda)$ and $W^{\textrm{s}}(\Lambda)$, which are tangent to
$E^{\textrm{u}}_{\Lambda}$ and $E^{\textrm{s}}_{\Lambda}$ respectively, and
$\mathscr{C}^{\ell-1}$-differentiable. They are foliated by $1$-dimensional
unstable and stable manifolds (fibers) of points, $W^{\textrm{u}}(z)$,
$W^{\textrm{s}}(z)$, $z\in\Lambda$, respectively, which are as smooth as the
flow, i.e., $\mathscr{C}^{r}$-differentiable. These fibers are equivariant in
the sense that

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

###  5.3. The NHIM of the unperturbed system

We now describe the geometric structures for the unperturbed system
corresponding to $\varepsilon=0$ and $\lambda=0$. Fix $0<I_{1}<I_{2}$.

The unperturbed system $H_{0}$ has a NHIM:

$\Lambda_{0}=\\{(0,0,I,\theta)\,|\,I\in[I_{1},I_{2}],\,\theta\in\mathbb{T}^{1}\\}.$

The flow restricted to $\Lambda_{0}$ corresponds to the equations of the
rotator subsystem:

(5.5) $\begin{cases}\dot{I}=0\\\ \dot{\theta}=I\\\ \end{cases}$

Hence every level set $I=\textrm{const.}$ is invariant under the flow. The
stable and unstable manifolds of the NHIM $\Lambda_{0}$ coincide, that is

$W^{\textrm{s}}(\Lambda_{0})=W^{\textrm{u}}(\Lambda_{0})=\\{(p,q,I,\theta)\,|\,h_{1}(p,q)=0,\,I\in[I_{1},I_{2}],\,\theta\in\mathbb{T}^{1}\\}.$

where $h_{1}$ is the Hamiltonian of the pendulum in (3.3). The
contraction/expansion rates along $E^{\textrm{s}}$ and $E^{\textrm{u}}$ are
$\mp 1$, respectively. For the time-$2\pi$ map, the corresponding
contraction/expansion rates are $e^{\mp 2\pi}$.

In the extended phase space, we have that
$\tilde{\Lambda}_{0}=\Lambda_{0}\times\mathbb{T}^{1}$ is a NHIM, and

$W^{\textrm{s}}(\tilde{\Lambda}_{0})=W^{\textrm{s}}(\Lambda_{0})\times\mathbb{T}^{1}=W^{\textrm{u}}(\Lambda_{0})\times\mathbb{T}^{1}=W^{\textrm{u}}(\tilde{\Lambda}_{0})$

for system (5.1).

Note that $\Lambda_{0}$ is also the NHIM for the time-$2\pi$ map $f_{0}$ of
the extended flow $\tilde{\Phi}^{t}_{0}$, which represents the first-return
map to the Poincaré section

$\Sigma=\\{(p,q,I,\theta,s)\,|\,s=0\\}.$

###  5.4. The inner map of the unperturbed system

Now, let us consider the time-$2\pi$ map for the Hamiltonian flow of the
rotator:

$\begin{cases}\dot{I}=0,\\\ \dot{\theta}=I.\\\ \end{cases}$

Solving, we have $I(t)=I_{0}$ and $\theta(t)=\theta_{0}+I_{0}t$, which gives
the time-$2\pi$-map $f_{0}$:

(5.6) $f_{0}(I,\theta)=(I^{\prime},\theta^{\prime})=(I,\theta+2\pi I).$

Note that $f_{0}$ satisfies the twist condition

(5.7) $\frac{\partial\theta^{\prime}}{\partial I}=2\pi>0.$

###  5.5. The model with small dissipation

From now on we will work with small dissipation. We will assume

(5.8) $\lambda=\varepsilon\rho,$

where $\rho$ is a free parameter. Consequently, the vector field (3.8) can be
written as

(5.9) $\dot{z}=\mathcal{X}^{0}(z)+\varepsilon\mathcal{X}^{1}(z,t;\rho)$

with $\mathcal{X}^{0}(z)=J\nabla H_{0}(z)$ is the unpertubed system (3.1), and

(5.10) $\mathcal{X}^{1}(z,t;\rho)=J\nabla H_{1}(z,t)+\mathcal{X}_{\rho},$

with $H_{1}$ given in (3.6) and $\mathcal{X}_{\rho}$ given in (3.7). Even when
we use $\lambda$ in the notation, we always assume that
$\lambda=\varepsilon\rho$.

###  5.6. The NHIM in the case of vanishing perturbation

In the case when the perturbation $H_{1}$ is of the form

$H_{1}(p,q,I,\theta,s)=(\cos q-1)\cdot g(\theta,s),$

then $H_{1}$ vanishes at $(p,q)=(0,0)$. When $\varepsilon H_{1}$ is added to
$H_{0}$, the NHIM $\tilde{\Lambda}_{0}$ persists as
$\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda}_{0}$ for the perturbed system
for $\varepsilon>0$, and the flow restricted to the NHIM is given by (5.5).
Consequently, each level set $\\{I=\textrm{constant}\\}$ in the NHIM persists.

When we add the dissipation $\mathcal{X}_{\lambda}$, where
$\lambda=\varepsilon\rho$, since the $(p,q)$-components of
$\mathcal{X}_{\lambda}$ vanish at $(p,q)=(0,0)$, then the NHIM survives for
the perturbed system (3.8) as
$\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda}_{0}$ (if we consider $\lambda$
as an independent parameter, then the perturbed NHIM in general depends on
both $\varepsilon$ and $\lambda$). The induced dynamics on
$\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda}_{0}$ is given by

(5.11) $\begin{cases}\dot{I}=-\lambda(I-\omega_{*})\\\ \dot{\theta}=I\\\
\dot{s}=1\end{cases}$

Note that $\dot{I}=0\Leftrightarrow I=\omega_{*}$. It follows that

(5.12) $\tilde{A}_{\varepsilon}=\\{(I,\theta,s),\ I=\omega_{*},\
(\theta,s)\in\mathbb{T}^{2}\\}\subset\tilde{\Lambda}_{\varepsilon}$

is a $2$-dimensional torus invariant under the flow restricted to
$\tilde{\Lambda}_{\varepsilon}$. This is the only invariant torus for the flow
on $\tilde{\Lambda}_{\varepsilon}$. On $\tilde{A}_{\varepsilon}$ we have
$\dot{\theta}=I=\omega_{*}$ and $\dot{s}=1$, so the flow along this level set
is a linear flow with frequency vector $(\omega_{*},1)$.

By integration of (5.11), we obtain the general solution with initial
condition $(I_{0},\theta_{0},s_{0})$ as:

(5.13) $\begin{split}I(t)&=(I_{0}-\omega_{*})e^{-\lambda t}+\omega_{*},\\\
\theta(t)&=\theta_{0}+\frac{1}{\lambda}(I_{0}-\omega_{*})(1-e^{-\lambda
t})+\omega_{*}t,\\\ s(t)&=s_{0}+t.\end{split}$

Using these explicit formulas, one can see that given
$(\omega_{*},\theta_{0},s_{0})\in\tilde{A}_{\varepsilon}$, if we consider
$(I,\theta,s)\in\tilde{\Lambda}_{\varepsilon}$, where $s=s_{0}$,
$\theta=\theta_{0}-\frac{1}{\lambda}(I-\omega_{*})$, then

(5.14)
$\|\tilde{\Phi}^{t}_{\varepsilon}(I,\theta,s_{0})-\tilde{\Phi}^{t}_{\varepsilon}(\omega_{*},\theta_{0},s_{0})\|\leq\left(1+\frac{1}{\lambda^{2}}\right)^{1/2}\left|I-\omega_{*}\right|e^{-\lambda
t}\rightarrow 0\textrm{ as }t\to\infty,$

showing that $\tilde{A_{\varepsilon}}$ is a global attractor for the flow on
$\tilde{\Lambda}_{\varepsilon}$. (Above we also denoted by
$\tilde{\Phi}^{t}_{\varepsilon}$ the flow restricted to
$\tilde{\Lambda}_{\varepsilon}$.)

###  5.7. The inner map in the case of vanishing perturbation

From the explicit solutions of $I(t)$ and $\theta(t)$ in (5.13) with $t=2\pi$,
we have that

(5.15)
$f_{\varepsilon}(I,\theta)=\bigg{(}(I-\omega_{*})e^{-2\pi\lambda}+\omega_{*},\,\theta+\frac{1}{\lambda}(I-\omega_{*})(1-e^{-2\pi\lambda})+2\pi\omega_{*}\bigg{)},$

which is the first-return map to the section
$\Lambda_{\varepsilon}=\tilde{\Lambda}_{\varepsilon}\cap\\{s=0\,(\textrm{mod}\,2\pi)\\}$.

In particular, for $I=\omega_{*}$ we have
$f_{\varepsilon}(I,\theta)=(\omega_{*},\theta+2\pi\omega_{*})=(I,\theta+2\pi\omega_{*})$.
That is,
$A_{\varepsilon}=\tilde{A}_{\varepsilon}\cap\\{s=0\,(\textrm{mod}\,2\pi)\\}$
is an invariant circle for $f_{\varepsilon}$ of irrational rotation number
$2\pi\omega_{*}$.

From (5.14) we have that, for $(I,\theta)$ with
$\theta=\theta_{0}-\frac{1}{\lambda}(I-\omega_{*})$,

(5.16)
$\|f^{k}_{\varepsilon}(I,\theta)-f^{k}_{\varepsilon}(\omega_{*},\theta_{0})\|\leq\left(1+\frac{1}{\lambda^{2}}\right)^{1/2}\left|I-\omega_{*}\right|e^{-2\pi\lambda
k}\rightarrow 0\textrm{ as }k\to\infty.$

This shows that $A_{\varepsilon}$ is a global attractor for the map
$f_{\varepsilon}$ on $\Lambda_{\varepsilon}$, and, moreover, the orbits of
$(I,\theta)$ and $(\omega_{*},\theta_{0})$ become asymptotically close to one
another as $n\to\infty$.

We have

$Df_{\varepsilon}(I,\theta)=\begin{pmatrix}e^{-2\pi\lambda}&0\\\
\frac{1}{\lambda}(1-e^{-2\pi\lambda})&1\end{pmatrix}$

with eigenvalues $e^{-2\pi\lambda}$ and $1$ and with corresponding
eigenvectors

$\begin{pmatrix}-\lambda\\\ 1\end{pmatrix}\textrm{ and }\begin{pmatrix}0\\\
1\end{pmatrix}$

respectively. The eigenvalue $1$ is associated to the dynamics along the
$\theta$-coordinate, and the eigenvalue of $e^{-2\pi\lambda}<1$ is associated
to the dynamics along the $I$-coordinate.

We conclude that $A_{\varepsilon}$ is a NHIM for
$(f_{\varepsilon})_{\mid\Lambda_{\varepsilon}}$, for which there is only
stable manifold $W^{\textrm{s}}(A_{\varepsilon})$ tangent to
$\begin{pmatrix}-\lambda\\\ 1\end{pmatrix}$, and no unstable manifold
$W^{\textrm{u}}(A_{\varepsilon})$. We note that for $\lambda\gtrsim 0$ (recall
that $\lambda=\varepsilon\rho$) the Lyapunov multipliers $1$ and
$e^{-2\pi\lambda}\lesssim 1$ for
$(f_{\varepsilon})_{\mid\Lambda_{\varepsilon}}$, are dominated by the
contraction rate of $Df_{\varepsilon}$ on the stable bundle $E^{\textrm{s}}$
of $\Lambda_{\varepsilon}$, which is $e^{-2\pi+O(\lambda)}$; see Section 5.3.

Since $|\det(Df_{\varepsilon})|=e^{-2\pi\lambda}<1$ we have that
$f_{\varepsilon}$ is area-contracting on $\Lambda_{\varepsilon}$, hence it is
conformally symplectic, i.e.

(5.17)
$(f_{\varepsilon})_{\mid\Lambda_{\varepsilon}}^{*}(\omega_{\mid\Lambda_{\varepsilon}})=e^{-2\pi\lambda}\omega_{\mid\Lambda_{\varepsilon}}.$

We now show that $f_{\varepsilon}$ is a $\lambda$-perturbation of $f_{0}$, a
time-$2\pi$ map for the rotator part of the unperturbed system given in (5.6).
Since

$\begin{split}e^{-2\pi\lambda}=&1-2\pi\lambda+O(\lambda^{2}),\end{split}$

we have

$\begin{split}f_{\varepsilon}(I,\theta)=&\left(I-2\pi\lambda(I-\omega_{*})+O(\lambda^{2}),\,\theta+2\pi
I-{2\pi^{2}\lambda}(I-\omega_{*})+O(\lambda^{2})\right).\end{split}$

Therefore $f_{\varepsilon}$ is a $\lambda$-perturbation of $f_{0}$ in (5.6),
i.e.

$\begin{split}f_{\varepsilon}(I,\theta)=&f_{0}(I,\theta)+O(\lambda)=f_{0}(I,\theta)+O(\varepsilon\rho).\end{split}$

###  5.8. The case of non-vanishing perturbation

In this case the time-periodic perturbation of the Hamiltonian in (3.5), is of
the form

(5.18) $H_{1}(p,q,I,\theta,s)=\cos q\cdot g(\theta,s).$

The perturbation $H_{1}$ does not vanish at the hyperbolic fixed point of the
pendulum $(p,q)=(0,0)$. The dissipative perturbation is given by the vector
field $\varepsilon\mathcal{X}_{\rho}$, where $\mathcal{X}_{\rho}$ is given by
(3.7), as before (see (5.9) and (5.10)).

