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unbounded energy growth, Arnold diffusion, slow time dependence
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\title{Unbounded energy growth in Hamiltonian systems
with a slowly varying parameter}
\author{Vassili Gelfreich$^1$\thanks{The authors thank the Royal Society which provided support
for D.T. to visit V.G. at the University of Warwick, where a
substantial part of this work had been completed. 
D.T. also acknowledges the support by grants ISF 926/04, 273/07 and MNTI-RFBR 06-01-72023.}
\and Dmitry Turaev$^{2}$\\[12pt]
\small
$^1$ Mathematics Institute, University of Warwick,\\ \small
Coventry CV4 7AL, United Kingdom\\ \small
Fax: +44 246 524 182
\\ \small
E-mail: {\tt v.gelfreich@warwick.ac.uk}\\[6pt] \small
$^{2}$
Department of Mathematics,
Ben Gurion University of the Negev\\ \small
Be'er Sheva 84105, Israel\\ \small
E-mail: {\tt turaev@math.bgu.ac.il}
}
\date{September 20, 2007}
\begin{document}

\maketitle

\begin{abstract}
We study Hamiltonian systems which depend slowly
on time. We show that if the corresponding frozen system
has a uniformly hyperbolic invariant set with chaotic behaviour,
then the full system has orbits with unbounded energy growth 
(under very mild genericity assumptions). We also provide formulas 
for the calculation of the rate of the fastest energy growth.
We apply our general theory to non-autonomous perturbations
of geodesic flows and Hamiltonian systems with billiard-like
and homogeneous potentials. In these examples, we show the existence
of orbits with the rates of energy growth that range, depending on the
type of perturbation, from linear to exponential in time.
Our theory also applies to non-Hamiltonian systems with a first integral.
\end{abstract}

\newpage

\section{Setting the problem}
Consider a Hamiltonian system
\begin{equation}\label{ham00}
H=H(p,q,\varepsilon t)
\end{equation}
with $\varepsilon$ small. It is natural to compare its dynamics
with the frozen system
\begin{equation}\label{frz0}
H=H(p,q,\nu),
\end{equation}
where $\nu$ is now treated as a constant parameter.
The Hamiltonian $H$ is a first integral of the frozen system
but not of the non-autonomous system described by (\ref{ham00}).
Let $(p(t),q(t))$ be a trajectory of (\ref{ham00})
and $H(t)\equiv H(p(t),q(t),\varepsilon t)$. Differentiating
with respect to time and using the Hamilton equations we see
that the rate of energy change is small
\[
\dot H(t)=\varepsilon\frac{\partial H}{\partial \nu}(p(t),q(t),\varepsilon t)\,.
\]
Adiabatic invariants play an important role in description of dynamics
for this class of systems \cite{Kasuga1961}.
It is also known that if the frozen system is integrable, then under certain 
assumptions the energy $H$ may oscillate at a bounded distance 
from its initial value for a very long time.

On the other hand, in the case of chaotic dynamics in the frozen system the
behaviour of the energy may be drastically different. Indeed, in the
mid nineties Mather discovered that adding a time-periodic perturbation to the Hamiltonian
of a uniformly hyperbolic flow creates orbits with an unbounded energy growth.
Moreover, the energy on the Mather's trajectories tends to infinity linearly, 
i.e., it changes at a much faster rate than
it could do if the unperturbed system were integrable.
This result and its generalisations were studied in \cite{BT99,T04,Pif,DLS00,DLS06,K03},
where the reader can e.g. find more detailed discussion on the history of the problem.
While some papers treat the problem of estimating the energy growth 
in nonautonomous Hamiltonian systems mostly as a simplified model for Arnold
diffusion, we think it has an independent interest and a wide range of applications
(see e.g. \cite{LRA,LR} where billiards with time-dependent boundaries were discussed
in connection with Fermi acceleration). 

In our paper we establish that the existence of orbits of unbounded
and rapid energy growth is a very
general phenomenon, typical for practically arbitrary slow non-autonomous
perturbation of a Hamiltonian system with chaotic behaviour. The construction
we employ is different from most of those used by the previous authors
and is applied to a wider class of systems.
Thus, we do not use variational constructions,
nor KAM theory, we do not build heteroclinic chains, and we do not assume any kind of periodicity
for the time-dependence. Finally, we provide formulas
for calculating the energy growth rates, and provide examples
for which the growth rates vary from linear to exponential one.

In short, the acceleration mechanism we discuss here is as follows. First,
by saying that the frozen system has a chaotic behaviour, we mean that there
exists $h^*$ such that the
frozen system has a uniformly-hyperbolic, compact, transitive, invariant
set $\Lambda_{h\nu}$ in every energy level $H=h\geq h^*$ for all~$\nu\geq 0$.
In every given energy level, the set $\Lambda_{h\nu}$ is in the closure of a set of
hyperbolic periodic orbits each of which has an orbit of a transverse heteroclinic
connection to any of the others. This means that orbits of (\ref{frz0}) may stay
close to any of the periodic orbits for an arbitrary number of periods, then
come close to another periodic orbit and stay there, and so on. Recall also that
periodic orbits of (\ref{frz0}) form families parametrized by the value of $H$
and by $\nu$. By a standard averaging procedure (see e.g.~\cite{Abook,LN94}), one establishes that
for the orbits of the original system (\ref{ham00}) close to a periodic family of
(\ref{frz0}) there exists an adiabatic invariant -- a function $J(h,\nu)$ such that
$J(H(t),\varepsilon t)$ stays almost constant for a very long time $\gg \varepsilon^{-1}$.
Now we take two periodic families, $L_a$ and $L_b$, of the frozen system.
For the orbits that stay near
$L_a$ the value of $J_a(H(t),\varepsilon t)$ will remain almost constant while
$J_b(H(t),\varepsilon t)$ may grow or decrease, and for the orbits that stay near
$L_b$ we will have $J_b$ nearly constant while $J_a$ changes. In this paper we show
that under some natural conditions one can arrange jumps between $L_a$ and $L_b$ in
such a way that one of the functions $J_{a}$ or $J_{b}$ will always grow while the second one
rests. Then the sum $J_a(H(t),\varepsilon t)+J_b(H(t),\varepsilon t)$ may tend to
infinity. Note that both $J_a$ and $J_b$ are monotonically increasing functions of $h$,
therefore the unbounded growth of $J_a(H(t),\varepsilon t)+J_b(H(t),\varepsilon t)$
implies, typically, the unbounded growth of $H(t)$.

These considerations do not depend on how the Hamiltonian depends on $\varepsilon t$.
Indeed, we treat the cases of periodic and non-periodic perturbations simultaneously,
and the results hold true for periodic, quasiperiodic and other settings.
We note that the invariant set $\Lambda_{h\nu}$ with the desired properties
exists,  provided the frozen system has a family of hyperbolic periodic orbits,
each one with a transversal homoclinic trajectory.

Technically, for the most part of this paper we do not use the Hamiltonian structure
of the system. Therefore, for a greater generality, instead of (\ref{ham00}) we consider
\begin{equation}\label{aham1}
\dot x=G(x,\varepsilon t)\,,
\end{equation}
where $x\in{\mathbb R}^m$. The corresponding frozen system has the form
\begin{equation}\label{aham0}
\dot x=G(x,\nu)\,.
\end{equation}
We assume that a function $H(x,\nu)$ is an integral, i.e.
\begin{equation}\label{bham0}
\frac{\partial H}{\partial x}(x,\nu)\cdot G(x,\nu)\equiv 0.
\end{equation}
We will continue calling $H$ the energy.

Assume that for all $\nu\geq 0$ in every energy level $H=h\geq h^*$
the frozen system (\ref{aham0}) has a heteroclinic cycle
composed of a pair of hyperbolic periodic orbits
$L_a$ and $L_b$, and a pair of transverse heteroclinic orbits,
$\Gamma_{ab}$ and $\Gamma_{ba}$, the first corresponds to a transverse
intersection of $W^u(L_a)$ and $W^s(L_b)$, while
the second one corresponds to a transverse intersection of $W^u(L_b)$ and $W^s(L_a)$.

We note that the set of all orbits that stay in a small neighbourhood of the heteroclinic
cycle in a given energy level is a locally maximal, uniformly hyperbolic,
compact, transitive, invariant set \cite{Shi67}. We denote this set by $\Lambda$.
It is well known that a hyperbolic periodic orbit
continues in a unique way as a smooth function of parameters $h$ and $\nu$.
The same holds true for a transverse heteroclinic.

So, $L_a$ and $L_b$, as well as the
transverse heteroclinic orbits $\Gamma_{ab}$ and $\Gamma_{ba}$, depend on $h$ and
$\nu$ in a smooth way.
Let $T_c(h,\nu)$ (where $c=a$ or $c=b$) be the period of the orbit $L_c(h,\nu):
x=x_c(t;h,\nu)$. Let us consider the average of $H'_\nu$ over the periodic orbit $L_c$:
\begin{equation}\label{vchn} v_c(h,\nu)=\frac{1}{T_c}\int_0^{T_c}
\left.\frac{\partial H}{\partial\nu}(x,\nu)\right|_{x=x_c(t;h,\nu)} dt\,.
\end{equation}

\begin{Theorem}\label{Thm1}
Assume that the differential equation
\begin{equation}\label{acc}
\frac{dh}{d\nu}=\max\{v_a(h,\nu),v_b(h,\nu)\}
\end{equation}
has a solution $h(\nu)$ such that $h(\nu)\geq h^*$ for all $\nu\geq 0$ and
$h(\nu)\rightarrow+\infty$ as $\nu\to+\infty$. Then given any $h_1\geq h_0\equiv h(0)$ there exists $t_1>0$
such that for every sufficiently small $\varepsilon$
there is a solution $x(t)$ of system {\rm (\ref{aham1})\/}
such that
$H(x(0),0)= h_0$ and $H(x(t),\varepsilon t)=h_1$ at a time $t\leq t_1/\varepsilon$.
\end{Theorem}

In Section~\ref{adi} we show that in the Hamiltonian setup
(i.e. in the case where system (\ref{aham1}) is Hamiltonian)
equation (\ref{acc}) possesses tending to infinity solutions
under very mild assumptions. Thus, for the case of periodic or quasiperiodic dependence
of the Hamiltonian on $\varepsilon t$ we show (Proposition \ref{pcaseq})
that the boundedness of solutions of (\ref{acc}) is a codimension infinity event.
Simple sufficient conditions for the unbounded energy growth are given 
for special classes of Hamiltonian systems in Section~\ref{examples}.