From (3.9), since $f^{\prime}(q)=-\sin q$ vanishes at $q=0$, we obtain that
the unperturbed NHIM $\tilde{\Lambda}_{0}$ survives the perturbation, that is
$\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda}_{0}$ for all $\varepsilon$.

When $\lambda\neq 0$, the perturbed dynamics restricted to
$\tilde{\Lambda}_{\varepsilon}=\tilde{\Lambda}_{0}$ is given by the following
equations:

(5.19) $\begin{cases}\dot{I}=-\lambda(I-\omega_{*})-\varepsilon\frac{\partial
g}{\partial\theta}(\theta,s)\\\ \dot{\theta}=I\\\ \dot{s}=1.\end{cases}$

Using the expression of $g$ in (3.11), this system can be reduced to the
second-order nonlinear differential equation

$\ddot{\theta}+\lambda\dot{\theta}-\varepsilon
a_{10}\sin\theta-\lambda\omega_{*}=0.$

Ignoring the last term, the remaining terms represent the equation of the
damped non-linear pendulum, for which explicit solutions are unknown; an
analytical approximation can be found in [Joh14]. Hence, we do not have an
explicit formula for the time-$2\pi$ map $f_{\varepsilon}$ in this case.

##  6\. Existence of a Transverse Homoclinic Intersection

In the sequel, we will identify vector fields with differential operators,
which is a standard operation in differential geometry (see, e.g., [BG05]).
That is, given a smooth vector field $\mathcal{X}$ and a smooth function $f$
on the manifold $M$, we denote:

(6.1) $(\mathcal{X}f)(z)=\sum_{j}(\mathcal{X})_{j}(z)\frac{\partial
f}{\partial z_{j}}(z),$

where $z_{j}$, $j\in\\{1,\ldots,\dim(M)\\}$, are local coordinates. Similarly,
a smooth time-dependent and parameter-dependent vector field acts as a
differential operator by

(6.2)
$(\mathcal{X}f)(z,t;\varepsilon)=\sum_{j}(\mathcal{X})_{j}(z,t;\varepsilon)\frac{\partial
f}{\partial z_{j}}(z).$

For the pendulum system, whose hamiltonian $h_{1}$ is given in (3.3), we
denote by $(p_{0}(t),q_{0}(t))$ a parametrization of a separatrix of the
pendulum, with $(p_{0}(0),q_{0}(0))=(p_{0},q_{0})$, where $(p_{0},q_{0})$ is
some initial point; this parametrization is explicitly given in (3.4). We
define a new locally defined system of symplectic coordinates $(y,x)$ in a
neighborhood of the separatrix – chosen away from the hyperbolic equilibrium
point – as follows. The coordinate $y$ is chosen to be equal to the energy of
the pendulum, i.e.,

(6.3) $y=h_{1}(p,q)=\frac{p^{2}}{2}+(\cos(q)-1)$

and is defined in a whole neighborhood of one of its separatrices. The
coordinate $x$ is defined by

$dx=\frac{dt}{||\nabla y||},$

where $dt=(dp^{2}+dq^{2})^{\frac{1}{2}}$ . It is immediate to see that $x$
equals the time $\tau$ it takes the solution $(p(t),q(t))$ to go along the
$y$-level set from one point to another (see [GdlLM21]). This coordinate
system $(y,x)$ constructed above is not defined in a neighborhood of the
separatrix that contains the hyperbolic equilibrium point, since this is a
critical point of the energy function. We define this coordinate system only
in some neighborhood $\mathcal{N}$ of a segment of the separatrix containing
$(p_{0},q_{0})$. On this neighborhood, we have $dy\wedge dx=dp\wedge dq$.
Relative to this new coordinate system, the separatrix is given by $y=0$.

An arbitrary point on the separatrix can be given in terms of the
$(p,q)$-coordinates as $(p_{0}(\tau),q_{0}(\tau))$ for some
$\tau\in\mathbb{R}$, and in terms of the $(y,x)$-coordinates as $(0,x)$ for
some $x\in\mathbb{R}$, where $x=\tau$.

Now let’s extend this coordinate system to a system of coordinates
$(y,x,I,\theta,s)$ on some neighborhood $\tilde{\mathcal{N}}$ of
$\\{(p_{0}(\tau),q_{0}(\tau),I,\theta,s)\\}$ in the extended phase space.

Relative to this coordinate system, in the unperturbed case, the
stable/unstable manifolds
$W^{\textrm{s}}(\tilde{\Lambda}_{0})=W^{\textrm{u}}(\tilde{\Lambda}_{0})$ are
locally given by $y=0$. A point $\tilde{z}_{0}\in
W^{\textrm{s}}(\tilde{\Lambda}_{0})=W^{\textrm{u}}(\tilde{\Lambda}_{0})$ can
be written in terms of the original coordinates $(p,q,I,\theta,s)$ as

$\tilde{z}_{0}=(p_{0}(\tau),q_{0}(\tau),I,\theta,s),\textrm{ for some
}\tau\in\mathbb{R},$

and in terms of the extended coordinates $(y,x,I,\theta,s)$ as

$\tilde{z}_{0}=(0,x,I,\theta,s),\textrm{ for }x=\tau\in\mathbb{R}.$

When we apply the flow to the point $\tilde{z}_{0}$ we obtain

$\tilde{\Phi}_{0}^{t}(\tilde{z}_{0})=(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t).$

Observe that if we denote by
$\tilde{z}^{\pm}_{0}:=(p,q,I,\theta,s)=(0,0,I,\theta,s)$, we have
$\tilde{\Phi}_{0}^{t}(\tilde{z}^{\pm}_{0})=(0,0,I,\theta+\omega(I)t,s+t)$,
therefore:

$\tilde{\Phi}_{0}^{t}(\tilde{z}_{0})-\tilde{\Phi}_{0}^{t}(\tilde{z}^{\pm}_{0})\to
0,\quad\mbox{as}\quad t\to\pm\infty.$

In the perturbed case, for $\varepsilon\neq 0$ small and
$\lambda=\rho\varepsilon$, we can locally describe both the stable and
unstable manifolds of $\tilde{\Lambda}_{\varepsilon}$ as graphs of
$C^{\ell-1}$-smooth functions $y_{\varepsilon}^{s},y_{\varepsilon}^{u},$ over
$(x,I,\theta,s)$, recalling that $x=\tau$, given by

$\begin{split}y_{\varepsilon}^{s}=&y_{\varepsilon}^{s}(x,I,\theta,s;\rho)=y_{\varepsilon}^{s}(\tau,I,\theta,s;\rho),\\\
y_{\varepsilon}^{u}=&y_{\varepsilon}^{u}(x,I,\theta,s;\rho)=y_{\varepsilon}^{u}(\tau,I,\theta,s;\rho),\end{split}$

respectively, for $(0,x,I,\theta,s)\in\tilde{\mathcal{N}}$. We stress the
dependence of $\rho$ of these functions because will be important in the
sequel.

Observe that, when $\varepsilon=0$ we have the equation of the separatrix of
the pendulum

$y_{0}^{s}(\tau,I,\theta,s;\rho)=y_{0}^{u}(\tau,I,\theta,s;\rho)=0.$

Consequently $y_{\varepsilon}^{u},y_{\varepsilon}^{s}=O(\varepsilon)$.

We recall the following Melnikov-type result for non-conservative
perturbations:

######  Theorem 6.1 (Splitting of the Stable and Unstable Manifolds
[GdlLM21]).

Fix $\rho_{0}>0$, then there exists $\varepsilon_{0}>0$ such that for any
$0\leq\rho\leq\rho_{0}$ and $0\leq|\varepsilon|\leq\varepsilon_{0}$ we have:
for $(0,\tau,I,\theta,s)\in\tilde{\mathcal{N}}$, the difference between
$y_{\varepsilon}^{s}(\tau,I,\theta,s;\rho)$ and
$y_{\varepsilon}^{\textrm{u}}(\tau,I,\theta,s;\rho)$ is given by

$\begin{split}y_{\varepsilon}^{\textrm{s}}-y_{\varepsilon}^{\textrm{u}}=&-\varepsilon\int_{-\infty}^{+\infty}((\mathcal{X}^{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(\mathcal{X}^{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}^{\pm}_{0})))dt+O(\varepsilon^{2}),\end{split}$

where we recall that $h_{1}(p,q)=\frac{p^{2}}{2}+(\cos q-1)$ and
$\mathcal{X}^{1}$ is given in (5.10).

######  Corollary 6.2 (Sufficient Conditions for the Existence of a Transverse
Homoclinic Intersection).

Fix $\rho_{0}>0$, then there exists $\varepsilon_{0}>0$ such that for any
$0\leq\rho\leq\rho_{0}$ and $|\varepsilon|\leq\varepsilon_{0}$ we have: for
$(0,\tau,I,\theta,s)\in\tilde{\mathcal{N}}$, the difference between
$y_{\varepsilon}^{\textrm{s}}(\tau,I,\theta,s;\rho)$ and
$y_{\varepsilon}^{\textrm{u}}(\tau,I,\theta,s;\rho)$ is given by

(6.4)
$\begin{split}y_{\varepsilon}^{\textrm{s}}-y_{\varepsilon}^{\textrm{u}}=&-\varepsilon\left[\int_{-\infty}^{+\infty}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)dt\right.\\\
&\left.\qquad-{\rho}\int_{-\infty}^{+\infty}p_{0}^{2}(t)dt\right]+O(\varepsilon^{2}),\end{split}$

where $\\{\cdot,\cdot\\}$ denotes the Poisson bracket.

If $\tau^{*}=\tau^{*}(I,\theta,s)$ is a non-degenerate zero of the mapping

(6.5)
$\begin{split}\tau\in\mathbb{R}\mapsto&-\int_{-\infty}^{+\infty}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)dt\end{split}$

then there exists $0<\rho_{1}\leq\rho_{0}$ such that for all
$0\leq\rho\leq\rho_{1}$

(6.6)
$\begin{split}\tau\in\mathbb{R}\mapsto&-\left[\int_{-\infty}^{+\infty}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)dt\right.\\\
&\left.\qquad-{\rho}\int_{-\infty}^{+\infty}p_{0}^{2}(t)dt\right]\end{split}$

has a non degenerate zero $\tau^{*}(I,\theta,s;\rho)$.
Moreover, there exists $0<\varepsilon_{1}\leq\varepsilon_{0}$ such that for
all $0\leq\rho\leq\rho_{1}$ and $0<|\varepsilon|\leq\varepsilon_{1}$,
$W^{\textrm{s}}(\tilde{\Lambda}_{\varepsilon})$ and
$W^{\textrm{u}}(\tilde{\Lambda}_{\varepsilon})$ have a transverse homoclinic
intersection which can be parametrized as

(6.7)
$(\tau^{*},y^{\textrm{s}}_{\varepsilon}(\tau^{*},I,\theta,s;\rho),I,\theta,s)=(\tau^{*},y^{\textrm{u}}_{\varepsilon}(\tau^{*},I,\theta,s;\rho),I,\theta,s),$

where
$\tau^{*}=\tau^{*}(I,\theta,s;\rho,\varepsilon)=\tau^{*}(I,\theta,s;\rho)+O(\varepsilon)=\tau^{*}(I,\theta,s)+O(\rho,\varepsilon)$,
for $(I,\theta,s)$ in some open set in
$\tilde{U}\subseteq\mathbb{R}\times\mathbb{T}^{1}\times\mathbb{T}^{1}$.

###### Proof.

From Theorem 6.1, we have

$\begin{split}y_{\varepsilon}^{\textrm{s}}-y_{\varepsilon}^{\textrm{u}}=&-\varepsilon\int_{-\infty}^{+\infty}((\mathcal{X}^{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(\mathcal{X}^{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm})))dt+O(\varepsilon^{2}).\end{split}$

As $\lambda=\varepsilon{\rho}$, the vector field $\mathcal{X}^{1}$ (see
(5.10)) is the sum of the Hamiltonian vector field $J\nabla H_{1}$ and the
dissipative vector field

$\mathcal{X}_{\rho}(p,q,I,\theta)=(-{\rho}p,0,-{\rho}(I-\omega_{*}),0),$

therefore

$\begin{split}y_{\varepsilon}^{\textrm{s}}-y_{\varepsilon}^{\textrm{u}}=&-\varepsilon\int_{-\infty}^{+\infty}\bigg{[}(J\nabla
H_{1}+\mathcal{X}_{\rho})h_{1}(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(J\nabla
H_{1}+\mathcal{X}_{\rho})h_{1}(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))\bigg{]}dt+O(\varepsilon^{2})\\\
=&-\varepsilon\bigg{[}\int_{-\infty}^{+\infty}(J\nabla
H_{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(J\nabla
H_{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt\\\
&\qquad+\int_{-\infty}^{+\infty}(\mathcal{X}_{\rho}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(\mathcal{X}_{\rho}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt\bigg{]}+O(\varepsilon^{2})\\\
:=&-\varepsilon(\mathcal{F}_{1}+\mathcal{F}_{2})+O(\varepsilon^{2}).\end{split}$

In the above,

$\begin{split}\mathcal{F}_{1}:=&\int_{-\infty}^{+\infty}(J\nabla
H_{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(J\nabla
H_{1}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt\\\
=&\int_{-\infty}^{+\infty}\\{h_{1},H_{1}\\}(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-\\{h_{1},H_{1}\\}(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt,\\\
=&\int_{-\infty}^{+\infty}\bigg{(}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)\\\
&\qquad-\\{h_{1},H_{1}\\}(0,0,I,\theta+\omega(I),s+t)\bigg{)}dt\\\
\mathcal{F}_{2}:=&\int_{-\infty}^{+\infty}(\mathcal{X}_{\rho}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(\mathcal{X}_{\rho}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt\\\
=&{\rho}\int_{-\infty}^{+\infty}(\mathcal{X}_{\omega_{*}}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))-(\mathcal{X}_{\omega_{*}}h_{1})(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}^{\pm}))dt\end{split}$

where we denote $\mathcal{X}_{\omega_{*}}=(-p,0,-(I-\omega_{*}),0)$. Since
$\mathcal{X}_{\omega_{*}}h_{1}=-p\frac{\partial h_{1}}{\partial p}=-p^{2}$,
and recalling that
$\tilde{\Phi}^{t}_{0}(\tilde{z}_{0})=(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)$
and $\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}^{\pm})=(0,0,I,\theta+\omega(I)t,s+t)$,
we obtain

$\begin{split}\mathcal{F}_{2}=&-{\rho}\int_{-\infty}^{+\infty}p_{0}^{2}(\tilde{\Phi}_{0}^{t}(\tilde{z}_{0}))dt=-\rho\int_{-\infty}^{+\infty}p_{0}^{2}(\tau+t)dt\end{split}$

Finally,

$\begin{split}y_{\varepsilon}^{\textrm{s}}-y_{\varepsilon}^{\textrm{u}}=&-\varepsilon\int_{-\infty}^{+\infty}\bigg{(}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)\\\
&\qquad-\\{h_{1},H_{1}\\}(0,0,I,\theta+\omega(I),s+t)\bigg{)}dt\\\
&+\varepsilon{\rho}\int_{-\infty}^{+\infty}p_{0}^{2}(\tau+t)dt+O(\varepsilon^{2}).\end{split}$

Note that $\\{h_{1},H_{1}\\}=-\sin q\frac{\partial H_{1}}{\partial
p}-p\frac{\partial H_{1}}{\partial q}$ hence
$\\{h_{1},H_{1}\\}(0,0,I,\theta+\omega(I),s+t)=0$. Also note that by the
change of variable formula
$\int_{-\infty}^{+\infty}p_{0}^{2}(\tau+t)dt=\int_{-\infty}^{+\infty}p_{0}^{2}(t)dt$.
Thus, we obtain the first part of Corollary 6.2.

The second part of Corollary 6.2 is as follows. First, if
$\tau^{*}=\tau^{*}(I,\theta,s)$ is a non-degenerate zero of the mapping (6.5),
there exists $0<\rho_{1}\leq\rho_{0}$ such that the function:

$\begin{split}\tau\in\mathbb{R}\mapsto&-\left[\int_{-\infty}^{+\infty}\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)dt-{\rho}\int_{-\infty}^{+\infty}p_{0}^{2}(t)dt\right]\end{split}$

also has a non-degenerate zero
$\tau^{*}(I,\theta,s;\rho)=\tau^{*}(I,\theta,s)+O(\rho)$ for any
$0\leq\rho\leq\rho_{1}$.
Now, we apply the implicit function theorem to find the zeroes of the
function:

$\tau\to
y_{\varepsilon}^{\textrm{s}}(\tau,I,\theta,s;\rho)-y_{\varepsilon}^{\textrm{u}}(\tau,I,\theta,s;\rho),$

obtaining a value $0<\tilde{\varepsilon}_{1}(\rho)\leq\varepsilon_{0}$, such
that, for any $0<\varepsilon\leq\tilde{\varepsilon}_{1}$, this map has a non-
degenerate zero
$\tau^{*}(I,\theta,s;\rho,\varepsilon)=\tau^{*}(I,\theta,s;\rho)+O(\varepsilon)=\tau^{*}(I,\theta,s)+O(\rho,\varepsilon)$.
An important observation is that $\tilde{\varepsilon}_{1}(0)\neq 0$, therefore
we set $\varepsilon_{1}=\min_{[0,\rho_{1}]}\tilde{\varepsilon}_{1}(\rho)>0$.
In this way, arguing as in [DLS06], the stable and unstable manifolds
$W^{\textrm{s}}(\tilde{\Lambda}_{\varepsilon})$ and
$W^{\textrm{u}}(\tilde{\Lambda}_{\varepsilon})$ have a transverse homoclinic
intersection which can be parametrized as in (6.7). ∎

Provided that the unperturbed stable and unstable manifolds of the NHIM
coincide, adding a generic Hamiltonian perturbation makes the stable and
unstable manifolds to intersect transversally; see, e.g., [GdlL18]. However,
non-conservative perturbations can in general destroy the homoclinic
intersection; this is for example the case of the dissipative pendulum shown
in Fig. 2b. In contrast, the Corollary 6.2 shows that for the system (3.8),
where the dissipation is of the same order as the forcing, that is
$\lambda=\varepsilon\rho$, the perturbed stable and unstable manifolds
intersect transversally for all sufficiently small perturbation parameter
values $\rho$. Later, in Section 8, we will be interested in taking
$\rho=\rho(\varepsilon)=\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$, but,
clearly, for $\varepsilon$ small enough, these values of $\rho$ satisfy the
hypotheses of Corollary 6.2. The result is summarized in next corollary.

######  Corollary 6.3 (Existence of Transverse Intersection in the Model).

Take any $\bar{\rho}>0$. Consider the perturbation $H_{1}$ given by (3.10) or
(3.11) and the dissipative pertubation as in (3.7) with
$\lambda=\varepsilon\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$. Then
there exists $\varepsilon_{0}$ sufficiently small such that for all
$0<|\varepsilon|<\varepsilon_{0}$,
$W^{\textrm{s}}(\tilde{\Lambda}_{\varepsilon})$ and
$W^{\textrm{u}}(\tilde{\Lambda}_{\varepsilon})$ have a transverse homoclinic
intersection $\tilde{\Gamma}_{\varepsilon}$ which can be parametrized as in
(6.7).

###### Proof.

The proof follows by the fact that in this case
$\rho=\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$ satisfies the conditions
of Corollary 6.2 if $\varepsilon$ is small enough. ∎

##  7\. Computation of the scattering map for the perturbed system

###  7.1. The scattering map

We give a brief description of the scattering map, following [DdlLS08].
Consider the general case of a normally hyperbolic invariant manifold
$\Lambda$ for a flow $\Phi^{t}$ on some smooth manifold $M$. Let
$W^{\textrm{s}}(\Lambda)$, $W^{\textrm{u}}(\Lambda)$ be the stable and
unstable manifolds of $\Lambda$. First, let

$\begin{split}\Omega^{+}:W^{\textrm{s}}(\Lambda)\to\Lambda,\\\
\Omega^{-}:W^{\textrm{u}}(\Lambda)\to\Lambda,\end{split}$

be the canonical projections along fibers, assigning to each point $x\in
W^{\textrm{s}}(\Lambda)$ its stable foot point $x^{+}=\Omega^{+}(x)$, uniquely
defined by $x\in W^{\textrm{s}}(x^{+})$, and, similarly, assigning to $x\in
W^{\textrm{u}}(\Lambda)$ its unstable footpoint $x^{-}=\Omega^{-}(x)$ uniquely
defined by $x\in W^{\textrm{u}}(x^{-})$.

Second, choose and fix a ‘homoclinic channel’, which is a homoclinic manifold
$\Gamma$ in $W^{\textrm{u}}(\Lambda)\cap W^{\textrm{s}}(\Lambda)$ that
satisfies the following strong transversality conditions:

$\begin{split}T_{x}\Gamma=&T_{x}W^{\textrm{s}}(\Lambda)\cap
T_{x}W^{\textrm{u}}(\Lambda),\\\ T_{x}M=&T_{x}\Gamma\oplus
T_{x}W^{\textrm{u}}(x^{-})\oplus T_{x}W^{\textrm{s}}(x^{+}),\end{split}$

for all $x\in\Gamma$, and such that

$\Omega^{\pm}_{\mid\Gamma}:\Gamma\to\Omega^{\pm}(\Gamma)\textrm{ is a
diffeomorphism. }$

Then, the scattering map associated to the homoclinic channel $\Gamma$ is the
mapping $\sigma:\Omega^{-}(\Gamma)\to\Omega^{+}(\Gamma)$ defined by

$\sigma=\Omega^{+}\circ(\Omega^{-})^{-1}.$

The map $\sigma$ is a locally defined diffeomorphism on $\Lambda$. Moreover,
$\sigma$ is symplectic provided that $M$, $\Lambda$, $\Phi^{t}$ are
symplectic.

######  Remark 7.1.

We have $\sigma(x^{-})=x^{+}$ if and only if

(7.1) $d(\Phi^{-T_{-}}(x),\Phi^{-T_{-}}(x^{-}))\to~{}0,\textrm{ and
}d(\Phi^{T_{+}}(x),\Phi^{T_{+}}(x^{+}))\to~{}0$

as $T_{-},T_{+}\to+\infty$, respectively, for some uniquely defined
$x\in\Gamma$. This means that for orbits in $\Lambda$ of the form
$x^{\prime}_{\textrm{end}}=\Phi^{T_{+}}\circ\sigma\circ\Phi^{T_{-}}(x^{\prime}_{\textrm{start}})$,
where $x^{\prime}_{\textrm{start}}=\Phi^{-T_{-}}(x^{-})$ and
$x^{\prime}_{\textrm{end}}=\Phi^{T_{+}}(x^{+})$, one can find homoclinic orbit
segments in $M$ of the form
$x_{\textrm{end}}=\Phi^{T_{+}+T_{-}}(x_{\textrm{start}})$, such that
$x_{\textrm{start}}$ is arbitrarily close to $x^{\prime}_{\textrm{start}}$ and
$x_{\textrm{end}}$ is arbitrarily close to $x^{\prime}_{\textrm{end}}$. See
Fig. 4.

Figure 4. Homoclinic orbit segment approximating an orbit obtained by applying
the scattering map and the inner map.

###  7.2. The scattering map of the perturbed system

Assuming that the conditions in Corollary 6.2 are satisfied, then
$W^{\textrm{s}}(\tilde{\Lambda}_{\varepsilon})$ and
$W^{\textrm{u}}(\tilde{\Lambda}_{\varepsilon})$ intersect transversally in the
homoclinic channel $\tilde{\Gamma}_{\varepsilon}$, which can be parametrized
as in Corollary 6.2, for all $0<|\varepsilon|<\varepsilon_{1}$. Let
$\tilde{z}_{\varepsilon}\in\tilde{\Gamma}_{\varepsilon}$ be a homoclinic point
for the perturbed extended flow $\tilde{\Phi}^{t}_{\varepsilon}$. In terms of
the coordinates from Section 6, we have

$\tilde{z}_{\varepsilon}=(\tau^{*}(I,\theta,s;\rho,\varepsilon),y^{\textrm{s}}_{\varepsilon}(\tau^{*}(I,\theta,s;\rho,\varepsilon),I,\theta,s;\rho),I,\theta,s)$

where
$\tau^{*}(I,\theta,s;\rho,\varepsilon)=\tau^{*}(I,\theta,s;\rho)+O(\varepsilon)$
and $\tau^{*}(I,\theta,s;\rho)=\tau^{*}(I,\theta,s)+O(\varepsilon)$ is a non-
degenerate zero of the mapping (6.6) near $\tau^{*}(I,\theta,s)$, a chosen
non-degenerate zero of the mapping (6.5).

Because of the smooth dependence of the NHIM and of its stable and unstable
manifolds on the perturbation parameter, to the homoclinic point
$\tilde{z}_{\varepsilon}=(z_{\varepsilon},s)$ for perturbed flow
$\tilde{\Phi}^{t}_{\varepsilon}$ it corresponds a homoclinic point
$\tilde{z}_{0}=(z_{0},s)$ for the unperturbed flow $\tilde{\Phi}^{t}_{0}$,
which is $O(\varepsilon)$-close to $\tilde{z}_{\varepsilon}$. In fact, going
back to the original coordinates, the point $\tilde{z}_{\varepsilon}$ becomes:

(7.2)
$\begin{split}\tilde{z}_{\varepsilon}&=(p_{0}(\tau^{*}(I,\theta,s;\rho,\varepsilon)),q_{0}(\tau^{*}(I,\theta,s;\rho,\varepsilon)),I,\theta,s)+O(\varepsilon)\\\
&=(p_{0}(\tau^{*}(I,\theta,s;\rho)),q_{0}(\tau^{*}(I,\theta,s;\rho)),I,\theta,s)+O(\varepsilon)\\\
&=\tilde{z}_{0}+O(\varepsilon),\mbox{ where }\\\
\tilde{z}_{0}&=(p_{0}(\tau^{*}(I,\theta,s;\rho)),q_{0}(\tau^{*}(I,\theta,s;\rho)),I,\theta,s)\end{split}$

Note that in the above the $O(\varepsilon)$-error only affects the $p,q$
components.