Theorem~\ref{Thm1} does not directly imply that system~(\ref{aham1})
has an orbit with unbounded energy. In order to prove the existence of
such an orbit we need an information on the behaviour of the system near
the hyperbolic set $\Lambda_{h\nu}$ at $\nu$ and $h$ tending to infinity,
i.e., for a non-compact set of values of $h$ and $\nu$. Therefore,
certain uniformity assumptions are necessary. As they are
quite technical, we postpone their precise
statements till Sections~\ref{horseshoe} and~\ref{average}.
In short, in condition [UA1] we require that the set $\Lambda_{h\nu}$ 
has a cross-section and that the right-hand sides of the
corresponding Poincar\'e map, when written in the so-called ``cross-form'' \cite{book}, are
uniformly bounded, along with their first derivatives, for all sufficiently large $h$
and $\nu$ and all small $\varepsilon$. In condition [UA2] we require
a certain uniformity for the times
of the first return to the cross-section and for the change in the energy 
between two consecutive returns. In Section~\ref{examples} we
check these uniformity assumptions for several classes of examples.

For a greater generality, we allow for the right-hand side of system (\ref{aham1})
under consideration to depend explicitly on $\varepsilon$, i.e., the system takes the form
\begin{equation}\label{aham2}
\dot x=G(x,\varepsilon t;\varepsilon)\,,
\end{equation}
with $G$ depending on $\varepsilon$ continuously.
Thus, the frozen system (\ref{aham0}) and the integral $H$ will also depend on $\varepsilon$,
as well as the functions $v_{a,b}$ in (\ref{vchn}).

\begin{Theorem}\label{Thm2}
Assume the uniformity assumptions\/ {\rm [UA1]} and~{\rm [UA2]}
hold true. Consider a differential equation
\begin{equation}\label{accb}
\frac{dh}{d\nu}=\max\{v_a(h,\nu),v_b(h,\nu)\} - \delta \beta(h,\nu)
\end{equation}
where the smooth function $\beta$ is defined by condition\/~{\rm  (\ref{betadef})}.
Suppose there exists $\delta>0$ such that equation\/ {\rm (\ref{accb})}
has, for all small $\varepsilon$, a solution $h_{\delta}(\nu)$ that satisfies $h_{\delta}(\nu)\geq h^*$
for all $\nu$ and tends to $+\infty$
as $\nu\to+\infty$.  Then for all sufficiently small $\varepsilon$
system\/ {\rm (\ref{aham2})} has an orbit $x(t)$ for which
$H(x(t),\varepsilon t;\varepsilon)\rightarrow+\infty$ as $t\rightarrow=\infty$.
\end{Theorem}

Theorem \ref{Thm2} above is an immediate corollary of the following comparison theorem.

\begin{Theorem}\label{Thm2a}
Assume the uniformity assumptions\/ {\rm [UA1]} and~{\rm [UA2]}
hold true, let $\delta>0$ and denote as $h_\delta$  a
solution of differential equation {\rm (\ref{accb})}.
Then for all sufficiently small $\varepsilon$
system\/~{\rm (\ref{aham2})}
has a solution $x(t)$ such that $H(x(0),0)=h_\delta(0)$
and
\[
H(x(t),\varepsilon t)\geq h_{\delta}(\varepsilon t)
\]
for all $t\ge0$.
\end{Theorem}

The proof of Theorem~\ref{Thm2a} is given in Sections~\ref{horseshoe} and~\ref{average}.
Note that the uniformity assumptions are automatically fulfilled
for any compact set of $h$ and $\nu$, hence Theorem~\ref{Thm1} is indeed extracted
from Theorem~\ref{Thm2a} by modifying, if necessary, the equations
outside a neighbourhood of the region $H(x,\nu)\in[h_0,h_1]$ and $\nu\in[0,t_1]$.

Note also that although the function $\beta$ in Theorem \ref{Thm2}
is defined in technical terms,
in the examples which we consider in Section~\ref{examples} this function is asymptotically (as $h\rightarrow+\infty$)
of the same order as the functions $v_{a,b}$. Therefore, the contribution
of the second term of equation (\ref{accb}) is not very important
(recall that $\delta$ in (\ref{accb}) can be taken arbitrarily small).
In other words, the energy growth rate is, essentially, given by the solution of equation (\ref{acc}).

\subsection{Scheme of the proof}

Let us now describe the scheme of the proof of Theorem~\ref{Thm2a}. Consider the family
$L_c: x=x_c(t;h,\nu)$ of the hyperbolic periodic orbits of the frozen system
(\ref{aham0}) (as usual, $c=a$ or $c=b$). This is a three-dimensional invariant
manifold of system (\ref{aham0}). Importantly, this manifold is normally-hyperbolic,
because of the hyperbolicity of the periodic orbits which comprise it. Therefore,
it persists for all small $\varepsilon$ \cite{Fen71} (the set of values of $(h,\nu)$ under consideration
is not compact and therefore we also need our uniformity assumptions to make such claim).
Thus, system (\ref{aham2}) has an invariant manifold
$x=\tilde x_c(\tau; h,\nu; \varepsilon)$ such that $\tilde x_c=x_c$ at
$\varepsilon=0$; moreover, $\tilde x_c$ depend periodically on the first argument $\tau$
(with the period depending on $h$, $\nu$ and $\varepsilon$). The dynamics on this
manifold is close to the dynamics of the frozen system, therefore the evolution of
$h$ and $\nu$ is slow, while the first argument $\tau$ is a fast rotating
phase. Hence, in the first order with respect to $\varepsilon$,
the evolution of $h$ and $\nu$ on the invariant manifold is described
by the system averaged with respect to the fast time
\begin{equation}\label{avr0}
\dot h = \varepsilon v_c(h,\nu), \qquad \dot\nu=\varepsilon\,,
\end{equation}
where $v_c$ is defined by equation~(\ref{vchn}).
Therefore if a trajectory stays close to $L_c$ its energy changes following
the equation
\[
\frac{d h}{d\nu}= v_c(h,\nu)+\mbox{h.o.t.}
\]
As we see, for given values of $h$ and $\nu$ the velocity of the change of $h$ depends
on the periodic orbit $L_c$. We will prove that the full system
has a trajectory which switches between small neighbourhoods
of $L_a$ and $L_b$ always choosing the periodic
orbit which gives larger velocity at the moment; clearly, this is the trajectory
which implements the optimal strategy for the acceleration.  For this trajectory
the rate of energy change is described
by the differential equation:
$$
\frac{d h}{d\nu} = \max\{\,v_a(h,\nu),v_b(h,\nu)\,\} + \mbox{ h.o.t. }
$$
Hence, equation (\ref{acc}) correctly describes the evolution 
of $h$ along the trajectory of the fastest energy growth.
The small $\delta$ term in (\ref{accb}) takes care of all higher order corrections
(we can neglect this term in the framework of Theorem~\ref{Thm1},
where the time of acceleration is finite).

As we see, in order to prove Theorem~\ref{Thm2a}, we just need to construct
an orbit which actually jumps between $L_a$ and $L_b$ in the above described way.
In order to do this, we code the orbits of the frozen system that stay in a
small neighbourhood of the heteroclinic cycle
$L_a\cup L_b\cup\Gamma_{ab}\cup\Gamma_{ba}$ by sequences of $a$'s and $b$'s.
Given any such sequence, the corresponding orbit depends smoothly on $h$ and $\nu$,
i.e. we have a normally-hyperbolic invariant manifold corresponding to any
of these sequences. Because of the uniform normal hyperbolicity, all of these
manifolds persist for all small $\varepsilon$ (we supply a proof in
Section \ref{horseshoe}; as a matter of fact, our approach is similar to that of \cite{Shi68}). 
We repeat that every sequence of $a$ and $b$ is a valid
code, i.e. for every itinerary of the jumps between $L_a$ and $L_b$ system
(\ref{aham2}) has an invariant manifold, orbits on which implement this particular
itinerary. In particular, it has an invariant manifold for the orbits on which
the growth of the energy is estimated from below by equation (\ref{accb}). The
rigorous construction is in Section~\ref{average}.


\subsection{Adiabatic invariant revisited}
\label{adi}
In this subsection we discuss the meaning of equation (\ref{acc}) in the Hamiltonian
setup and conditions which imply that all its solutions tend to infinity.
This section is of independent interest and
the proofs of our main theorems do not rely on its results.

Note that Theorems~\ref{Thm1}, \ref{Thm2}  and~\ref{Thm2a}
do not assume that system~(\ref{aham1}) is Hamiltonian.
However, as we will show in a moment, in the Hamiltonian
case equation (\ref{acc}) indeed has a tending to infinity
solution under almost no assumptions.

We first recall that hyperbolic periodic solutions $L_c$ comprise,
at every fixed $\nu$, a one-parameter family parametrized by the energy $h$. Thus,
they fill a certain two-dimensional cylinder in the phase space. As usual in the
theory of slow perturbations, we may introduce ``action-angle'' variables on
this surface. The ``action'' is defined by
\begin{equation}\label{acj}
J_c(h,\nu)=\oint_{L_c} p \;dq
\end{equation}
in the case of the standard symplectic form. In a more general case, where the
Hamiltonian system (\ref{frz0}) is defined on a manifold with a symplectic form
$\Omega$, let us assume that the symplectic form is exact, i.e. $\Omega=d\vartheta$
where $\vartheta$ is an $1$-form. Then the action is defined as
\begin{equation}\label{acjn}
J_c(h,\nu)=\oint_{L_c} \vartheta.
\end{equation}
It is well-known that
\begin{equation}\label{actder}
\frac{\partial J_c}{\partial h} (h,\nu)= T_c(h,\nu),\qquad
\frac{\partial J_c}{\partial \nu} (h,\nu)=
-\int_0^{T_c} \left.\frac{\partial H} {\partial \nu}\right|_{x=x_c(t;h,\nu)} dt.
\end{equation}
In order to see this, note that, by definition 
of the action, the difference between the actions corresponding
to two close closed curves $L_c(h,\nu)$ and $L_c(h+\Delta h,\nu+\Delta\nu)$ is, essentially, the
area of the surface spanned by these two curves. Therefore 
in the case of standard symplectic form $\Omega$ we obtain
\begin{eqnarray*}
\frac{\partial J_c}{\partial \nu}&=&\int_0^{T_c(h,\nu)}
\left(\dot q_c(t;h,\nu) \frac{\partial p_c(t;h,\nu)}{\partial\nu}-
\dot p_c(t;h,\nu) \frac{\partial q_c(t;h,\nu)}{\partial\nu}\right) dt,
\\
\frac{\partial J_c}{\partial h}&=&\int_0^{T_c(h,\nu)}
\left(\dot q_c(t;h,\nu) \frac{\partial p_c(t;h,\nu)}{\partial h}-
\dot p_c(t;h,\nu) \frac{\partial q_c(t;h,\nu)}{\partial h}\right) dt.
\end{eqnarray*}
In the general case we have
\begin{eqnarray*}
\frac{\partial J_c}{\partial \nu}&=&\int_0^{T_c}
\Omega\left(\dot x_c(t;h,\nu)\;,\; \frac{\partial x_c(t;h,\nu)}{\partial\nu}\right) dt,\\
\frac{\partial J_c}{\partial h}&=&\int_0^{T_c}
\Omega\left(\dot x_c(t;h,\nu)\;,\; \frac{\partial x_c(t;h,\nu)}{\partial h}\right)dt.
\end{eqnarray*}
Taking into account the definition of the frozen Hamiltonian vector field
we see $\Omega(\dot x,\cdot) =dH(\cdot)$, which implies in the coordinates 
\begin{eqnarray*}
\frac{\partial J_c}{\partial \nu}&=&\int_0^{T_c}\left.
\frac{\partial H(x,\nu)}{\partial x}\right|_{x=x_c(t;h,\nu)} \frac{\partial x_c(t;h,\nu)}{\partial\nu}dt,
\\
\frac{\partial J_c}{\partial h}&=&\int_0^{T_c}
\left.\frac{\partial H(x,\nu)}{\partial x}\right|_{x=x_c(t;h,\nu)} \frac{\partial x_c(t;h,\nu)}{\partial h} dt.
\end{eqnarray*}
Since $H(x_c(t;h,\nu),\nu)\equiv h$ for all $h$ and $\nu$ due to the definition, 
these formulas imply (\ref{actder}) immediately.