We denote the stable- and unstable-footpoints of $\tilde{z}_{\varepsilon}$ and
of $\tilde{z}_{0}$ by $\tilde{z}_{\varepsilon}^{\pm}$ and
$\tilde{z}_{0}^{\pm}$, respectively. Recall that we already know that
$\tilde{z}_{0}^{\pm}=(0,0,I,\theta,s)$. Summarizing the notation:

  * •

$\tilde{z}_{\varepsilon}\in\tilde{\Gamma}_{\varepsilon}\subset
W^{\textrm{s}}(\tilde{\Lambda}_{\varepsilon})\pitchfork
W^{\textrm{u}}(\tilde{\Lambda}_{\varepsilon})$;

  * •

$\tilde{z}^{\pm}_{\varepsilon}=\Omega^{\pm}(\tilde{z}_{\varepsilon})\in\tilde{\Lambda}_{\varepsilon}$;

  * •

$\tilde{z}_{0}\in\tilde{\Gamma}_{0}\subset
W^{\textrm{s}}(\tilde{\Lambda}_{0})\pitchfork
W^{\textrm{u}}(\tilde{\Lambda}_{0})$;

  * •

$\tilde{z}^{\pm}_{0}=\Omega^{\pm}(\tilde{z}_{0})\in\tilde{\Lambda}_{0}$;

Under the above assumptions, we have
$\tilde{\sigma}_{\varepsilon}(\tilde{z}^{-}_{\varepsilon})=\tilde{z}^{+}_{\varepsilon}$,
and $\tilde{\sigma}_{0}(\tilde{z}^{-}_{0})=\tilde{z}^{+}_{0}$. We recall that,
in our model, for the unperturbed system,
$\tilde{z}^{-}_{0}=\tilde{z}^{+}_{0}=\tilde{z}_{0}^{\pm}$ and therefore the
scattering map is the identity: $\tilde{\sigma}_{0}=$Id.

The perturbed scattering map $\tilde{\sigma}_{\varepsilon}$ can be expanded in
terms of powers of $\varepsilon$, with the zero-th order term being the
unperturbed scattering map $\tilde{\sigma}_{0}$, as follows

$\begin{split}\tilde{\sigma}_{\varepsilon}(I,\theta,s)=&\tilde{\sigma}_{0}(I,\theta,s)+\varepsilon\mathcal{S}(I,\theta,s)+O(\varepsilon^{2})\\\
=&(I,\theta,s)+\varepsilon\mathcal{S}(I,\theta,s)+O(\varepsilon^{2}),\end{split}$

where $\mathcal{S}=(\mathcal{S}^{I},\mathcal{S}^{\theta},\textrm{Id}^{s})$.

In the sequel, we evaluate the components $\mathcal{S}^{I}$ and
$\mathcal{S}^{\theta}$ in order to compute the change in action $I$ and the
change in angle $\theta$ by the scattering map. We follow the approach in
[GdlLM21, GdlLM22].

###  7.3. Change in Action by the Scattering Map

We use the following result:

######  Theorem 7.2 (Change in Action by the Scattering Map [GdlLM21]).

For a general non-conservative perturbation $\mathcal{X}^{1}$ of (3.1) like in
(5.9), the change in action $I$ by the scattering map
$\tilde{\sigma}_{\varepsilon}$ is given by:

(7.3)
$\begin{split}I\left(\tilde{z}^{+}_{\varepsilon}\right)-I\left(\tilde{z}^{-}_{\varepsilon}\right)=&\varepsilon\int_{-\infty}^{+\infty}\left(\mathcal{X}^{1}I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X}^{1}I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\right)dt\\\
&+O\left(\varepsilon^{2}\right),\end{split}$

where $\tilde{z}_{\varepsilon}$ and $\tilde{z}_{0}$ are given in (7.2), and we
denote $I_{0}=I(\tilde{z}_{0})=I(\tilde{z}^{\pm}_{0})$.

Denote $I\left(\tilde{z}^{\pm}_{\varepsilon}\right)=I^{\pm}_{\varepsilon}$.
Applying Theorem 7.2 in the case of (5.10):

$\begin{split}\mathcal{X}^{1}=&J\nabla
H_{1}+(-{\rho}p,0,-{\rho}(I-\omega_{*}),0)\\\ =&J\nabla
H_{1}+\mathcal{X}_{\rho}\end{split}$

we obtain

$\begin{split}I^{+}_{\varepsilon}-I^{-}_{\varepsilon}=&\varepsilon\int^{\infty}_{-\infty}\bigg{(}(J\nabla
H_{1}+\mathcal{X_{\rho}})I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-(J\nabla
H_{1}+\mathcal{X_{\rho}})I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt+O(\varepsilon^{2})\\\
=&\varepsilon\int^{\infty}_{-\infty}\bigg{(}(J\nabla
H_{1}I)(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-(J\nabla
H_{1}I)(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
&+\varepsilon\int^{\infty}_{-\infty}\bigg{(}\mathcal{X_{\rho}}I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X_{\rho}}I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt+O(\varepsilon^{2})\\\
=&\varepsilon(\mathcal{S}_{1}^{I}+\mathcal{S}_{2}^{I})+O(\varepsilon^{2})\end{split}$

where

$\begin{split}\mathcal{S}_{1}^{I}:=&\int^{\infty}_{-\infty}\bigg{(}\\{I,H_{1}\\}((\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\\{I,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
=&\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.,\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\\\
\mathcal{S}_{2}^{I}=&\int^{\infty}_{-\infty}\bigg{(}\mathcal{X_{\rho}}I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X_{\rho}}I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt,\end{split}$

where $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ is a non-degenerate zero of the
function (6.6).

Since $\mathcal{X}_{\rho}I=-{\rho}(I-\omega_{*})$,
$\tilde{\Phi}^{t}_{0}(\tilde{z}_{0})=(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)$
and $\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}^{\pm})=(0,0,I,\theta+\omega(I)t,s+t)$,
we have

$\begin{split}\mathcal{S}_{2}^{I}:=&\int^{\infty}_{-\infty}\bigg{(}-{\rho}I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))+{\rho}I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt=0.\end{split}$

Thus, we have proved the following result:

######  Corollary 7.3.

For the perturbation $\mathcal{X}^{1}=J\nabla H_{1}+\mathcal{X}_{\rho}$,

(7.4)
$\begin{split}I^{+}_{\varepsilon}-I^{-}_{\varepsilon}=&\varepsilon\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt+O(\varepsilon^{2})\end{split}$

where $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ is a non-degenerate zero of the
function (6.6).

In the case when $H_{1}$ is as in (3.10) or (3.11),
$\\{I,H_{1}\\}=-\frac{\partial H_{1}}{\partial\theta}=a_{10}f(q)\sin\theta$,
where $f(q)=\cos q-1$ or $f(q)=\cos q$, so

(7.5) $\begin{split}I^{+}_{\varepsilon}-I^{-}_{\varepsilon}=&\varepsilon
a_{10}\int^{\infty}_{-\infty}(\cos(q_{0}(\tau^{*}+t))-1)\sin(\theta+\omega(I)t)dt+O(\varepsilon^{2}).\end{split}$

###  7.4. Change in Angle by the Scattering Map

We use the following result:

######  Theorem 7.4 (Change in Angle by the Scattering Map [GdlLM21]).

For a general non-conservative perturbation $\mathcal{X}^{1}$ of (3.1) like in
(5.9), the change in angle $\theta$ by the scattering map
$\tilde{\sigma}_{\varepsilon}$ is given by:

(7.6)
$\begin{split}\theta(\tilde{z}^{+}_{\varepsilon})-\theta(\tilde{z}^{-}_{\varepsilon})=&\varepsilon\int_{-\infty}^{+\infty}\mathcal{X}^{1}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X}^{1}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))dt\\\
&-\varepsilon\int_{-\infty}^{+\infty}(\mathcal{X}^{1}I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X}^{1}I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0})))tdt\cdot\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\right)\\\ &+O(\varepsilon^{2}),\end{split}$

where $\tilde{z}_{\varepsilon}$ and $\tilde{z}_{0}$ are given in (7.2), and we
denote $I_{0}=I(\tilde{z}_{0})=I(\tilde{z}^{\pm}_{0})$.

Denote
$\theta\left(\tilde{z}^{\pm}_{\varepsilon}\right)=\theta^{\pm}_{\varepsilon}$.
Applying Theorem 7.4 in the case of (5.10), i.e., $\mathcal{X}^{1}=J\nabla
H_{1}+\mathcal{X}_{\rho}$ we obtain

(7.7)
$\begin{split}\theta^{+}_{\varepsilon}-\theta^{-}_{\varepsilon}=&\varepsilon\int^{\infty}_{-\infty}\bigg{(}(J\nabla
H_{1}+\mathcal{X_{\rho}})\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-(J\nabla
H_{1}+\mathcal{X_{\rho}})\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
&-\varepsilon\int^{\infty}_{-\infty}\left((J\nabla
H_{1}+\mathcal{X_{\rho}})I(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))\right.\\\
&\left.\qquad\qquad-(J\nabla
H_{1}+\mathcal{X_{\rho}})I(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\right)tdt\cdot\bigg{(}\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I)\bigg{)}\\\ &+O(\varepsilon^{2})\end{split}$

We simplify the first integral above by splitting into two integrals:

$\begin{split}&\varepsilon\bigg{[}\int^{\infty}_{-\infty}\bigg{(}(J\nabla
H_{1}\theta)(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-(J\nabla
H_{1}\theta)(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
&\,+\int^{\infty}_{-\infty}\bigg{(}\mathcal{X_{\rho}}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X_{\rho}}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\bigg{]}\\\
&=\varepsilon(\mathcal{S}_{1}^{\theta}+\mathcal{S}_{2}^{\theta}),\end{split}$

where

$\begin{split}\mathcal{S}_{1}^{\theta}:=&\int^{\infty}_{-\infty}\bigg{(}\\{\theta,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0})-\\{\theta,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
=&\int^{\infty}_{-\infty}\bigg{(}\\{\theta,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\\\
&-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\bigg{)}dt,\\\
\mathcal{S}_{2}^{\theta}=&\int^{\infty}_{-\infty}\bigg{(}\mathcal{X_{\rho}}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\mathcal{X_{\rho}}\theta(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt,\\\
\end{split}$

where $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ is a non-degenerate zero of the
function (6.6).

Since $\mathcal{X}_{\rho}\theta=0$ we obtain $\mathcal{S}_{2}^{\theta}=0.$

The second integral in (7.7) can be simplified as we did in Section 7.3 to
analyze the change in actions, and thus, combining both parts of (7.7) proves
the following result:

######  Corollary 7.5.

For the perturbation $\mathcal{X}^{1}=J\nabla H_{1}+\mathcal{X}_{\rho}$ given
in (5.9),

(7.8)
$\begin{split}\theta^{+}_{\varepsilon}-\theta^{-}_{\varepsilon}=&\varepsilon\int^{\infty}_{-\infty}\bigg{(}\\{\theta,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\\{\theta,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}dt\\\
&-\varepsilon\int^{\infty}_{-\infty}\bigg{(}\\{I,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}_{0}))-\\{I,H_{1}\\}(\tilde{\Phi}^{t}_{0}(\tilde{z}^{\pm}_{0}))\bigg{)}tdt\cdot\bigg{(}\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I)\bigg{)}\\\ &+O(\varepsilon^{2})\\\
&=\varepsilon\int^{\infty}_{-\infty}\bigg{(}\\{\theta,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\\\
&\qquad-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\bigg{)}dt\\\
&-\varepsilon\int^{\infty}_{-\infty}\bigg{(}\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\\\
&\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\bigg{)}t\,dt\cdot\bigg{(}\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I_{0})\bigg{)}\\\ &+O(\varepsilon^{2}),\end{split}$

where $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ is a non-degenerate zero of the
function (6.6).

In the case when $h_{0}(I)=\frac{I^{2}}{2}$ and $H_{1}$ is as in (3.10) or
(3.11), we have $\frac{\partial^{2}h_{0}}{\partial I^{2}}=1$,
$\\{\theta,H_{1}\\}=0$, and $\\{I,H_{1}\\}=-\frac{\partial
H_{1}}{\partial\theta}=a_{10}f(q)\sin\theta$, where $f(q)=\cos q-1$ or
$f(q)=\cos q$, so

(7.9)
$\begin{split}\theta^{+}_{\varepsilon}-\theta^{-}_{\varepsilon}=&-\varepsilon
a_{10}\int^{\infty}_{-\infty}\bigg{(}\cos(q_{0}(\tau^{*}+t)-1)\sin(\theta+\omega(I)t)\bigg{)}t\,dt+O(\varepsilon^{2}).\end{split}$

######  Remark 7.6.

We remark that both components $\mathcal{S}^{I}$, $\mathcal{S}^{\theta}$ of
the vector field generating the scattering map up to $O(\varepsilon^{2})$ only
depends on the dissipation $\mathcal{X}_{\lambda}$ and, therefore, on the
parameter $\rho$, through the value $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$. In
fact, in the next section we will see that the vector field generating the
scattering map is a Hamiltonian vector field in the variables $(I,\theta)$ up
to $O(\varepsilon^{2})$, even though the system (3.8) is not symplectic but
conformally symplectic. We will show that the scattering map is symplectic in
the variables $(I,\theta)$ up to $O(\varepsilon^{2})$. Moreover, in the case
when $H_{1}$ is as in (3.10) or (3.11), we will provide an explicit formula
for the Hamiltonian vector field that generates the scattering map up to
$O(\varepsilon^{2})$.

###  7.5. Symplecticity of the scattering map up to $O(\varepsilon^{2})$

In the case that the perturbation is Hamiltonian (which in our case
corresponds to $\rho=0$), it was proven in [DdlLS08] that the scattering map
is symplectic and is given by

(7.10)
$\tilde{\sigma}_{\varepsilon}(I,\theta,s)=\tilde{\sigma}_{0}(I,\theta,s)+\varepsilon\left(\frac{\partial
L^{*}}{\partial\theta}(I,\theta,s),-\frac{\partial L^{*}}{\partial
I}(I,\theta,s),s\right)+O(\varepsilon^{2})$

for some function (Melnikov potential) $L^{*}$ which depends on the effect of
the Hamiltonian perturbation on the homoclinic orbits of the unperturbed
system. More precisely, let

(7.11)
$\begin{split}\mathcal{L}(I,\theta,s)=-\int_{-\infty}^{+\infty}&\left(H_{1}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.-H_{1}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\end{split}$

where $\omega(I)=\frac{\partial h_{0}}{\partial I}(I)$. Let
$\tau^{*}=\tau^{*}(I,\theta,s)$ be a non-degenerate critical point of the
function

$\tau\mapsto\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)$

Then the function $L^{*}$ referred to in (7.10) is defined by

(7.12) $L^{*}(I,\theta,s)=\mathcal{L}(I,\theta-\omega(I)\tau^{*},s-\tau^{*}).$

An auxiliary function that will be referred to later is the reduced Melnikov
potential defined by

(7.13) $\mathcal{L}^{*}(I,\bar{\theta})=L^{*}(I,\bar{\theta},0)\textrm{ for
}\bar{\theta}=\theta-\omega(I)s.$

In our case the perturbation is not Hamiltonian, but we will see that,
nevertheless, the scattering map is symplectic up to $O(\varepsilon^{2})$, and
is given by

(7.14) $\begin{split}\mathcal{S}^{I}=&\frac{\partial
L^{*}_{\rho}}{\partial\theta}\\\ \mathcal{S}^{\theta}=&-\frac{\partial
L^{*}_{\rho}}{\partial I},\end{split}$

for some function $L^{*}_{\rho}$ that depends on the effect of the Hamiltonian
perturbation on the homoclinic orbits of the unperturbed system and also on
the dissipation. Our computation is similar to [DdlLS08].

######  Proposition 7.7.

The vector field $\mathcal{S}$ generating the scattering map
$\tilde{\sigma}_{\varepsilon}$ up to $O(\varepsilon^{2})$ is of the form

(7.15)
$\mathcal{S}(I,\theta,s)=\left({-}J\nabla_{(I,\theta)}L^{*}_{\rho}(I,\theta,s),s\right)$

for the function $L^{*}_{\rho}:V\subset\tilde{\Lambda}_{0}\to\mathbb{R}$
defined below. Let

(7.16)
$\begin{split}\mathcal{L}(I,\theta,s)=-\int_{-\infty}^{+\infty}&\left(H_{1}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.-H_{1}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\end{split}$

where $\omega(I)=\frac{\partial h_{0}}{\partial I}(I)$.
Let

(7.17) $A=\int_{-\infty}^{+\infty}p_{0}^{2}(t)dt.$

Let $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ be a non-degenerate critical point of
the function

$\tau\mapsto\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)+\rho(s-\tau)A.$

Let $L^{*}$ be defined by

$L^{*}(I,\theta,s)=\mathcal{L}(I,\theta-\omega(I)\tau^{*},s-\tau^{*}).$

Then the function $L^{*}_{\rho}$ is defined by

$L^{*}_{\rho}(I,\theta,s)=L^{*}(I,\theta,s)+\rho(s-\tau^{*})A.$

###### Proof.

We claim that

(7.18) $\begin{split}\mathcal{S}^{I}=&\frac{\partial
L^{*}_{\rho}}{\partial\theta}=\frac{\partial
L^{*}}{\partial\theta}-\rho\frac{\partial\tau^{*}}{\partial\theta}A\\\
\mathcal{S}^{\theta}=&-\frac{\partial L^{*}_{\rho}}{\partial
I}=-\frac{\partial L^{*}}{\partial I}+\rho\frac{\partial\tau^{*}}{\partial
I}A.\end{split}$

The first observation is that the non-degenerate zeroes of the function (6.6)
are the non-degenerate critical points of the function

(7.19)
$\tau\in\mathbb{R}\mapsto\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)+\rho(s-\tau)A,$

where $\mathcal{L}$ is given by (7.16) and $A$ is given by (7.17). To see
this, first note that by a change of variables $t-\tau\mapsto t$, we can
express $\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)$ as

(7.20)
$\begin{split}\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)=-\int_{-\infty}^{+\infty}&\left(H_{1}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.-H_{1}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\end{split}$

Differentiating (7.19) with respect to $\tau$ we obtain

$\begin{split}\int_{-\infty}^{+\infty}&\left(\\{h_{1},H_{1}\\}(p_{0}(\tau+t),q_{0}(\tau+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.-\\{h_{1},H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt-\rho
A,\end{split}$

so the non-degenerate zeroes of this function are the non-degenerate critical
points of (7.19).
If $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$ is a non-degenerate critical point of
the function (7.19), then by the chain rule it follows that

(7.21)
$\begin{split}0=&\frac{d}{d\tau}\left[\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)+\rho(s-\tau)A\right]_{\mid\tau=\tau^{*}}\\\
=&-\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\omega(I)-\frac{\partial\mathcal{L}}{\partial
s}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})-\rho A.\end{split}$

To compute $\frac{\partial
L^{*}_{\rho}}{\partial\theta}=\frac{\partial}{\partial\theta}(L^{*}(I,\theta,s)+\rho(s-\tau^{*})A)$
in (7.14) we use the chain rule and (7.21) to obtain

(7.22) $\begin{split}\frac{\partial
L^{*}_{\rho}}{\partial\theta}=&\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\left(1-\omega(I)\frac{\partial\tau^{*}}{\partial\theta}\right)\\\
&+\frac{\partial\mathcal{L}}{\partial
s}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\left(-\frac{\partial\tau^{*}}{\partial\theta}\right)-\rho\frac{\partial\tau^{*}}{\partial\theta}A\\\
=&\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*}).\end{split}$

Applying the latter formula to (7.16), using $\frac{\partial
H_{1}}{\partial\theta}=-\\{I,H_{1}\\}$, and making the change of variable
$t-\tau^{*}\mapsto t$ we obtain

(7.23) $\begin{split}\frac{\partial
L^{*}_{\rho}}{\partial\theta}=&\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)dt\\\
=&\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt.\end{split}$

This integral is the same as the integral (7.4) that appears in the formula
for the change of action by the scattering map up to $O(\varepsilon^{2})$.
Therefore, we conclude that:

$I_{\varepsilon}^{+}-I_{\varepsilon}^{-}=\varepsilon\frac{\partial
L^{*}}{\partial\theta}(I,\theta,s)+O(\varepsilon^{2})$

To compute $-\frac{\partial L^{*}_{\rho}}{\partial
I}=-\frac{\partial}{\partial I}(L^{*}(I,\theta,s)+\rho(s-\tau^{*})A)$ in
(7.14), we use the chain rule and (7.21) to obtain

(7.24) $\begin{split}-\frac{\partial L^{*}_{\rho}}{\partial
I}=&-\frac{\partial\mathcal{L}}{\partial
I}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\\\
&-\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\left(-\frac{\partial\omega}{\partial
I}\tau^{*}-\omega(I)\frac{\partial\tau^{*}}{\partial I}\right)\\\
&-\frac{\partial\mathcal{L}}{\partial
s}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\left(-\frac{\partial\tau^{*}}{\partial
I}\right)\\\ &+\rho\frac{\partial\tau^{*}}{\partial I}A\\\
=&-\frac{\partial\mathcal{L}}{\partial
I}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\\\
&+\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})\frac{\partial\omega}{\partial
I}\tau^{*}.\end{split}$

We express the two terms in (7.24) as integrals

(7.25) $\begin{split}&-\frac{\partial\mathcal{L}}{\partial
I}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})=\int_{-\infty}^{+\infty}\left(\\{\theta,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)dt\\\
&-\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)\left(\frac{\partial\omega}{\partial
I}(I)t\right)dt\end{split}$ (7.26)
$\begin{split}\frac{\partial\mathcal{L}}{\partial\theta}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})=&\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)dt\\\
\end{split}$

Above we used that $\frac{\partial H_{1}}{\partial\theta}=-\\{I,H_{1}\\}$ and
$\frac{\partial H_{1}}{\partial I}=\\{\theta,H_{1}\\}$.
Combining (7.25) and (7.26) in (7.24) we obtain

(7.27) $\begin{split}-\frac{\partial L^{*}_{\rho}}{\partial
I}(I,\theta,s)=&\int_{-\infty}^{+\infty}\left(\\{\theta,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)dt\\\
&-\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(t),q_{0}(t),I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t-\omega(I)\tau^{*},s+t-\tau^{*})\right)\left(\frac{\partial\omega}{\partial
I}(I)(t-\tau*)\right)dt\end{split}$

Making the change of variable $t-\tau^{*}\mapsto t$ and writing
$\frac{\partial\omega}{\partial I}(I)=\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I)$ we obtain

(7.28) $\begin{split}-\frac{\partial L^{*}}{\partial
I}(I,\theta,s)=&\int_{-\infty}^{+\infty}\left(\\{\theta,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\\\
&-\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I)t\right)dt\end{split}$

Since the integrals in (7.28) are computed in terms of the effect of the
perturbation on orbits of the unperturbed system, we have that $I$ in is
constant and equal to $I=I(\tilde{z}_{0})=I(\tilde{z}^{\pm}_{0})$, and
therefore $\left(\frac{\partial^{2}h_{0}}{\partial I^{2}}(I)\right)$ can be
taken outside of the second integral obtaining:

(7.29) $\begin{split}-\frac{\partial L^{*}_{\rho}}{\partial
I}(I,\theta,s)=&\int_{-\infty}^{+\infty}\left(\\{\theta,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{\theta,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)dt\\\
&-\int_{-\infty}^{+\infty}\left(\\{I,H_{1}\\}(p_{0}(\tau^{*}+t),q_{0}(\tau^{*}+t),I,\theta+\omega(I)t,s+t)\right.\\\
&\left.\qquad-\\{I,H_{1}\\}(0,0,I,\theta+\omega(I)t,s+t)\right)tdt\cdot\left(\frac{\partial^{2}h_{0}}{\partial
I^{2}}(I)\right)\end{split}$

This integral is the same as the integral (7.8) that appears in the formula
for the change of angle by the scattering map.

Therefore, we conclude that:

$\theta_{\varepsilon}^{+}-\theta_{\varepsilon}^{-}=-\varepsilon\frac{\partial
L_{\rho}^{*}}{\partial I}(I,\theta,s)+O(\varepsilon^{2})$

∎

Consider the mapping

$L^{*}_{\rho}(I,\theta,s)=\mathcal{L}(I,\theta-\omega(I)\tau^{*},s-\tau^{*})+\rho(s-\tau^{*})A.$

for $\tau^{*}=\tau^{*}(I,\theta,s;\rho)$. Since $\tau^{*}$ is a critical point
for

$\tau\mapsto\mathcal{L}(I,\theta-\omega(I)\tau,s-\tau)+\rho(s-\tau)A,$

then for every $t^{\prime}\in\mathbb{R}$, $\tau^{*}-t^{\prime}$ is a critical
point for

$\tau\mapsto\mathcal{L}(I,\theta-\omega(I)(\tau+t^{\prime}),s-(\tau+t^{\prime}))+\rho(s-(\tau+t^{\prime}))A.$

Then, denoting $Z=(I,\theta,s;\rho)$ and
$Z^{\prime}=(I,\theta-\omega(I)t^{\prime},s-t^{\prime};\rho)$, we have

$\tau^{*}(Z^{\prime})=\tau^{*}(Z)-t^{\prime}.$

Therefore

(7.30)
$\begin{split}L^{*}_{\rho}(I,\theta-\omega(I)t^{\prime},s-t^{\prime})=&\mathcal{L}(I,\theta-\omega(I)(\tau^{*}(Z^{\prime})+t^{\prime}),s-(\tau^{*}(Z^{\prime})+t^{\prime}))\\\
&+\rho(s-(\tau^{*}(Z^{\prime})+t^{\prime})A\\\
=&\mathcal{L}(I,\theta-\omega(I)(\tau^{*}(Z)-t^{\prime}+t^{\prime}),s-(\tau^{*}(Z)-t^{\prime}+t^{\prime}))\\\
&+\rho(s-(\tau^{*}(Z)-t^{\prime}+t^{\prime}))A\\\
=&L^{*}_{\rho}(I,\theta,s).\end{split}$

Making $t^{\prime}=s$ in (7.30) we obtain

$L^{*}_{\rho}(I,\theta,s)=L^{*}_{\rho}(I,\theta-\omega(I)s,0).$

This says that, while the function $L^{*}_{\rho}$ nominally depends on three
variables $(I,\theta,s)$, in fact it depends on the variable $I$ and the
linear combination $\theta-\omega(I)s$, and is therefore a function of two
independent variables $I$ and $\bar{\theta}=\theta-\omega(I)s$. Thus, we
define the reduced Melnikov potential by:

(7.31)
$\mathcal{L}^{*}_{\rho}(I,\bar{\theta})=\mathcal{L}^{*}_{\rho}(I,\bar{\theta},0)=\mathcal{L}(I,\bar{\theta}-\omega(I)\tau^{*},-\tau^{*})-\rho\tau^{*}A,\textrm{
for }\bar{\theta}=\theta-\omega(I)s.$

The reduced Melnikov potential allows to compute the scattering map associated
to the time-$2\pi$ map associated to a surface of section $\\{s=s^{*}\\}$;
more precisely, the trajectories of the scattering map are given by the
$\varepsilon$-time of the Hamiltonian $-\mathcal{L}^{*}_{\rho}$ up to order
$O(\varepsilon^{2})$, as we shall see below.