Let us consider the Hamiltonian system with one degree of freedom
defined by the Hamiltonian function $J_c(h,\nu)$:
\[h'=-\frac{\partial J_c}{\partial \nu}(h,\nu)\,,\qquad
\nu'=\frac{\partial J_c}{\partial h}(h,\nu)\,.
\]
Taking into account (\ref{actder}), we conclude that
\begin{equation}\label{avr1}
h' = \int_0^{T_c}
\frac{\partial H}{\partial\nu}(x,\nu)|_{x=x_c(t;h,\nu)} dt, \qquad
\nu'= T_c(h,\nu)\,,
\end{equation}
which coincides with equation (\ref{avr0}) up to a time change.
Consequently $J_c(h,\nu)$ is an integral of (\ref{avr0}), i.e.,
$J_c(h(\nu),\nu)=J(h(0),0)$ for every its solution.

This gives us a leading order model for an orbit of the full system
which stays close to $L_c$: the action $J_c$ is an adiabatic invariant
and the energy oscillates like a trajectory of a Hamiltonian system with
one degree of freedom described by the Hamilton function $J_c$.

Thus, when the orbit is close to the invariant manifold that corresponds to $L_a$,
the function $J_a(H(t),\varepsilon t)$ remains almost constant for a long time, while
the evolution of $J_b(H(t),\varepsilon t)$ is, in the first order,
described by the equation
\begin{equation}\label{jbpo}
T_a \dot J_b = \varepsilon \{J_b,J_a\} = \varepsilon
\left(\frac{\partial J_b}{\partial h} \; \frac{\partial J_a}{\partial \nu}\;-\;
\frac{\partial J_b}{\partial \nu} \; \frac{\partial J_a}{\partial h}\right)\,,
\end{equation}
where the factor $T_a$ is due to the change of the time variable.
Analogously, when the orbit is near the
invariant manifold that corresponds to $L_b$, the function $J_b(H(t),\varepsilon t)$
remains nearly constant, while the evolution of $J_a$ is, in the first order,
given by
\begin{equation}\label{japo}
T_b \dot J_a = \varepsilon \{J_a,J_b\}= \varepsilon
\left(\frac{\partial J_a}{\partial h} \; \frac{\partial J_b}{\partial \nu}\;-\;
\frac{\partial J_a}{\partial \nu} \; \frac{\partial J_b}{\partial h}\right).
\end{equation}
As we see from (\ref{jbpo}) and (\ref{japo}), by virtue of the anti-symmetricity of the
Poisson bracket, if $\{J_a,J_b\}$ is not identically zero one can
always choose between $L_a$ and $L_b$
in such a way that one of the functions $J_a$ or $J_b$
will be increasing, while the other is constant.
Thus, for an orbit of (\ref{ham00}) that stays near the invariant
manifold corresponding to $L_a$ when $\{J_b,J_a\}>0$ and near the invariant
manifold corresponding to $L_b$ when $\{J_b,J_a\}<0$, we will have the ``total action'' $J:=J_a+J_b$
steadily growing with time (in the first order of our approximations).
Since $\partial J/\partial h \equiv T_a+T_b$ is always positive, the growth of $J$ allows
$h(t)\equiv H(x(t),\varepsilon t)$ to grow (we will make this statement more precise below;
see  Propositions \ref{pcase} and~\ref{pcaseq}).

It is remarkable that the above described itinerary of the switching between $L_a$ and $L_b$ coincides with that
employed in equation (\ref{acc}), due to the following relation:
\begin{equation}\label{Eq:poissonvavb}
\{J_a,J_b\}=T_a T_b (v_a-v_b)\,,
\end{equation}
which directly follows from equations (\ref{actder}) and~(\ref{vchn})
and implies, obviously, that the Poisson bracket changes its sign at the
same time as $(v_a-v_b)$ does.

Note that $v_c$ defined by (\ref{vchn}) has a simple geometrical
meaning: since (\ref{actder}) implies
\begin{equation}\label{Eq:vcJc}
v_c(h,\nu)=-\;\frac{\partial J_c}{\partial \nu}(h,\nu)\left/\frac{\partial J_c}{\partial h} (h,\nu)\right.,
\end{equation}
$v_c(h,\nu)$ describes the angle between the direction of the Hamiltonian
vector field of $J_c$ at a point $(h,\nu)$ and the direction
of the $\nu$-axis. Now, equation (\ref{acc}) can be interpreted in the following way.
On the plane $(h,\nu)$ there are two Hamiltonian vector fields generated by
the Hamiltonian functions $J_a$ and $J_b$. A solution of (\ref{acc})
follows the level lines of $J_a$ and $J_b$: at every point there are two
level lines and the solution chooses the one which leads to larger $h$
in the immediate future.

Because of the monotone dependence of $J_c$ on $h$, such choice implies that both
functions $J_c(h,\nu)$ are non-decreasing along the solutions of equation (\ref{acc}). Indeed,
by (\ref{actder}) and~(\ref{vchn}), if $h(\nu)$ is a solution of (\ref{acc}), then
$$\frac{1}{T_c} \frac{d}{d\nu} J_c(h(\nu),\nu) = \max\{v_a,v_b\}-v_c\geq 0.$$

Now we are ready to formulate a general criterion for the unbounded growth of the
solutions of (\ref{acc}). Note that since $\partial J_c/\partial h\neq 0$, every level line
of $J_c(h,\nu)$ is a graph of a certain function $h$ of $\nu$. We will say that a certain level
line of $J_a$ is asymptotic to a level line of $J_b$ if the difference in $h$ between these two lines
tends to zero as $\nu\rightarrow+\infty$.

\begin{Proposition}\label{pcase}
Let the actions $J_a$ and $J_b$ be defined at $(h\geq h^*, \nu\geq 0)$. Assume that
$\lim_{h\rightarrow+\infty} J(h,\nu) =+\infty$ uniformly for all $\nu\geq 0$.
Suppose also that the actions $J_{a,b}(h,\nu)$ remain bounded from above
and the periods $T_{a,b}(h,\nu)\equiv\frac{\partial J_{a,b}}{\partial h}(h,\nu)$ 
remain bounded away from zero on any bounded set
of values of $h$, uniformly for all $\nu\geq 0$. Under these assumptions,
if none of the level lines of $J_a$
is asymptotic to a level line of $J_b$, then every solution
of equation (\ref{acc}) that starts with a sufficiently large $h_0$ stays in the region $h\geq h^*$
and, if defined for all $\nu>0$, tends to infinity as $\nu\rightarrow+\infty$.
\end{Proposition}

\noindent{\em Proof.} Let $c=a$ or $c=b$.
Since $J_c(h,\nu)$ uniformly tends to infinity as $h\rightarrow +\infty$, for every finite value of $J_c$
the corresponding level line is defined for all $\nu$ and the corresponding values of $h$ remain
uniformly bounded. Let $\ell$ be a level line which corresponds to the value
of $J_c$ greater than $\sup_{\nu\geq 0} J(h^*,\nu)$, so this line stays
entirely above $h=h^*$ (recall that $J(h,\nu)$ is an increasing function of $h$ for a fixed $\nu$).
Since $J_c$ is non-decreasing along the orbits of equation (\ref{acc}),
any solution of (\ref{acc}) that starts above $\ell$ at $\nu=0$ remains above it for all $\nu\geq 0$,
i.e. it remains above $h=h^*$. If $h(\nu)$ is such solution, then the monotonicity of $J_c(h(\nu),\nu)$
implies that there exists $\lim_{\nu\rightarrow+\infty} J_c(h(\nu),\nu)$, finite or infinite.