###  7.6. Growth of action by the scattering map

We reduce the dynamics of the flow $\tilde{\Phi}^{t}_{\varepsilon}$ to the
dynamics of the Poincaré first return map to the surface of section

$\Sigma=\\{(p,q,I,\theta,s)\,|s=s^{*}\\}$

for some choice of $s^{*}\in\mathbb{T}^{1}$.

The NHIM $\tilde{\Lambda}_{\varepsilon}$ for $\tilde{\Phi}^{t}_{\varepsilon}$
in the extend phase space yields the NHIM $\Lambda_{\varepsilon}$ for
$f_{\varepsilon}$ in $\Sigma$. In particular $\Lambda_{\varepsilon}$ is
invariant under $f_{\varepsilon}$.

The scattering map $\tilde{\sigma}_{\varepsilon}$, which is defined on the
domain $\tilde{U}\subseteq\tilde{\Lambda}_{\varepsilon}$, yields a scattering
map $\sigma_{\varepsilon}$ defined on the following domain in
$\Lambda_{\varepsilon}=\Lambda_{0}$:

$U=\\{(I,\bar{\theta})\,|\,(I,\theta,s^{*})\in\tilde{U}\textrm{ for
}\theta=\bar{\theta}+\omega(I)s^{*}\\},$

The scattering map $\sigma_{\varepsilon}$ is given in the variables
$(I,\bar{\theta})$ by (see [DS18]):

(7.32)
$\sigma_{\varepsilon}(I,\bar{\theta})=\sigma_{0}(I,\bar{\theta})-\varepsilon
J\nabla\mathcal{L}^{*}_{\rho}(I,\bar{\theta})+O(\varepsilon^{2}),$

where $\sigma_{0}=\textrm{Id}$. In particular, the scattering map
$\sigma_{\varepsilon}$ is symplectic up to $O(\varepsilon^{2})$.

By Theorem 3.11 in [GdlLMS20], whenever
$J\nabla\mathcal{L}^{*}_{\rho}(z_{0})\neq 0$ for some point
$z_{0}\in\Lambda_{0}$, there exists a $O(1)$-family of solutions
$\gamma_{z}(t)$ of the differential equation

$\dot{z}=-J\nabla\mathcal{L}^{*}_{\rho}(z)$

for $z$ in a $O(1)$-neighborhood of $z_{0}\in U\subseteq\Lambda_{0}$, and $t$
in some interval $[T_{1}(z),T_{2}(z)]\subset\mathbb{R}$ depending on $z$, such
that for each path $\gamma_{z}$, there is an orbit of the scattering map
$\sigma_{\varepsilon}$ that follows closely that path.

If, in addition, we have that
$\mathcal{S}^{I}(z_{0})=\frac{\partial\mathcal{L}^{*}_{\rho}}{\partial\bar{\theta}}(z_{0})>0$,
then the family of paths $\gamma_{z}$ can be chosen so that the corresponding
orbits of the scattering map $\sigma_{\varepsilon}$ along $\gamma_{z}$ have
the property that $I$ increases by $O(\varepsilon)$ for each application of
$\sigma_{\varepsilon}$.

Consequently, letting $z_{0}=(I_{0},\theta_{0})$ there exist
$\theta_{1}<\theta_{0}<\theta_{2}$, $I_{1}<I_{0}<I_{2}$, and a ‘strip’ of the
form

(7.33)
$S=\\{\gamma_{z}(t)\,|\,z=(I_{0},\theta)\,|\,\theta\in[\theta_{1},\theta_{2}],\,t\in[T_{1}(z),T_{2}(z)]\\}\subseteq
U$

with $\gamma_{z}(T_{1}(z))=I_{1}$ and $\gamma_{z}(T_{2}(z))=I_{2}$, such that
the following properties hold: There exist $c>0$, such that for every
$\delta=O(\varepsilon)>0$ and every path $\gamma_{z}(t)$ contained in $S$,
there exists an orbit $(z_{n})_{n=0,\ldots,N}$ of $\sigma_{\varepsilon}$ and
$0=t_{0}<t_{1}<\ldots<t_{N}=T$ with $t_{i}=\varepsilon i$ for all $i$, such
that

(7.34) $\begin{split}z_{i+1}=\sigma_{\varepsilon}(z_{i}),\\\
I(z_{i+1})-I(z_{i})>c\varepsilon,\textrm{ for }i=0,\ldots,N-1,\\\
d(z_{i},\gamma_{z}(t_{i}))<\delta,\textrm{ for }i=0,\ldots,N.\end{split}$

###  7.7. Scattering map in the case of vanishing and non-vanishing
perturbation

For both the vanishing and non-vanishing perturbations:

$\begin{split}H_{1}(p,q,I,\theta,s)=&(\cos(q)-1)(a_{00}+a_{10}\cos(\theta)+a_{01}\cos(s))\\\
H_{1}(p,q,I,\theta,s)=&\cos(q)(a_{00}+a_{10}\cos(\theta)+a_{01}\cos(s))\end{split}$

we have the same expression for the Melnikov potential

(7.35)
$\begin{split}\mathcal{L}(I,\theta,s)=&-\int_{-\infty}^{+\infty}(\cos(q_{0}(t))-1)(a_{00}+a_{10}\cos(\theta+It)+a_{01}\cos(s+t))dt\\\
=&-\int_{-\infty}^{+\infty}(\cos(\arctan
e^{2t})-1)(a_{00}+a_{10}\cos(\theta+It)+a_{01}\cos(s+t))dt.\end{split}$

Above we have used the parametrization (3.4) of the separatrix. Since
$\frac{p_{0}^{2}}{2}+(\cos q_{0}(t)-1)=0$, in the above integral we can
alternatively write
$\cos(q_{0}(t))-1=-\frac{p_{0}^{2}}{2}=-\frac{2}{\cosh^{2}(t)}$. It turns out
that (see [DG00, DS17])

(7.36)
$\begin{split}\mathcal{L}(I,\theta,s)=&A_{00}+A_{10}(I)\cos(\theta)+A_{01}(I)\cos(s),\textrm{
where }\\\ A_{00}=&4a_{00},\quad A_{10}=\frac{2\pi Ia_{10}}{\sinh(\frac{\pi
I}{2})},\quad A_{01}=\frac{2\pi a_{01}}{\sinh(\frac{\pi}{2})}.\end{split}$

In [DS17] the reduced Melnikov function $\mathcal{L}^{*}$ defined by (7.13),
which corresponds to the Hamiltonian perturbation only, is computed
explicitly. The level curves of $\mathcal{L}^{*}$ are shown in Fig. 5. One can
find explicitly regions of size $O(1)$ in $\Lambda_{\varepsilon}$ where
$\frac{\partial\mathcal{L}^{*}}{\partial\bar{\theta}}(I,\bar{\theta})>c_{1}>0$,
for some $c_{1}>0$.
In our case, when the system is also subject to the dissipative perturbation
$\mathcal{X}_{\lambda}$, the reduced Melnikov potential
$\mathcal{L}^{*}_{\rho}$, given by (7.31), is $O(\rho)$-close to the reduced
Melnikov potential $\mathcal{L}^{*}$ corresponding to the Hamiltonian
perturbation. This implies that, for $\rho$ sufficiently small, there exists a
region of $O(1)$ in $\Lambda_{\varepsilon}$ where
$\frac{\partial\mathcal{L}^{*}_{\rho}}{\partial\bar{\theta}}(I,\bar{\theta})>c_{2}>0$,
for some $0<c_{2}<c_{1}$. In the case when
$\rho=\frac{\bar{\rho}}{\log(1/\varepsilon)}$, it follows that there exists
$\varepsilon_{2}>0$ such that, for all $0<\varepsilon<\varepsilon_{2}$, we
have that
$\frac{\partial\mathcal{L}^{*}_{\rho}}{\partial\bar{\theta}}(I,\bar{\theta})>c_{2}>0$
on the aforementioned region. This region can be used to define a strip $S$ as
in (7.33), where the scattering map increases the action $I$ by $c\varepsilon$
at each step, as in (7.34), for some $0<c<c_{2}$.

Figure 5. Level sets of $\mathcal{L}^{*}$. (Credit A. Delshams and R.
Schaefer)

##  8\. Proof of Theorem 4.1

###  8.1. The case of vanishing perturbation

Choose $\omega_{*}$ such that $I_{1}<\omega_{*}<I_{2}$, where $I_{1}$, $I_{2}$
are as in Section 7.6. There is an invariant circle
$A_{\varepsilon}=\\{I=\omega_{*}\\}$ in $\Lambda_{\varepsilon}$, as defined in
Section 5.7, which is a global attractor for $f_{\varepsilon}$ on
$\Lambda_{\varepsilon}$. The circle $A_{\varepsilon}$ is a NHIM for
$f_{\varepsilon}$ restricted to $\Lambda_{\varepsilon}$, and has only stable
manifold $W^{\textrm{s}}_{\Lambda_{\varepsilon}}(A_{\varepsilon})$ which is
the whole manifold $\Lambda_{\varepsilon}$.
$W^{\textrm{s}}_{\Lambda_{\varepsilon}}(A_{\varepsilon})$ is foliated by
stable leaves

$W^{\textrm{s}}_{\Lambda_{\varepsilon}}(A_{\varepsilon})=\bigcup_{y\in
A_{\varepsilon}}W^{\textrm{s}}_{\Lambda_{\varepsilon}}(y)\textrm{ with
}y=(\omega_{*},\theta)\in A_{\varepsilon}.$

From (5.16) we have that each stable leaf is a slanted line

$W^{\textrm{s}}_{\Lambda_{\varepsilon}}(\omega_{*},\theta_{0})=\left\\{(I,\theta(I))\in\Lambda_{\varepsilon}\,|\,\theta(I)=\theta_{0}-\frac{1}{\lambda}(I-\omega_{*})\right\\}$

Since $\frac{1}{\lambda}\gg 1$, the slope of these lines (as a function of
$\theta$) is $-\lambda$, so the stable leaves are nearly horizontal lines. See
Fig. 6.

Figure 6. The attractor $A_{\varepsilon}$ inside the NHIM
$\Lambda_{\varepsilon}$

For $z\in W^{\textrm{s}}_{\Lambda_{\varepsilon}}(y)$, by the equivariance
property of the stable fibers we have $f_{\varepsilon}^{k}(z)\in
W^{\textrm{s}}_{\Lambda_{\varepsilon}}(f_{\varepsilon}^{k}(y))$, for all
$k>0$. Given some initial point $(I_{0},\theta_{0})$, let
$f_{\varepsilon}^{k}(I_{0},\theta_{0})=(I_{k},\theta_{k})$. From (5.15), we
deduce

(8.1)
$\begin{split}\theta_{k}=&\theta_{0}+\frac{1}{\lambda}(I_{0}-\omega_{*})(1+\ldots+e^{-2\pi(k-1)\lambda})(1-e^{-2\pi\lambda})+2\pi
k\omega_{*}\\\ =&\theta_{0}+\frac{1}{\lambda}(I_{0}-\omega_{*})(1-e^{-2\pi
k\lambda})+2\pi k\omega_{*}\\\ =&\theta_{0}+(I_{0}-\omega_{*})2\pi k+2\pi
k\omega_{*}+O(k^{2}\lambda)\\\ =&\theta_{0}+2\pi
kI_{0}(1+O(k\lambda)).\end{split}$

Recall that
$\lambda=\varepsilon\rho(\varepsilon)=\varepsilon\frac{\bar{\rho}}{\log(\frac{1}{\varepsilon})}$,
hence the relative error term in (8.1) is
$O(k\frac{\bar{\rho}\varepsilon}{\log(\frac{1}{\varepsilon})})$, so $k$
iterations of the inner map change the angle coordinate by approximately $2\pi
kI_{0}\,(\textrm{mod}\,2\pi)$ provided that
$k\frac{\bar{\rho}\varepsilon}{\log(\frac{1}{\varepsilon})}\ll 1$.

Consider the strip $S$ defined in (7.33), where the scattering map is
increasing $I$ by $O(\varepsilon)$ at each step. Provided that $|I_{1}-I_{2}|$
is suitably small (but independent of $\varepsilon$), there exists
$k_{\textrm{max}}>0$ such that, whenever $z\in S$ we have
$f_{\varepsilon}^{k}(z)\in S$ for some $k\leq k_{\textrm{max}}$. That is, each
point $z$ in the strip returns to the strip in a maximum of $k_{\textrm{max}}$
iterates. This implies that for a time $T=T_{0}\log(1/\varepsilon)$, with
$\varepsilon>0$ small, each point $z$ in the strip returns to the strip for at
least $\lfloor\frac{T_{0}\log(1/\varepsilon)}{k_{\textrm{max}}}\rfloor$ times.

One easily obtains from (5.15):

$I_{k}=(I_{0}-\omega_{*})e^{-2\pi k\lambda}+\omega_{*}.$

Consequently, if $z=(I_{0},\theta_{0})\in
W^{\textrm{s}}_{\Lambda_{\varepsilon}}(y)$, $y=(\omega_{*},\theta)\in
A_{\varepsilon}$, is a point at a $d_{I}$-distance $d_{0}\leq
d_{\textrm{max}}$ from $y$, where
$d_{\textrm{max}}=\max\\{|I_{1}-\omega_{*}|,|I_{2}-\omega_{*}|\\}$, then
$f_{\varepsilon}^{k}(z)\in
W^{\textrm{s}}_{\Lambda_{\varepsilon}}(f_{\varepsilon}^{k}(y))$ is at a
$d_{I}$-distance at most

$d_{0}\cdot e^{-2\pi\lambda k}=d_{0}\cdot
e^{-2\pi\frac{\varepsilon}{\log(1/\varepsilon)}\bar{\rho}k}$

from $A_{\varepsilon}$ after $k$-iterates. For points with initial $I_{0}$
above $\omega_{*}$ the $I$-coordinate decreases at each iterate, and for
points with initial $I$ below $\omega_{*}$ the $I$-coordinate increases at
each iterate. Therefore the loss in $I$ after $k$-iterates, for points with
initial $I_{0}>\omega_{*}$, is

$I_{0}-I_{k}=d_{0}\cdot(1-e^{-2\pi\lambda
k})=d_{0}\cdot(1-e^{-2\pi\frac{\varepsilon}{\log(1/\varepsilon)}\bar{\rho}k}).$

Hence the maximum loss in the action coordinate $I$ of a point $z$ after $k$
iterates, where $2\pi k\leq T_{0}\log(1/\varepsilon)$, is

$d_{\textrm{max}}\cdot(1-e^{-\varepsilon\bar{\rho}T_{0}}).$

On the other hand, for each $z\in S$ we can apply a scattering map
$\sigma_{\varepsilon}$ to $z$. The effect of the scattering map is an increase
in the action coordinate $I$ by $O(\varepsilon)$. We recall that there exists
$c>0$ such that

$I(\sigma_{\varepsilon}(z))-I(z)>c\varepsilon,\textrm{ for all }z\in S.$

Thus, starting with a point $z\in S$, applying the scattering map
$\sigma_{\varepsilon}$ to $z$, and then applying the inner map
$(f_{\varepsilon})_{\mid\Lambda_{\varepsilon}}$ for $k$ iterates with $2\pi
k\leq T_{0}\log(1/\varepsilon)$, until
$(f_{\varepsilon})^{k}(\sigma_{\varepsilon}(z))\in S$, we obtain a net growth
in $I$ that is at least

$c\varepsilon-d_{\textrm{max}}\cdot(1-e^{-\varepsilon\bar{\rho}T_{0}}).$

We want that the net growth is at least $c\varepsilon/2$, that is

$c\varepsilon-
d_{\textrm{max}}\cdot(1-e^{-\varepsilon\bar{\rho}T_{0}})>\frac{c\varepsilon}{2}.$

This is equivalent to

$e^{-\varepsilon\bar{\rho}T_{0}}>1-\frac{c\varepsilon}{2d_{\textrm{max}}}.$

Taking the logarithm of both sides we obtain

$T_{0}<\frac{\log\left(1-\frac{c\varepsilon}{2d_{\textrm{max}}}\right)}{-\varepsilon\bar{\rho}}.$

By L’Hopital rule

$\lim_{\varepsilon\to
0}\frac{\log\left(1-\frac{c\varepsilon}{2d_{\textrm{max}}}\right)}{-\varepsilon\bar{\rho}}=\frac{c}{2d_{\textrm{max}}\bar{\rho}}.$

This means that, in order to be able to achieve a growth in $I$ of at least
$c\varepsilon/2$ per step, for all $\varepsilon$ sufficiently small, we need
to choose $\bar{\rho}$ small enough so that

(8.2) $\bar{\rho}<\frac{c}{2d_{\textrm{max}}T_{0}}.$

With these choices, we obtain orbits of the iterated function system (IFS)
generated by $\\{f_{\varepsilon},\sigma_{\varepsilon}\\}$ of the form
$z_{n+1}=f_{\varepsilon}^{k(n)}\circ\sigma_{\varepsilon}(z_{n})$ with
$k(n)=O(\log(1/\varepsilon))$, such that $I$ increases by $c\varepsilon/2$
from $z_{n}$ to $z_{n+1}$. In
$O(\frac{1}{\varepsilon}\log\left(\frac{1}{\varepsilon}\right))$ steps, such
orbits increase $I$ by $O(1)$. In Section 8.3 we use these orbits of the IFS
to produce diffusing pseudo-orbits as claimed as in Theorem 4.1. ∎

######  Remark 8.1.

When a point $z\in\Lambda_{\varepsilon}$ is of action coordinate
$I_{0}<\omega_{*}$ below that of the attractor $A_{\varepsilon}$, applying the
inner dynamics $(f_{\varepsilon})_{\Lambda_{\varepsilon}}$ to $z$ moves the
point towards the attractor, and hence increases $I$. Thus, the effect of the
inner dynamics and the effect of the scattering map concur towards increasing
$I$. The situation reverses when the action coordinate of $z$ is above that of
the attractor, in which case the effect of the scattering map is opposed to
the effect of the inner dynamics.

###  8.2. The case of non-vanishing perturbation

The evolution of the $I$\- and $\theta$-variables under the inner dynamics on
$\tilde{\Lambda}_{\varepsilon}$ is given by:

(8.3)
$\begin{cases}\dot{I}_{\lambda}(t)=-\lambda({I_{\lambda}}(t)-\omega_{*})+\varepsilon
a_{10}\sin\theta_{\lambda}(t)\\\
\dot{\theta}_{\lambda}(t)=I_{\lambda}(t).\end{cases}$

Denote the general solution of the system (8.3) by
$z_{\lambda}(t)=(I_{\lambda}(t),\theta_{\lambda}(t))$.
Setting $\lambda=0$ yields:

(8.4) $\begin{cases}\dot{I}_{0}(t)=\varepsilon a_{10}\sin\theta_{0}(t)\\\
\dot{\theta}_{0}(t)=I_{0}(t),\end{cases}$

with general solution denoted $z_{0}(t)=(I_{0}(t),\theta_{0}(t))$; see Fig. 7.
Note that (8.4) is a Hamiltonian system with Hamiltonian (energy function)

(8.5) $K(I,\theta)=\frac{I^{2}}{2}+\varepsilon a_{10}(\cos\theta-1).$

Figure 7. The inner dynamics without dissipation.

The solutions $(I_{0}(t),\theta_{0}(t))$ of the system (8.4) satisfy

$K(I_{0}(t),\theta_{0}(t))=K(I_{0}(0),\theta_{0}(0))$

therefore, the application of the inner dynamics does not change the level
sets $\\{K=\textrm{const.}\\}$ in the case $\lambda=0$. On the other hand, the
variable $I$ may change by up to $O(\varepsilon^{1/2})$ by one application of
the inner dynamics (8.4). At the same time, the change in $I$ by one
application of the scattering map is $O(\varepsilon)\ll O(\varepsilon^{1/2})$.
Therefore, instead of comparing the effects on the action $I$ by the
scattering map and by the inner dynamics, as in Section 8.1, in this case we
want to compare the effects on the energy $K$ by the scattering map and by the
inner dynamics. The next lemma gives the change in the energy $K$ that we
obtain after one application of the scattering map:

######  Lemma 8.2.

Let
$\tilde{z}_{\varepsilon}^{+}=\sigma_{\varepsilon}(\tilde{z}_{\varepsilon}^{-})$,
where
$\tilde{z}_{\varepsilon}^{+}=(I^{+}_{\varepsilon},\theta^{+}_{\varepsilon},s)$
and
$\tilde{z}^{-}_{\varepsilon}=(I^{-}_{\varepsilon},\theta^{-}_{\varepsilon},s)$.
Then:

(8.6)
$\begin{split}K(\tilde{z}_{\varepsilon}^{+})-K(\tilde{z}^{-}_{\varepsilon})&=I_{0}(I^{+}_{\varepsilon}-I^{-}_{\varepsilon})+O(\varepsilon^{2})\\\
&=\varepsilon
I_{0}\frac{\partial\mathcal{L}^{*}_{\rho}}{\partial\theta}(I,\theta,s)+O(\varepsilon^{2})\\\
&=\varepsilon
a_{10}I_{0}\int^{\infty}_{-\infty}(\cos(q_{0}(\tau^{*}+t))-1)\sin(\theta+\omega(I)t)dt+O(\varepsilon^{2}),\end{split}$

where $I_{0}=I(\tilde{z}_{0})$.

###### Proof.

The proof of this lemma is a similar computation to the ones done to compute
the change in actions of the scattering map. In fact, using the formulas for
the change in actions and angles, Corollaries 7.3 and 7.5, particularly that
$I^{+}_{\varepsilon}-I^{-}_{\varepsilon}=O(\varepsilon)$ and
$\theta^{+}_{\varepsilon}-\theta^{-}_{\varepsilon}=O(\varepsilon)$, we have:

(8.7)
$\begin{split}K(\tilde{z}_{\varepsilon}^{+})-K(\tilde{z}^{-}_{\varepsilon})&=\frac{(I^{+}_{\varepsilon})^{2}}{2}-\frac{(I^{-}_{\varepsilon})^{2}}{2}+\varepsilon
a_{10}\left(\cos\theta^{+}_{\varepsilon}-\cos\theta^{-}_{\varepsilon}\right)\\\
&=\frac{(I^{+}_{\varepsilon}+I^{-}_{\varepsilon})}{2}(I^{+}_{\varepsilon}-I^{-}_{\varepsilon})+\varepsilon
a_{10}\left(\cos\theta^{+}_{\varepsilon}-\cos\theta^{-}_{\varepsilon}\right)\\\
&=(I_{0}+O(\varepsilon))(I^{+}_{\varepsilon}-I^{-}_{\varepsilon})+O(\varepsilon^{2})\\\
&=I_{0}(I^{+}_{\varepsilon}-I^{-}_{\varepsilon})+O(\varepsilon^{2}).\end{split}$

Applying Proposition 7.7 yields the desired result. ∎

The next lemma compares the actions and angles of solutions of systems (8.3)
and (8.4).

######  Lemma 8.3.

Let $(I_{\lambda}(t),\theta_{\lambda}(t))$ a solution of system (8.3) with
initial condition $(I_{0},\theta_{0})$ and $(I_{0}(t),\theta_{0}(t))$ the
solution of system (8.4) with the same initial condition. Then, there exists
$d^{\prime}>d_{\textrm{max}}>0$ such that, for $|\varepsilon|$ small enough,
and for $0\leq t\leq T_{0}\log\left(\frac{1}{\varepsilon}\right)$ we have:

(8.8) $\begin{split}|I_{\lambda}(t)-I_{0}(t)|&\leq d^{\prime}(1-e^{-\lambda
t})\\\ |\theta_{\lambda}(t)-\theta_{0}(t)|&\leq
d^{\prime}\lambda\frac{t^{2}}{2}.\end{split}$

###### Proof.

Calling

$u(t)=I_{\lambda}(t)-I_{0}(t),\quad\alpha(t)=\theta_{\lambda}(t)-\theta_{0}(t),$

one can easily see that:

(8.9) $\begin{split}\dot{u}&=\lambda(I_{0}(t)-\omega_{*}-u)+\varepsilon
a_{10}\left(\sin(\theta_{0}(t)+\alpha)-\sin\theta_{0}(t)\right)\\\
\dot{\alpha}&=u.\end{split}$

Therefore:

(8.10)
$\begin{split}u(t)&=\int_{0}^{t}e^{-\lambda(t-s)}\left[\lambda(I_{0}(s)-\omega_{*})+\varepsilon
a_{10}\left(\sin(\theta_{0}(s)+\alpha(s))-\sin\theta_{0}(s)\right)\right]ds\\\
\alpha(t)&=\int_{0}^{t}u(s)ds.\end{split}$

For the first equation in (8.10) we have used the method of variation of
constants.
We will bound $u$ in four steps:

  1. (1)

First we will use the first equation of (8.10) to obtain a weak bound for $u$;

  2. (2)

Second, we will use the obtained bound on $u$ in the second equation of (8.10)
to obtain a bound for $\alpha$;

  3. (3)

Third, we will use the obtained bound on $\alpha$ in the first equation of
(8.10) to obtain a sharper bound for $u$;

  4. (4)

Finally, using the sharper bound on $u$ in the second equation of (8.10) to
obtain a new bound for $\alpha$.