Now suppose that $h(\nu)$ does not tend to infinity as $\nu\rightarrow+\infty$. Then there exists at least
a sequence of values of $\nu=\nu_k\rightarrow+\infty$ such that the corresponding values of $h(\nu_k)$
remain all bounded from above by the same constant. By assumption, the values of, say, $J_a(h(\nu_k),\nu_k)$
also remain uniformly bounded for all $k$, therefore
$\bar J_a:=\lim_{\nu\rightarrow+\infty} J_a(h(\nu),\nu)$ is finite. Moreover, the line $h=h(\nu)$
stays entirely below the level line $J_a(h,\nu)=\bar J_a$, hence $h(\nu)$ is uniformly bounded for all $\nu$.
Since both curves $J_a(h,\nu)=\bar J_a$ and $h=h(\nu)$ stay in the region of bounded $h$, the value of
$\partial J_a/\partial h = T_a$ remains bounded away from zero between these curves. Therefore,
the fact that $J_a(h(\nu),\nu)\rightarrow \bar J_a$ as $\nu\rightarrow+\infty$ implies that the line
$h=h(\nu)$ tends to the level line $J_a(h,\nu)=\bar J_a$. As the same arguments are equally applied to the action
$J_b$, we find that by assuming that $h(\nu)$ does not tend to infinity we obtain the existence of
two level lines, $J_a(h,\nu)=\bar J_a$ and $J_b(h,\nu)=\bar J_b$, that are asymptotic to each other. \qed

\bigskip

In the case of periodic or quasiperiodic dependence of $H$ on $\varepsilon t$
the periodic orbits $L_c$ of the frozen system do not necessarily depend periodically,
or quasiperiodically, on $\nu$. However, let us assume that $L_a$ and $L_b$
are periodic or quasiperiodic functions of $\nu$, or at least 
the corresponding actions $J_{a,b}$ and, hence, the periods $T_{a,b}$
are (this is always the case in many settings, e.g. for the classes of systems 
we consider in the Examples Section). Then all the uniformity assumptions of
Proposition \ref{pcase} are automatically fulfilled. Moreover, 
two level lines of (quasi)periodic functions may be asymptotic to each other only if these lines
coincide (this is, of course, a very rare event). Thus, we arrive at the following result.

\begin{Proposition}\label{pcaseq}
Let the actions $J_a(h,\nu)$ and $J_b(h,\nu)$ be defined at $(h\geq h^*, \nu\geq 0)$,
and let them depend periodically or quasiperiodically on $\nu$. Assume that
$\lim_{h\rightarrow+\infty} J(h,\nu) =+\infty$ uniformly for all $\nu\geq 0$.
Then, if $J_a$ and $J_b$ do not have a common level line, then every solution
of equation (\ref{acc}) that starts with a sufficiently large $h_0$ stays 
in the region $h\geq h^*$ and tends to infinity as $\nu\rightarrow+\infty$.
\end{Proposition}

As we see, conditions of Theorem \ref{Thm1} are almost always fulfilled if the
system under consideration is Hamiltonian. Thus, the phenomenon of an unbounded
energy growth in slowly perturbed chaotic Hamiltonian systems has a universal nature,
practically independent of a particular perturbation shape, or of the structure of
the frozen system. It is caused by some basic properties of Hamiltonian dynamics,
namely by the existence of adiabatic invariants for slowly perturbed one-degree-of-freedom 
systems and by the fact that the adiabatic invariant is the Hamiltonian 
of the corresponding averaged motion. In other words, this phenomenon
is a direct consequence of the Hamiltonian structure of the problem.

\section{Description of a horseshoe \\and normally-hyperbolic invariant manifolds}
\label{horseshoe}

Consider the frozen system
\begin{equation}\label{aham}
\dot x=G(x,\nu;\varepsilon),
\end{equation}
and assume that a function $H(x,\nu;\varepsilon)$ is an integral of system (\ref{aham}), i.e.
\begin{equation}\label{bham}
\frac{\partial H}{\partial x} \cdot G(x,\nu)\equiv 0
\end{equation}
(we suppress, notationally, the dependence on $\varepsilon$ in the frozen system from now on).

Let system (\ref{aham}) have a pair of saddle periodic orbits
$L_a: x=x_a(t;h,\nu)$ and $L_b: x=x_b(t,h,\nu)$ at all $\nu\geq\nu^*$ (for some
$\nu^*<0$) in every energy
level $H=h\geq h^*$. Take a pair of small smooth cross-sections,
$\Sigma_a$ and $\Sigma_b$,
to $L_a$ and $L_b$ respectively. As $L_a$ and $L_b$ depend smoothly on $h$ and $\nu$,
the cross-sections $\Sigma_{a,b}$ can also be taken to depend smoothly on $h$ and
$\nu$.
Denote the Poincar\'e map on $\Sigma_c$ near $L_c$ as $\Pi_{cc}$ ($c=a,b$);
the Poincar\'e map is smooth and depends smoothly on $h$ and $\nu$.

We assume that the frozen system has, at all $\nu\geq\nu^*$ in every energy
level $H=h\geq h^*$, a pair of heteroclinic orbits:
$\Gamma_{ab}\subseteq W^u(L_a)\cap W^s(L_b)$ and
$\Gamma_{ba}\subseteq W^u(L_b)\cap W^s(L_a)$.
Let $\Pi_{ab}$ and $\Pi_{ba}$ be maps on $\Sigma_a$ and on $\Sigma_b$
defined by the orbits close to $\Gamma_{ab}$
and $\Gamma_{ba}$, respectively; $\Pi_{ab}$ acts from some open set in $\Sigma_a$
into an open set in $\Sigma_b$, while $\Pi_{ba}$ acts from an open set
in $\Sigma_b$ into an open set in $\Sigma_a$. There is a certain freedom in the
definition of the maps $\Pi_{ab}$ and $\Pi_{ba}$: each of these maps acts from
a neighbourhood of one point of a heteroclinic orbit to a neighbourhood of another
point of the same orbit, and different choices of the pairs of points lead to
different maps. When a definite choice of the maps is made (we will do it in a moment),
we find for every orbit that lies entirely in a sufficiently small
neighbourhood of the heteroclinic cycle $L_a\cup L_b\cup \Gamma_{ab}\cup\Gamma_{ba}$
a uniquely defined sequence of points $M_i\in\Sigma_a\cup\Sigma_b$ such that
$$M_{i+1}=\Pi_{\xi_i\xi_{i+1}} M_i$$
where
$$\xi_i=c \;\mbox{ if }\; M_i\in\Sigma_c \;\;(c=a,b).$$
The sequence $\{\xi_i\}_{i=-\infty}^{i=+\infty}$ is called the code of the orbit.

The periodic orbits $L_a$ and $L_b$ are saddle, and the intersections
of the stable and unstable manifolds of $L_a$ and $L_b$ that create the heteroclinic orbits are
transverse, by the assumption of the theorem.

This implies (cf. \cite{AfShi73}) that one can choose the maps $\Pi_{ab}$ and $\Pi_{ba}$ and
define coordinates $(u,w)$ in $\Sigma_a$ and  $\Sigma_b$ in such a way that the following holds.
\begin{itemize}
\item
In the given coordinates, $\Sigma_c=U_c\times W_c$ where $U_{a,b}$ and $W_{a,b}$
are certain balls in $\mathbb{R}^{m-1}$ (we assume that
the dimension of the $x$-space equals to $2m$);
so we may choose some constant $R$ such that
\begin{equation}\label{diamr}
\max\bigl\{\,\mathrm{diam}\, U_a, \mathrm{diam}\, U_b, \mathrm{diam}\,
W_a, \mathrm{diam}\, W_b\,\bigr\}\leq R.
\end{equation}
\item For each pair $c$ and $c'$ the Poincar\'e map $\Pi_{cc'}$ can be written in
the ``cross-form'' \cite{book}; namely, there exist smooth functions
$f_{cc'},g_{cc'}: U_c\times W_{c'}\rightarrow U_{c'}\times W_{c}$
such that a point $M(u,w)\in \Sigma_c$ is mapped
into $\bar M(\bar u,\bar w)\in \Sigma_{c'}$ by the map $\Pi_{cc'}$ if
and only if
\begin{equation}\label{crossmap}
\bar u = f_{cc'}(u,\bar w), \qquad w = g_{cc'}(u,\bar w).
\end{equation}
\item There exists $\lambda<1$ such that
\begin{equation}\label{hyperb}
\left\|\frac{\partial (f_{\sigma\sigma'},g_{\sigma\sigma'})}{\partial (u,\bar w)}
\right\| \leq \lambda <1
\end{equation}
(where we define the norm in $U\times W$ as $\max\{\|u\|,\|w\|\}$).
\end{itemize}

Inequality (\ref{hyperb}) means that the set $\Lambda_{h\nu}$ of all the orbits that
lie entirely in a sufficiently small neighbourhood of the heteroclinic
cycle $L_a\cup L_b\cup \Gamma_{ab}\cup\Gamma_{ba}$ in the energy level $H=h$
at the given value of $\nu$ is hyperbolic, a horseshoe.
Thus, one can show that $\Lambda_{h\nu}$ is in one-to-one correspondence with the set
of all sequences of $a$'s and $b$'s, i.e. for every sequence
$\{\xi_i\}_{i=-\infty}^{i=+\infty}$ there exists one and only one orbit
in $\Lambda_{h\nu}$
which has this sequence as its code. Indeed, by (\ref{crossmap}),
an orbit from $\Lambda_{h\nu}$ has code
$\{\xi_i\}_{i=-\infty}^{i=+\infty}$ if and only if the intersection points
$M_i(u_i,w_i)$ of the orbit with the cross-section satisfy
$$u_{i+1}=f_{\xi_i\xi_{i+1}}(u_i,w_{i+1}),
\qquad w_i = g_{\xi_i\xi_{i+1}}(u_i,w_{i+1}),$$
i.e. the sequence $\{\,(u_i,w_i)\,\}_{i=-\infty}^{+\infty}$ is a fixed point of the operator
$$
\{\,(u_i,w_i)\,\}_{i=-\infty}^{+\infty}\mapsto \{\,(f_{\xi_{i-1}\xi_i}(u_{i-1},w_i),
g_{\xi_i\xi_{i+1}}(u_i,w_{i+1})\,\}_{i=-\infty}^{+\infty}.
$$
By (\ref{hyperb}), this operator is a contracting map of the space
$\prod_{i=-\infty}^{+\infty} U_{\xi_i}\times W_{\xi_i}$, hence the
existence and uniqueness of the orbit with the code $\{\xi_i\}_{i=-\infty}^{i=+\infty}$
follows (see e.g. \cite{Shi67}). Moreover, as the fixed point of
a smooth contracting map depends smoothly on parameters,
the orbit depends smoothly on $h$ and $\nu$, so the derivatives of
$(u_i(h,\nu,\xi),w_i(h,\nu,\xi))$ with respect to $(h,\nu)$ are bounded uniformly
for all $i$ and $\xi$.

It also follows from (\ref{diamr}),(\ref{hyperb}) that
\begin{equation}\label{mix}
\|(u_i(h,\nu,\xi^{(1)})-u_i(h,\nu,\xi^{(2)}),
 w_i(h,\nu,\xi^{(1)})-w_i(h,\nu,\xi^{(2)}))\|\leq
R \lambda^{n-|i|}
\end{equation}
for any two code sequences
$\xi^{(1)}=\{\xi^{(1)}_i\}_{i=-\infty}^{+\infty}$,
$\xi^{(2)}=\{\xi^{(2)}_i\}_{i=-\infty}^{+\infty}$
which coincide at $|i|\leq n$ (i.e. $\xi^{(1)}_i=\xi^{(2)}_i$ at $|i|\leq n$);
the constants $R>0$ and $\lambda\in(0,1)$ are given by (\ref{diamr}) and
(\ref{hyperb}) and are independent of $\xi^{(1,2)}$.