We will use that the solution $I_{0}(t)$ is a bounded function, in fact
$|I_{0}(t)-\omega_{*}|\leq d_{\textrm{max}}$, where we recall
$d_{\textrm{max}}=\max\\{|I_{1}-\omega_{*}|,|I_{2}-\omega_{*}|\\}$. From the
first equation of (8.10) we obtain:

(8.11) $\begin{split}|u(t)|&\leq\int_{0}^{t}e^{-\lambda(t-s)}\left(\lambda
d_{\textrm{max}}+2\varepsilon a_{10}\right)ds\\\
&\leq\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)(1-e^{-\lambda
t}).\end{split}$

Using this bound in the second equation of (8.10) and that $0\leq
1-e^{-\lambda t}\leq\lambda t$, we obtain:

(8.12)
$\begin{split}|\alpha(t)|&\leq\int_{0}^{t}\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)\lambda
s\,ds\\\
&=\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)\lambda\frac{t^{2}}{2}.\end{split}$

Finally, we will use the obtained bound on $\alpha$, and the fact that
$|\sin(\theta_{0}(t)+\alpha)-\sin\theta_{0}(t)|\leq|\alpha|$ in the first
equation of (8.10) to obtain:

(8.13) $\begin{split}|u(t)|&\leq d_{\textrm{max}}(1-e^{-\lambda
t})+\varepsilon a_{10}\int_{0}^{t}e^{-\lambda(t-s)}|\alpha(s)|ds\\\ &\leq
d_{\textrm{max}}(1-e^{-\lambda t})+\varepsilon
a_{10}\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)\frac{\lambda
t^{2}}{2}\int_{0}^{t}e^{-\lambda(t-s)}ds\\\
&\leq\left[d_{\textrm{max}}+\varepsilon
a_{10}\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)\frac{t^{2}}{2}\right](1-e^{-\lambda
t}).\end{split}$

Observe that, as $0\leq t\leq T=T_{0}\log\left(\frac{1}{\varepsilon}\right)$,
and
$\lambda=\varepsilon\frac{\bar{\rho}}{\log\left(\frac{1}{\varepsilon}\right)}$
we have that:

$\varepsilon
a_{10}\left(d_{\textrm{max}}+2\frac{\varepsilon}{\lambda}a_{10}\right)\frac{t^{2}}{2}\leq\varepsilon\log^{2}\left(\frac{1}{\varepsilon}\right)\left(a_{10}d_{\textrm{max}}\frac{T_{0}^{2}}{2}\right)+\varepsilon\log^{3}\left(\frac{1}{\varepsilon}\right)\left(\frac{a_{10}^{2}}{\bar{\rho}}T_{0}^{2}\right)$

which is arbitrarily small if $\varepsilon$ is small (indeed,
$\varepsilon\log^{p}\left(\frac{1}{\varepsilon}\right)\ll\varepsilon^{\nu}$
for $p>0$ and $\nu\in(0,1)$). Therefore there exist
$d^{\prime}>d_{\textrm{max}}$ such that:

(8.14) $|u(t)|\leq d^{\prime}(1-e^{-\lambda t})$

and consequently we have:

$|I_{\lambda}(t)-I_{0}(t)|\leq d^{\prime}(1-e^{-\lambda t})$

for $0\leq t\leq T_{0}\log\left(\frac{1}{\varepsilon}\right)$. Note that
$d^{\prime}$ can be chosen arbitrarily close to $d_{\textrm{max}}$ provided
that $\varepsilon$ is small enough.
Using this new bound in the second equation of (8.10) and that $0\leq
1-e^{-\lambda t}\leq\lambda t$, we obtain:

(8.15) $\begin{split}|\theta_{\lambda}(t)-\theta_{0}(t)|&\leq
d^{\prime}\lambda\frac{t^{2}}{2}.\end{split}$

∎

The next lemma estimates the change in the energy $K$ by the inner dynamics
over time intervals of order
$O\left(\log\left(\frac{1}{\varepsilon}\right)\right)$.

######  Lemma 8.4.

There exists $d^{\prime\prime}>0$, such that for $|\varepsilon|$ small enough,
and for $0\leq t\leq T_{0}\log\left(\frac{1}{\varepsilon}\right)$ we have:

(8.16) $|K(z_{\lambda}(t))-K(z_{0}(t))|\leq d^{\prime\prime}(1-e^{-\lambda
t}).$

###### Proof.

From (8.5) and Lemma 8.3 we obtain

(8.17)
$\begin{split}|K(z_{\lambda}(t))-K(z_{0}(t))|&=\Big{|}\frac{1}{2}(I^{2}_{\lambda}(t)-I^{2}_{0}(t))+\varepsilon
a_{10}(\cos(\theta_{\lambda}(t))-\cos(\theta_{0}(t)))\Big{|}\\\
&\leq\Big{|}\frac{1}{2}(I_{\lambda}(t)-I_{0}(t))(I_{\lambda}(t)+I_{0}(t))\Big{|}+\varepsilon
a_{10}|\theta_{\lambda}(t)-\theta_{0}(t)|\\\
&\leq\Big{|}\frac{1}{2}(I_{\lambda}(t)-I_{0}(t))(2I_{0}(t)+(I_{\lambda}(t)-I_{0}(t)))\Big{|}+\varepsilon
a_{10}|\theta_{\lambda}(t)-\theta_{0}(t)|\\\
&\leq\frac{1}{2}d^{\prime}(1-e^{-\lambda
t})\left(2(d_{\textrm{max}}+\omega_{*})+d^{\prime}(1-e^{-\lambda
t})\right)+\varepsilon a_{10}d^{\prime}\lambda\frac{t^{2}}{2}\\\ &\leq
d^{\prime}(d_{\textrm{max}}+\omega_{*})(1-e^{-\lambda
t})+\frac{(d^{\prime})^{2}}{2}(1-e^{-\lambda t})^{2}+\varepsilon
a_{10}d^{\prime}\lambda\frac{t^{2}}{2}.\end{split}$

Taking into account that
$\lambda=\frac{\varepsilon\bar{\rho}}{\log\left(\frac{\varepsilon}{2}\right)}$,
for $|\varepsilon|$ sufficiently small we have

$\begin{split}(1-e^{-\lambda t})^{2}&\leq(1-e^{-\lambda t}),\\\
\varepsilon\lambda\frac{t^{2}}{2}&\leq(1-e^{-\lambda
t}),\end{split}\quad\text{for}\quad 0\leq t\leq
T_{0}\log{(\frac{1}{\varepsilon})}.$

Therefore, using (8.17) we conclude that there exists
$d^{\prime\prime}>d^{\prime}$, such that

$|K(z_{\lambda}(t))-K(z_{0}(t))|\leq d^{\prime\prime}(1-e^{-\lambda t}).$

We note that $d^{\prime\prime}$ can be chosen arbitrarily close to
$d^{\prime}(d_{\textrm{max}}+\omega_{*})+\frac{1}{2}(d^{\prime})^{2}$ provided
that $|\varepsilon|$ is sufficiently small. ∎

We now continue with the proof of Theorem 4.1. By Lemma 8.2, given that
$0<I_{1}<I_{2}$, the effect of the scattering map is an increase in the energy
$K$ by $O(\varepsilon)$. Let $S$ be a strip as in (7.33) and $c>0$ such that

$K(\sigma_{\varepsilon}(z))-K(z)>c\varepsilon,\textrm{ for all }z\in S.$

From Lemma 8.4, the maximum loss in $K$ by the inner flow over a time $0\leq
t\leq T_{0}\log\left(\frac{1}{\varepsilon}\right)$ is

$d^{\prime\prime}(1-e^{-\varepsilon\bar{\rho}T_{0}}).$

Switching from the flow to the time-$2\pi$ map, it follows that the maximum
loss in $K$ by the inner map after $k$ iterates of $f_{\varepsilon}$, with
$2\pi k\leq T_{0}\log(1/\varepsilon)$, is also

$d^{\prime\prime}(1-e^{-\varepsilon\bar{\rho}T_{0}}).$

Note that the level sets of $K$ are $O(\varepsilon^{1/2})$-close to level sets
of $I$. Therefore we can choose $0<I_{1}<I_{2}$ such that the growth in $K$ by
repeated applications of the scattering map corresponds to a change in $I$
from below $I_{1}$ to above $I_{2}$. Provided that $|I_{1}-I_{2}|$ is suitably
small, there exists $k_{\textrm{max}}$ such that whenever $z\in S$ we have
$f_{0}^{k}(z)\in S$ for some $k\leq k_{\textrm{max}}$. By Lemma 8.8, for
$0\leq t\leq T_{0}\log(1/\varepsilon)$,

$|\theta_{\lambda}(t)-\theta_{0}(t)|<d^{\prime}\lambda\frac{t^{2}}{2}\leq
d^{\prime}\varepsilon\log\left(\frac{1}{\varepsilon}\right)\frac{\bar{\rho}T_{0}^{2}}{2},$

so, for $\varepsilon$ sufficiently small, $\theta_{\lambda}(t)$ is
$O(\varepsilon^{\nu})$-close to $\theta_{0}(t)$, for some $\nu\in(0,1)$. This
implies that each point $z$ in the strip returns to the strip in a maximum of
$k_{\textrm{max}}$ iterates of $f_{\varepsilon}$. Thus, starting with a point
$z\in S$, applying the scattering map $\sigma_{\varepsilon}$ to $z$, and then
applying the inner map $(f_{\varepsilon})_{\mid\Lambda_{\varepsilon}}$ for $k$
times, where $2\pi k\leq T_{0}\log(1/\varepsilon)$, until
$(f_{\varepsilon})^{k}(\sigma_{\varepsilon}(z))\in S$, we obtain a net growth
in $K$ that is at least

$c\varepsilon-d^{\prime\prime}(1-e^{-\varepsilon\bar{\rho}T_{0}}).$

We require that this net growth in $K$ is at least $c\varepsilon/2$, that is

$c\varepsilon-d^{\prime\prime}(1-e^{-\varepsilon\bar{\rho}T_{0}})>\frac{c}{2}\varepsilon.$

Similarly to the proof in Section 8.1, in order to be able to achieve a growth
in $K$ of at least $c\varepsilon/2$ per step, for all $\varepsilon$
sufficiently small, we need to choose $\bar{\rho}$ small enough so that

$\bar{\rho}<\frac{c}{2d^{\prime\prime}T_{0}}.$

We obtain orbits of the iterated function system (IFS) generated by
$\\{f_{\varepsilon},\sigma_{\varepsilon}\\}$, of the form
$z_{n+1}=f_{\varepsilon}^{k(n)}\circ\sigma_{\varepsilon}(z_{n})$ with
$k(n)=O(\log(1/\varepsilon))$, such that $K$ increases by $c\varepsilon/2$
from $z_{n}$ to $z_{n+1}$. In
$O(\frac{1}{\varepsilon}\log\left(\frac{1}{\varepsilon}\right))$ steps, such
orbits increase $K$, as well as $I$, by $O(1)$. By our choice of
$I_{1},I_{2}$, these orbits go from below $I_{1}$ to above $I_{2}$. In Section
8.3 we use these orbits of the IFS to produce diffusing pseudo-orbits as
claimed in Theorem 4.1.

###  8.3. Existence of diffusing pseudo-orbits

In Sections 8.1 and 8.2 we obtained diffusing orbits of the iterated function
system (IFS) generated by $\\{f_{\varepsilon},\sigma_{\varepsilon}\\}$
consisting of orbit segments of the form
$z_{n+1}=f_{\varepsilon}^{k(n)}\circ\sigma_{\varepsilon}(z_{n})$ in
$\Lambda_{\varepsilon}$, $n=1,\ldots,m-1$, where $k(n)=O(\log(1/\varepsilon))$
and $m=O(1/\varepsilon)$, such that $I(z_{0})<I_{1}$ and $I(z_{m})>I_{2}$. We
can rearrange these orbits into orbit segments of the form
$x_{i+1}=f^{m_{i}}\circ\sigma_{\varepsilon}\circ f^{n_{i}}(x_{i})$, with
$m_{i},n_{i}=O(\log(1/\varepsilon))$, for $i=0,\ldots,m-1$, such that
$I(x_{0})<I_{1}$ and $I(x_{m})>I_{2}$. Each such orbit segment can be
approximated up to $O(\varepsilon)$ by a true orbit of the Poincaré map of the
form $y^{i}_{\textrm{end}}=f^{k_{i}}(y^{i}_{\textrm{start}})$,
$k_{i}=O(\log(1/\varepsilon))$, with
$d(y^{i}_{\textrm{end}},y^{i+1}_{\textrm{start}})<O(\varepsilon)$; see Remark
7.1. The diffusion time is
$O(\frac{1}{\varepsilon}\log(\frac{1}{\varepsilon}))$. Finally, each such
orbit of $f_{\varepsilon}$ gives rise to an orbit segment $z^{i}(t)$,
$t\in[t_{i},t_{i+1}]$, of the flow, satisfying the requirements of Theorem
4.1.

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