Let us now switch to the system with a slowly changing parameter
$\nu=\varepsilon t$. This means that we augment system (\ref{aham}) by the equation
\begin{equation}\label{cham}
\dot\nu=\varepsilon,
\end{equation}
while (\ref{aham}) remains unchanged. Although relation (\ref{bham}) still holds true,
the conservation of energy no longer follows: indeed, by (\ref{aham}),(\ref{bham}),(\ref{cham})
\begin{equation}\label{dham}
\frac{d}{dt} H(x(t),\nu(t))=\varepsilon \frac{\partial H}{\partial\nu}(x(t),\nu(t)).
\end{equation}

By continuity, for system (\ref{aham}),(\ref{cham}) the Poincar\'e maps
$\Pi_{cc'}:\cup_{h,\nu} \Sigma_c\rightarrow \cup_{h,\nu} \Sigma_{c'}$
are still defined at small $\varepsilon$. Denoting $z=(h,\nu)$,
for any compact set of $z$ values we may write the maps in the following form:
\begin{equation}\label{crosseps}
\left\{\begin{array}{ll}\displaystyle
\bar u = f_{cc'}(u,\bar w,z,\varepsilon), \qquad w = g_{cc'}(u,\bar w,z,\varepsilon)\\ \\
\displaystyle \bar z= z+\varepsilon \phi_{cc'}(u,\bar w,z,\varepsilon),
\end{array}\right.
\end{equation}
where $f,g,\phi$ are bounded along with the
first derivatives and $f,g$ satisfy (\ref{hyperb}). Clearly, any smooth transformation
of the $z$-variables will not change the form of map (\ref{crosseps}).

As the set of values of $\nu$ and $h$ under consideration is not compact
(we are interested in the behaviour of the system for $\nu$ and $h$ tending to
infinity), we need certain uniformity assumptions. We require the following.

\begin{itemize}
\item[{\bf[UA1]}] {\em For all $h\geq h^*$ and $\nu\geq\nu^*$, one can introduce
coordinates $(u,w)$ on $\Sigma_a$ and $\Sigma_b$
and define $z=(\alpha(h,\varepsilon),\nu)$ with a smooth function
$\alpha$ such that $\alpha'(h)>0$, in such a way that for all small} $\varepsilon$
\item[(i)] {\em formula (\ref{crosseps}) holds for the Poincar\'e maps $\Pi_{cc'}$,
and the functions $f,g,\phi$ along with the first
derivatives are uniformly bounded and uniformly continuous
with respect to $\varepsilon$, for all $h\geq h^*$ and $\nu\geq\nu^*$};
\item[(ii)] {\em estimate (\ref{hyperb}) holds with the constant $\lambda<1$ the same
for all $h\geq h^*$, $\nu\geq\nu^*$ and all small $\varepsilon$};
\item[(iii)] {\em the diameter of the balls $U_c$ and $W_c$ is uniformly bounded, i.e.
(\ref{diamr}) holds with the constant $R$ the same for all $h\geq h^*$,
$\nu\geq\nu^*$ and all small $\varepsilon$.}
\end{itemize}

It also makes no harm to assume that $\phi\equiv 0$ if $h=h^*$ or $\nu=\nu^*$,
i.e. the region $\{h\geq h^*,\nu\geq\nu^*\}$ is invariant with respect
to the Poincar\'e map. If this is not the case, then we can modify $\phi$
in a small neighbourhood of $h=h^*$ and in a small neighbourhood of $\nu=\nu^*$:
as we are interested
in the orbits for which $h\rightarrow+\infty$, they will never enter the region
of $h$ close to $h^*$; and $\nu=\varepsilon t$ is a growing function of $t$ anyway.

Now we are ready to formulate the main technical result beneath Theorems~\ref{Thm1}-\ref{Thm2a}. 
It has a general nature and has little to do with the Hamiltonian
structure of the equations.
Rather we notice that by fixing any code $\xi$ and varying $h$ and $\nu$
we obtain at $\varepsilon=0$ a sequence of smooth two-dimensional surfaces,
the $i$-th surface is the set run, as $h$ and $\nu$ change, by the point $M_i$ of the uniquely
defined orbit with the code $\xi$; this sequence is invariant with respect to the
corresponding Poincar\'e maps and is uniformly normally-hyperbolic --- hence it
persists at all $\varepsilon$ sufficiently small.

\begin{Lemma}\label{Lemma1}
Given any sequence $\xi$ of $a$'s and $b$'s,
there exists a uniquely defined sequence of smooth surfaces
\begin{equation}\label{seq}
{\cal L}_i(\xi,\varepsilon):
(u,w)=(u_i(z,\xi,\varepsilon),w_i(z,\xi,\varepsilon))
\end{equation}
such that
\begin{equation}\label{inva}
\Pi_{\xi_i\xi_{i+1}} {\cal L}_i={\cal L}_{i+1}.
\end{equation}
The functions $(u_i,w_i)$ are defined for all small $\varepsilon$ and all
$h\geq h^*,\nu\geq\nu^*$, they are
uniformly bounded along with their derivatives with respect to $z$
and satisfy~{\rm (\ref{mix})}.
\end{Lemma}

\noindent{\em Proof.} Take a sufficiently large $K$ and consider any sequence
of surfaces of form (\ref{seq}) with
\begin{equation}\label{uwk}
\left\|\frac{\partial(u_i,w_i)}{\partial z}\right\|\leq K
\end{equation}
(we further suppress notationally the dependence of $u_i$ and $w_i$ of $\xi$
and $\varepsilon$).
Define functions $\eta_i(z)$ and $\bar \eta_i(z)$ by the relations
\begin{equation}\label{vperjod}
z = \eta_i+\varepsilon \phi_{\xi_i\xi_{i+1}}(u_i(\eta_i),w_{i+1}(z),\eta_i,\varepsilon),
\end{equation}
and
\begin{equation}\label{nazad}
\bar \eta_i = z+\varepsilon \phi_{\xi_i\xi_{i+1}}(u_i(z),w_{i+1}(\bar \eta_i),z,\varepsilon).
\end{equation}
As all the derivatives of $\phi$, $u_i$ and $w_{i+1}$ are uniformly bounded,
equations (\ref{vperjod}) and (\ref{nazad}) define the functions $\eta_i(z)$
and $\bar\eta_i(z)$ uniquely.

By (\ref{crosseps}), the sequence of surfaces will satisfy (\ref{inva})
if and only if the sequence of functions $\{u_i(z),w_i(z)\}$ is a fixed point
of the operator
$${\cal F}: \{u_i(z),w_i(z)\}\mapsto \{\tilde u_i(z),\tilde w_i(z)\}$$
defined by
\begin{equation}\label{incross}
\begin{array}{l}\displaystyle
\tilde u_{i+1}(z) = f_{\xi_i\xi_{i+1}}(u_i(\eta_i(z)),w_{i+1}(z),\eta_i(z),\varepsilon),\\
\displaystyle
\tilde w_i(z) = g_{\xi_i\xi_{i+1}}(u_i(z),w_{i+1}(\bar \eta_i(z)),z,\varepsilon).
\end{array}
\end{equation}
Let $\Psi_K$ be the space of sequences of functions
$\psi=\{u_i(z),w_i(z)\}_{i=-\infty}^{+\infty}$ satisfying
(\ref{uwk}). Endow $\Psi_K$ with the norm
\begin{equation}\label{ynrm}
\|\psi\|=\sup_{i,z}\max\{\|u_i(z)\|,\|w_i(z)\|\}.
\end{equation}
It is easy to see that
${\cal F}(\Psi_K)\subset \Psi_K$ provided $K$ is large enough,
and that $\cal F$ is contracting
on $\Psi_K$, for all small $\varepsilon$. Indeed, let us check
this claim at $\varepsilon=0$. In this case we have $\eta_i\equiv\bar \eta_i\equiv z$
(see (\ref{vperjod}),(\ref{nazad})). Therefore,
$$
\begin{array}{l}\displaystyle
\frac{\partial\tilde u_{i+1}}{\partial z} =
\frac{\partial f}{\partial (u,w)} \frac{\partial (u_i,w_{i+1})}{\partial z}+
\frac{\partial f}{\partial z}\\ \\ \displaystyle
\frac{\partial\tilde w_i}{\partial z} =
\frac{\partial g}{\partial (u,w)} \frac{\partial (u_i,w_{i+1})}{\partial z}+
\frac{\partial g}{\partial z},
\end{array}
$$
which gives
\begin{equation}\label{fkink}
\left\|\frac{\partial(\tilde u_{i+1},\tilde w_i)}{\partial z}\right\| \leq \lambda
\left\|\frac{\partial (u_i,w_{i+1})}{\partial z}\right\|+\sup
\left\|\frac{\partial (f,g)}{\partial z}\right\|
\end{equation}
(see (\ref{hyperb})). Thus, for any
$$K>\frac{1}{1-\lambda} \sup \left\|\frac{\partial (f,g)}{\partial z}\right\|,$$
we have ${\cal F}(\Psi_K)\subset \Psi_K$ indeed. To prove the contractivity
of $\cal F$ at $\varepsilon=0$ just note that it follows immediately from
(\ref{incross}),(\ref{hyperb}) that
\begin{equation}\label{fkc}
\|(\tilde u_{i+1}^{(1)}-\tilde u_{i+1}^{(2)},\tilde w_i^{(1)}-\tilde w_i^{(2)})\|
\leq \lambda \|(u_i^{(1)}-\tilde u_i^{(2)}, w_{i+1}^{(1)}-w_{i+1}^{(2)})\|.
\end{equation}

At $\varepsilon\neq 0$ inequalities (\ref{fkink}),(\ref{fkc}) change to
\begin{equation}\label{fkinke}
\left\|\frac{\partial(\tilde u_{i+1},\tilde w_i)}{\partial z}\right\| \leq
(\lambda+O(\varepsilon))\left\|\frac{\partial (u_i,w_{i+1})}{\partial z}\right\|+\sup
\left\|\frac{\partial (f,g)}{\partial z}\right\|+O(\varepsilon)
\end{equation}
and
\begin{equation}\label{fkce}
\|(\tilde u_{i+1}^{(1)}-\tilde u_{i+1}^{(2)},\tilde w_i^{(1)}-\tilde w_i^{(2)})\|
\leq (\lambda+O(\varepsilon))\;
\|(u_i^{(1)}-\tilde u_i^{(2)}, w_{i+1}^{(1)}-w_{i+1}^{(2)})\|.
\end{equation}
Hence, at all small $\varepsilon$ the operator $\cal F$ remains a contracting
map $\Psi_K\rightarrow \Psi_K$. Thus, it has a fixed point in the closure of $\Psi_K$
in the norm (\ref{ynrm}). This gives us the existence of the invariant sequence
of Lipshitz continuous invariant surfaces --- the smoothness is standard (see e.g. Theorem 4.4 of
\cite{book}).

Finally, the estimate (\ref{mix}) follows immediately from (\ref{fkce}) and~(\ref{diamr}).~\qed

\bigskip

According to this lemma, for all sufficiently small $\varepsilon$,
for every code $\xi=\{\xi_i\}_{i=-\infty}^{i=+\infty}$ system
(\ref{aham}),(\ref{cham}) in the space
of $(x,\nu)$  has a smooth three-dimensional invariant
manifold ${\cal M}_{\xi}$ that corresponds to this code,
i.e. the manifold depends continuously on $\varepsilon$ and, at $\varepsilon=0$, it is
the union, over all $h\geq h^*, \nu\geq \nu^*$ of the orbits with the code $\xi$
(recall that for each $h,\nu$ we have exactly one such orbit).
The intersection of ${\cal M}_\xi$ with the cross-section $\cup_{h,\nu}(\Sigma_a\cup\Sigma_b)$
is exactly the sequence of surfaces
$\{{\cal L}_i(\xi,\varepsilon)\}_{i=-\infty}^{+\infty}$. Thus,
dynamics on ${\cal M}_\xi$ is described by
the Poincar\'e map on the cross-section. The Poincar\'e map is obtained by plugging
$u=u_i(z,\varepsilon;\xi), w=w_i(z,\varepsilon;\xi)$ into (\ref{crosseps}). Namely,
$z_i$ is the sequence of the points of intersection with the cross-section of
an orbit on the invariant manifold ${\cal M}_\xi$ if and only if
\begin{equation}\label{dynman}
z_{i+1}= z_i+\varepsilon
\phi_{\xi_i\xi_{i+1}}(u_i(z_i,\varepsilon;\xi),
w_{i+1}(z_{i+1},\varepsilon;\xi),z_i,\varepsilon).
\end{equation}
Recall that in our notations $z$ is a vector of two components: $y:=\alpha(h,\varepsilon)$
and $\nu$. So we will write
\begin{equation}\label{dynman2}
\begin{array}{l}\displaystyle
y_{i+1}= y_i+\varepsilon \theta_{\xi_i\xi_{i+1}}(u_i(z_i,\varepsilon;\xi),
w_{i+1}(z_{i+1},\varepsilon;\xi),z_i,\varepsilon), \\ \\ \displaystyle
\nu_{i+1}= \nu_i+\varepsilon \tau_{\xi_i\xi_{i+1}}(u_i(z_i,\varepsilon;\xi),
w_{i+1}(z_{i+1},\varepsilon;\xi),z_i,\varepsilon),
\end{array}
\end{equation}
i.e., $\theta$ and $\tau$ denote the two components of the function $\phi$ in (\ref{dynman}).

Note that for the codes $\xi=a^{\omega}$ (i.e. $\xi_i=a$ for all $i$) and $\xi=b^{\omega}$
we have $u_{i+1}\equiv u_i$ and $w_{i+1}\equiv w_i$ for all $i$.
We denote $u_c(z,\varepsilon):=u_i(z,\varepsilon;c^{\omega})$ and
$w_c(z,\varepsilon):=w_i(z,\varepsilon;c^{\omega})$ (where $c=a \mbox{ or } b$).
By construction, the manifold $(u,w)=(u_c(z,0),w_c(z,0))$ is the set of the
intersection points of the periodic orbit $L_c:\{x=x_c(t;h,\nu)\}$ of
the frozen system with the cross-section (we have one intersection
point for every value of $z=(\alpha(h,\varepsilon),\nu)$).

By (\ref{cham}), the function $\tau$ in (\ref{dynman2}) is just the time of one return
onto the cross-section. Therefore, for the orbits on the manifold
${\cal M}_{c^{\omega}}$ we have
\begin{equation}\label{tauee}
\tau_{cc}|_{\varepsilon=0}=T_c(h,\nu),
\end{equation}
where $T_c$ is the period of $L_c$. Analogously, for the function $\theta_{cc}$ in
(\ref{dynman2}) we have
$$\theta_{cc}|_{\varepsilon=0}=
\lim_{\varepsilon\rightarrow 0} \frac{y_{i+1}-y_i}{\varepsilon}=
\lim_{\varepsilon\rightarrow 0} \frac{1}{\varepsilon}\int_0^{\tau} \dot y dt,$$
where $\dot y$ is the time derivative of $y$ by virtue of system (\ref{aham}),(\ref{cham}).
As $y=\alpha(h)$, we find from (\ref{dham}),(\ref{cham}) that
\begin{equation}\label{yvel}
\dot y =\varepsilon \alpha'(H) H'_{\nu},
\end{equation}
so
\begin{equation}\label{thetee}
\theta_{cc}|_{\varepsilon=0}=\alpha'(h) \int_0^{T_c}
\left.\frac{\partial H}{\partial\nu}(x,\nu)\right|_{x=x_c(t;h,\nu)} dt
\end{equation}
for the orbits on the manifold ${\cal M}_{c^{\omega}}$.

\section{Proof of Theorem~\ref{Thm2a}}
\label{average}

Before proceeding to the proof we will formulate
the second uniformity assumption. It is automatically
satisfied for any compact set of values of $z=(y,\nu)$,
hence for any compact set of values of $h$ and $\nu$.

Denote
\begin{equation}\label{defq}
\begin{array}{l}\displaystyle
\Theta_{\rho}(z,\varepsilon)=\sup_{u,w,c,c',\zeta} \left\{ |\alpha'(H)H'_\nu|\;\tau_{cc'}(u,w,\zeta,\varepsilon)
+\left\|\frac{\partial}{\partial (u,w)}\theta_{cc'}(u,w,\zeta,\varepsilon)\right\|\right\},\\ \\
\displaystyle
{\cal T}_{\rho}(z,\varepsilon)=\sup_{u,w,c,c',\zeta} \left\{ \tau_{cc'}(u,w,\zeta,\varepsilon)
+\left\|\frac{\partial}{\partial (u,w)}\tau_{cc'}(u,w,\zeta,\varepsilon)\right\|\right\},\end{array}
\end{equation}
where $\zeta$ runs a centered at $z$ ball of some small radius $\rho$ -- we
take $\rho$ as small as we want, but independent of $\varepsilon$. The supremum
of $|\alpha'(H)H'_\nu|$ is taken over the piece of the orbit that starts on the
cross-section at the point $(u,w,\zeta)$ and continues until the next hit with the
cross-section (i.e. $\tau_{cc'}(u,w,\zeta,\varepsilon)$ gives the length of the
corresponding time-interval). Thus, by (\ref{yvel}), $\varepsilon\Theta_\rho$
estimates the maximal change in $y$ between the two intersections with the cross-section.
In particular,
\[\Theta_\rho(z,\varepsilon)\geq |\theta_{cc'}(u,w,\zeta,\varepsilon)|.\]
The function $\cal T$, obviously, estimates the return times to the cross-section.

Recall that $T_c(h,\nu)$ denotes the period of the saddle periodic orbit $L_c$ ($c=a,b$)
of the frozen system (\ref{aham}) for a given value of $z=(\alpha(h,\varepsilon),\nu)$.
Assume that the following holds.

\begin{itemize}
\item[{\bf [UA2]} (i)]
{\em There exist a constant $C>0$ and a small $\rho>0$ such that}
\begin{equation}\label{allweneed}
\max\{\Theta_{\rho}(\alpha(h),\nu,\varepsilon),\; {\cal T}_{\rho}(\alpha(h),\nu,\varepsilon)\}\leq
C \min\{T_a(h,\nu),T_b(h,\nu)\}
\end{equation}
{\em for all $h\geq h^*$ and $\nu\geq\nu^*$, and for all small $\varepsilon$}.

\item[(ii)] {\em The functions}
$$\frac{1}{T_c(h,\nu)} \phi_{cc}(u_c(z,\varepsilon),w_c(z,\varepsilon),z,\varepsilon)
\quad (c=a,b)$$
{\em are uniformly continuous with respect to $\varepsilon$ and $z=(y,\nu)$
for all $h=\alpha^{-1}(y)\geq h^*$ and $\nu\geq\nu^*$.}
\end{itemize}

Denote as $\beta(h,\nu,\varepsilon)$ any smooth function such that
\begin{equation}\label{betadef}
\beta(h,\nu,\varepsilon)\geq\frac{\Theta_{\rho}(\alpha(h),\nu,\varepsilon)}
{\alpha^{\prime}(h) \min\{T_a(h,\nu),T_b(h,\nu)\}}.
\end{equation}
By construction, $\Theta/T_{a,b}$ estimates the velocity of the change of $y=\alpha(h)$,
therefore the function $\beta$ estimates the velocity of the change of $h$. Indeed,
in the examples we consider in Section~\ref{examples}, it is of the same order as
$v_{a,b}$.

Now we can prove Theorem~\ref{Thm2a}.
Take any independent of $\varepsilon$, arbitrarily large $N$ and consider any code
$\xi$ such that for some $i$ we have $\xi_i=\xi_{i+1}=\dots=\xi_{i+N-1}=c$
(where $c=a \mbox{ or } b$). By (\ref{mix}),
\begin{equation}\label{clos}
\|(u_{i+j}(z,\varepsilon,\xi)-u_c(z,\varepsilon),\;
w_{i+j}(z,\varepsilon,\xi)-w_c(z,\varepsilon,\xi))\|\leq
R \lambda^{\min\{j,N-1-j\}}.
\end{equation}
Take any orbit on the invariant manifold ${\cal M}_{\xi}$ that corresponds to this
code $\xi$, and let $z_i$ be the
sequence of intersection points of the orbit with the cross-section
$\cup_{h,\nu}(\Sigma_a\cup\Sigma_b)$.
It follows from (\ref{clos}) and from the uniform $\varepsilon$-closeness of
the right-hand side of (\ref{dynman}) to identity that for all sufficiently small $\varepsilon$
\begin{equation}\label{averma}
\begin{array}{l}\displaystyle
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\|z_{i+N}-z_i-
\varepsilon N \phi_{cc}(u_c(z_i,\varepsilon),w_c(z_i,\varepsilon),z_i,\varepsilon)\|\leq
\\ \\ \displaystyle \qquad\qquad
\leq K (\Theta_{\rho}(z_i,\varepsilon)+{\cal T}_{\rho}(z_i,\varepsilon))\;
(\varepsilon + (\varepsilon N)^2),
\end{array}
\end{equation}
where $K$ is a constant.

By (\ref{averma}),(\ref{tauee}) and the uniformity
assumption,
\begin{equation}\label{nuch}
\lim_{\varepsilon\rightarrow 0} (\nu_{i+N}-\nu_i)/\varepsilon = N T_c(h_i,\nu_i) + O(1),
\end{equation}
uniformly for all $z$. It also follows immediately that uniformly for all $z$
\begin{equation}\label{derwr}
\lim_{\varepsilon\rightarrow 0} \frac{y_{i+N}-y_i}{\nu_{i+N}-\nu_i}= \hat v_c(z_i)
 + O(N^{-1}\beta(h_i,\nu_i)\alpha'(h_i)),
\end{equation}
where we denote
\begin{equation}\label{vbig}
\hat v_c(z)=\frac{\theta_{cc}(z,c^{\omega},0)}{T_c(h,\nu)}\equiv \alpha'(h) v_c(h,\nu)
\end{equation}
(see (\ref{dynman2}),(\ref{thetee}),(\ref{allweneed}),(\ref{betadef}),(\ref{vchn})).

Thus, for every $\delta>0$ there exists $N$ such that
for all sufficiently small $\varepsilon$ and for every code $\xi$ such that
$[\xi_i\xi_{i+1}\dots\xi_{i+N-1}]=c^N$ the change in $y$ after $N$ iterations of the
Poincar\'e map is greater than the change in the solution $y$ of the equation
\begin{equation}\label{eqyv}
\frac{dy}{d\nu}=\hat v_c(y,\nu)-\delta \beta\alpha'
\end{equation}
on the interval $\nu\in[\nu_i,\nu_{i+N}]$, with the initial conditions $y(\nu_i)=y_i$.
Indeed, as the function $\hat v_c$ is uniformly bounded and
uniformly continuous (according to [UA2]),
we have for the solution $y(\nu)$ of (\ref{eqyv})
\begin{equation}\label{diffy}
\lim_{\varepsilon\rightarrow 0} \frac{y(\nu_{i+N})-y(\nu_i)}{\nu_{i+N}-\nu_i}=
\hat v_c(z_i)-\delta \beta \alpha',
\end{equation}
uniformly for all $z$, so the claim follows from the comparison of (\ref{diffy})
with (\ref{derwr}).

It follows immediately that if $N$ is taken sufficiently large,
then for all sufficiently small $\varepsilon$, given any code sequence $\xi$
built of length $N$ blocks of equal symbols, i.e.
$\xi_{jN}=\xi_{jN+1}=\dots=\xi_{jN+N-1}=c_j$ where
$\{c_j\}_{j=-\infty}^{+\infty}$ is an arbitrary sequence of $a$'s and
$b$'s, for every orbit $\ell$ in the invariant manifold ${\cal M}_{\xi}$
\begin{equation}\label{key}
y_{\stackrel{}{jN}}> y(\nu_{\stackrel{}{jN}}) \mbox{  for all } j>0,
\end{equation}
where $\{(y_i,\nu_i)\}_{i=-\infty}^{+\infty}$ is the sequence of the
intersection points of $\ell$ with the cross-section, and $y(\nu)$ is the
solution of the equation
\begin{equation}\label{keyeq}
\frac{dy}{d\nu}=\hat v_{c_j}(y,\nu)-\frac{1}{2}\delta \beta(h,\nu)\alpha'(h)
\mbox{  at } \nu\in[\nu_{jN},\nu_{(j+1)N}]
\end{equation}
with the initial condition $y(\nu_0)=y_0$.

Let us now construct a particular code sequence $\xi^*$ by the following rule.
Fix some $(y_0,\nu_0)$ such that $y_0>\alpha(h^*)$, $\nu_0>\nu^*$.
At $i<0$ we put $\xi_i^*=a$. At $i\geq 0$ we put
$\xi_{jN}^*=\xi_{jN+1}^*=\dots=\xi_{jN+N-1}^*=c_j^*$, where the symbols $c_j^*$
are defined inductively, as follows.
Denote as $\xi^{*(j)}$ the code sequence such that
$\xi^{*(j)}_i=\xi^*_i$ at $i<jN$ and $\xi^{*(j)}_i=a$ at $i\geq jN$. Denote
as ${\cal M}_j$ the invariant manifold with the code $\xi^{*(j)}$. Let
$\ell^{*(j)}$ be the orbit on ${\cal M}_j$ with the initial conditions
$z_0^{*(j)}=(y_0,\nu_0)$. Let $z^{*(j)}_i=(y^{*(j)}_i,\nu^{*(j)}_i)$
denote the $i$-th point of intersection of the orbit $\ell^{*(j)}$ with the cross-section.
Define
\begin{equation}\label{atlast}
c_j^*=\left\{\begin{array}{ll} a & \mbox{ if } \hat v_a(z^{*(j)}_{jN})>\hat v_b(z^{*(j)}_{jN}),\\
b & \mbox{ if } \hat v_a(z^{*(j)}_{jN})\leq \hat v_b(z^{*(j)}_{jN}).\end{array}\right.
\end{equation}
By construction, the value of $c_j^*$ is completely determined by the segment
of $\xi^*$ with $i<jN$, so we indeed can inductively define $\xi^*$ in this way.

Let $\ell^*$ be the orbit on the manifold ${\cal M}_{\xi^*}$ with the same
initial values of $(y_0,\nu_0)$, as we have chosen for the orbits
$\ell^{*(j)}$, and let $z^*_i$ be the points of the intersection of $\ell^*$
with the cross-section. As the code $\xi^*$ coincides with the code
$\xi^{*(j)}$ for all $i<jN$, it follows from (\ref{mix}) that
$$\|(u_i(z,\xi^*)-u_i(z,\xi^{*(j)}),w_i(z,\xi^*)-w_i(z,\xi^{*(j)})\|\leq
R \lambda^{jN-i}.$$
Plugging this into (\ref{dynman}) gives, uniformly for all j,
$$z^*_{jN}-z^{*(j)}_{jN}=O(\varepsilon)$$
(we use here that $z^*_0-z^{*(j)}_0=0$).
Therefore, it follows from the uniform continuity of $\hat v(z)$ that $\hat v_{c^*_j}(z^{*(j)})$
is uniformly close to $\max\{\,\hat v_a(z^{*(j)}),\hat v_b(z^{*(j)})\,\}$
(see (\ref{atlast})). This implies (see (\ref{key}),(\ref{keyeq})) that
\begin{equation}\label{keyag}
y^*_{jN}> y(\nu^*_{jN}) \mbox{  for all } j>0,
\end{equation}
where $y(\nu)$ denotes here the solution of the equation
\begin{equation}\label{kage}
\frac{dy}{d\nu}=\max\{\hat v_a(y,\nu),\hat v_b(y,\nu)\}-\delta \beta(h,\nu) \alpha'(h)
\end{equation}
with the initial condition $y(\nu_0)=y_0$. As the change in $y$ between $N$
intersections with the cross-section is $O(\varepsilon N \Theta_{\rho})$, i.e. 
it is uniformly small in comparison with $\beta \alpha' \min\{T_a,T_b\}$ (see (\ref{betadef})),
we find from (\ref{keyag}) that for every point on the orbit
$\ell^*$ the value of $y$ is larger than the value of $y$ for the solution of
(\ref{kage}) at the same value of $\nu$.

Now recall that $y=\alpha(h)$ with an increasing function $\alpha$.
Thus, it follows from (\ref{kage}) that for every point on the orbit
$\ell^*$ the value of $h$ is larger than the value of $h$ for the solution of
$$\frac{dh}{d\nu}=\frac{1}{\alpha'(h)}\max\{\hat v_a(y,\nu), \hat v_b(y,\nu)\}-
\delta\beta(h,\nu,\varepsilon)$$
at the same value of $\nu$, which completes the proof of Theorem~\ref{Thm2a} (see (\ref{vbig})).

\section{Examples}
\label{examples}
\subsection{Non-autonomous perturbation of a geodesic flow}

We begin with the Mather problem: 
a geodesic flow on an $m$-dimensional manifold ($m\geq 2$), with the Hamiltonian $H_g$, 
subject to a non-autonomous perturbation
$V(q,t)$. Here $q$ denotes position in the configuration
space, i.e. $V$ does not depend on momenta. Assume the uniform hyperbolicity for
the geodesic flow (i.e. strictly negative curvature; recall that the uniform
hyperbolicity implies that periodic trajectories are dense in the phase space 
of the geodesic flow \cite{Anosov}), and assume
uniform boundedness and continuity for $V$ and its first and second derivatives.

The trajectories of the unperturbed geodesic flow are the same in every energy level, just
the velocity of motion grows as the square root of the energy. Namely,
the flow does not change with the following scaling of time, energy, and momenta:
\begin{equation}\label{scalg}
t\rightarrow t/\sqrt{s}, \quad H\rightarrow H s, \quad
p\rightarrow p \sqrt{s}\,.
\end{equation}
At the same time, this transformation changes the perturbation.
If we let $s=\varepsilon^{-2}$,
then the perturbation term $V(q,t)$ is replaced by $\varepsilon^2 V(q,\varepsilon t)$.
Therefore at large energies, adding $V(q,t)$ to $H_g$ is, effectively,
a small and slow perturbation of the geodesic flow.
Thus, this example belongs to the class of systems~(\ref{ham00}).

\begin{Theorem}\label{Thm3}
Let $L_a: q=q_a(t)$ and $L_b: q=q_b(t)$ be two periodic
trajectories of the geodesic flow $H=H_g$ in the energy level $H_g=1$. Denote
\begin{equation}\label{brvc}
\bar V_c(\nu)=\frac{1}{T_c}\int_0^{T_c} V(q_c(t),\nu) dt,
\end{equation}
where $T_c$ ($c=a,b$) is the period of $L_c$. Assume
\begin{equation}\label{cong}
\liminf_{s\rightarrow+\infty} \frac{1}{s}
\int_0^s \left|\frac{d}{d\nu} (\bar V_a(\nu)-\bar V_b(\nu))\right| d\nu >0.
\end{equation}
Then the Hamiltonian system
\begin{equation}\label{matherh} H=H_g+V(q,t)\end{equation}
has orbits for which $H$ linearly grows to infinity.
\end{Theorem}

\noindent
{\em Proof.} After the scaling transformation (\ref{scalg}) with
$s=\varepsilon^{-2}$ the Hamiltonian function recasts as
\begin{equation}\label{epsform}
\hat H=H_g(p,q)+\varepsilon^2 V(q,\varepsilon t)\,,
\end{equation}
which has the form~(\ref{ham00}).
We check that the assumptions of Theorem~\ref{Thm2a}
are all satisfied. First we note that if $\varepsilon$ is
sufficiently small, the frozen system
\[
\hat H=H_g + \varepsilon^2 V(q,\nu)
\]
has a heteroclinic cycle close to the heteroclinic cycle of the geodesic flow
in every energy level $\hat H=h\geq 1$. Indeed, after the scaling
transformation (\ref{scalg}) with $s=h$, the system in the energy level $\hat H=h$
is transformed into
\[
\tilde H=H_g+\frac{\varepsilon^2}{h} V(q,\nu)
\]
in the energy level $\tilde H=1$, i.e. it is uniformly close to
the geodesic flow in the level $H_g=1$.
Therefore the heteroclinic cycle of the frozen system exists
for all $h\ge1$.

It is useful to note that according to (\ref{scalg}) the periods
of the periodic orbits $L_a(h,\nu)$ and $L_b(h,\nu)$ in the heteroclinic cycle
behave as
\begin{equation}\label{pertsc}
T_{a,b}(h,\nu)= h^{-1/2}(T_{a,b}^{(g)} + O(\varepsilon^2)),
\end{equation}
where $T_{a,b}^{(g)}$ denotes here the $\nu$-independent period of the corresponding
orbit of the geodesic flow in the level $H_g=1$.

In order to apply Theorem~\ref{Thm2} we have to check
that Poincar\'e maps near the heteroclinic cycle
satisfy the uniformity assumptions [UA1] and [UA2].
Let $(u_0,w_0)$ be the coordinates for which the Poincar\'e map
for the geodesic flow in the level $H_g=1$ has the form
which satisfies (\ref{crossmap}) and (\ref{hyperb}).
Then we define coordinates $(u,w)$ on the cross-section
in the following way:
\begin{equation}\label{uvvw}
(u,w)=(u_0(q,p/\sqrt{\hat H(p,q,\nu)}),w_0(q,p/\sqrt{\hat H(p,q,\nu)}))
\end{equation}
and $z=(h,\nu)$ with $h=\hat H(p,q,\nu)$.

Now we need uniform estimates for the Poincar\'e map represented
in these coordinates. Let us take any sufficiently large $s$ and consider
the part of the phase space that corresponds to
\begin{equation}\label{hsp1}
s\leq \hat H(p,q,\nu)\leq 2s.
\end{equation}
The scaling (\ref{scalg}) transforms the system to
$\tilde H\equiv H_g+\frac{\varepsilon^2}{s} V(q,\varepsilon t/\sqrt{s})$ in the energy levels
$1\leq \tilde H\leq 2$.
The scaled system is uniformly $O(\varepsilon^2/s)$-close to
the geodesic flow in these energy levels. Therefore, in the coordinates
\begin{equation}\label{uvvws}
(u,w)=(u_0(q,p/\sqrt{s}),w_0(q,p/\sqrt{s})), \quad h=\hat H(p,q,\nu), \; \nu=\varepsilon t,
\end{equation}
the following formulas hold for the Poincar\'e map
\begin{equation}\label{crossinf}
\begin{array}{rclrcl}
\bar u &=& f_{cc'}(u,\bar w) + O(\varepsilon^2/h), \quad&
w &=& g_{cc'}(u,\bar w) +O(\varepsilon^2/h),\\
\bar h&=&h+O(\varepsilon^3/h^{1/2}),\qquad &\bar \nu&=&\nu+O(\varepsilon /h^{1/2})\,,
\end{array}
\end{equation}
uniformly for all $h\geq 1$ and $\nu$; the functions $f,g$ in (\ref{crossinf})
define the Poincar\'e map for the geodesic flow. The equations
for $\bar h$ and $\bar \nu$ are obtained immediately from the fact that the time of return
to the cross-section behaves as $O(h^{-1/2})$ (see e.g. (\ref{pertsc})),
while the time derivative of $\hat H$ along the orbit is
given by $\varepsilon^3\partial V/\partial \nu$, i.e it is uniformly $O(\varepsilon^3)$.

As $s/\hat H$ is uniformly bounded and separated from zero,
it is easy to check that the Poincar\'e map written
in the coordinates $(u,w)$ defined by (\ref{uvvw})
also has the form (\ref{crossinf}).
Recall that $(u,w)$ run over balls of finite radii by construction,
so the validity of the uniformity assumptions (with $\alpha(h,\varepsilon)\equiv h$)
follows immediately from (\ref{crossinf}) and~(\ref{pertsc}).

From the last line of (\ref{crossinf}) we see that the function $\beta$ defined by
(\ref{betadef}) is uniformly $O(\varepsilon^2)$.
Now, according to Theorem~\ref{Thm2a}, it remains to check
that for some sufficiently small $\delta>0$ solutions of the equation
$$
h'(\nu)=\max\{v_a(h,\nu),v_b(h,\nu)\} - \varepsilon^2 \delta
$$
tend to infinity asymptotically linearly with time.
Recall that $v_c$ is the
average change in $H$ along the periodic solution of the frozen system
(\ref{vchn}). As the frozen system is $O(\varepsilon^2)$-close to the geodesic flow
in our case, we find that
$$
v_c(h,\nu)=\varepsilon^2\frac{d}{d\nu}\bar V_c(\nu)+O(\varepsilon^4)
$$
(see (\ref{brvc}),(\ref{pertsc})).
For small $\varepsilon$ the $O(\varepsilon^4)$-term is absorbed
by $\varepsilon^2\delta$, so after scaling $h$
we are left to examine the behaviour of solutions of
$$
h'(\nu)=\max\left\{\,\bar V_a'(\nu), \bar V_b'(\nu)\,\right\} -\delta.$$
By taking integral of both parts we find
$$h(\nu)-h_0=\frac{1}{2}\left(\bar V_a(\nu)+\bar V_b(\nu) - \bar V_a(0)-\bar V_b(0)
+ \int_{0}^\nu \left|\frac{d}{d\nu} \left(\bar V_a-\bar V_b\right)\right| d\nu-2\delta\;\nu\right),$$
i.e. condition (\ref{cong}) ensures the existence of linearly
tending to $+\infty$ solutions indeed.~\qed

\bigskip


Note that in (\ref{cong}) we take an integral of a non-negative function,
therefore (\ref{cong}) is not very restrictive. For example, if
$V(q,t)$ is periodic or quasi-periodic in time, condition (\ref{cong})
is equivalent to
\begin{equation}\label{qper}
\bar V_a(\nu) - \bar V_b(\nu) \neq {\rm const}.
\end{equation}
Thus, for a (quasi)periodic in $t$ potential $V(q,t)$ the only case where
there may be no trajectories of unbounded energy is that when
the average (\ref{brvc}) of $V(q(t),\nu)$ is the same (up to a constant)
function of $\nu$ for every
periodic trajectory of the geodesic flow.

\medskip

Note also that we do not, in fact, need the hyperbolicity of the flow in the whole
phase space. It is sufficient to have a locally-maximal, uniformly-hyperbolic,
transitive, compact, invariant set $\Lambda$ in the energy level $H_g=1$. Theorem~\ref{Thm3}
then holds true, provided the periodic orbits $L_a$ and $L_b$ belong to $\Lambda$.

\subsection{Exponential energy growth}

In the previous example we found trajectories with the energy growth
which is asymptotically linear in time. Such estimate is essentially optimal
in that case: because of the uniform boundedness of the time derivative 
of the perturbation $V(q,t)$ there can be no trajectories
with energy growing faster than linearly.

In this section we describe a different class of perturbations for geodesic flows
which have trajectories whose energy tends to infinity at a much faster rate.
These are obtained by a ``parametric'' perturbation of the geodesic flow.
Namely, consider the following Hamiltonian
\begin{equation}\label{geoex}
H=\frac{1}{2}\; p \; g(q,t)\; p,
\end{equation}
where $g^{-1}$ is the metric tensor. We assume that the corresponding curvature remains strictly
negative for all $t$. We also assume the uniform boundedness and continuity
of $g$ with all the derivatives up to the second order. The scaling (\ref{scalg}) with
$s=\varepsilon^{-2}$ changes the Hamiltonian to
$$
\tilde H=\frac{1}{2}\; p\; g(q,\varepsilon t)\; p\,.
$$
We see that at large energies the original system belongs to the class of Hamiltonian systems
with slowly varying parameter.

By the assumed strict negativity of the curvature at every fixed $t$, the frozen system
is hyperbolic in every energy level
except for $H=0$. Therefore, we may take a pair of saddle periodic orbits
$L_a(h,\nu)$ and $L_b(h,\nu)$ and a heteroclinic cycle that contains them.
As in the case of Theorem~\ref{Thm3}, the period of $L_{a,b}$
behaves as $\sim \varepsilon h^{-1/2}$,
while the change of $h$ during one period is given by
$$
\Delta h\sim \varepsilon \sqrt{h}\,.
$$
Indeed, by the scaling invariance of the frozen system, we find that
\begin{eqnarray*}
\lefteqn{\int_0^{T_c(h,\nu)}
\left.\frac{\partial H}{\partial\nu}(p,q,\nu)\right|_{(p,q)=(p_c,q_c)(t;h,\nu)} dt=}&&
\\ &&=
\frac{1}{2} \sqrt{h} \int_0^{T_c(1,\nu)} p_c(t;1,\nu) g'_{\nu}(q_c(t;1,\nu),\nu) p_c(t;1,\nu) dt,
\end{eqnarray*}
where $(p,q)=(p_c,q_c)(t;h,\nu)$ is the equation of the periodic orbit $L_c(h,\nu)$.

Like we
did it in the proof of Theorem~\ref{Thm3}, by using the fact that the scaling (\ref{scalg})
can map a neighbourhood of an arbitrarily high energy level into a neighbourhood
of the energy level $\hat H=1$, we find that the Poincar\'e map has the following
form (uniformly for all $h$ and $\nu$)
\begin{equation}\label{cross5}
\begin{array}{l}
\bar u = f_{cc'}(u,\bar w,\nu) + O(\varepsilon/\sqrt{h}), \qquad
w = g_{cc'}(u,\bar w,\nu) +O(\varepsilon/\sqrt{h}),\\
\bar h=h+O(\varepsilon\sqrt{h}),\qquad \bar \nu=\nu+O(\varepsilon/\sqrt{h})
\end{array}
\end{equation}
where the functions $f,g$ define the Poincar\'e map for the frozen geodesic
flow in the energy level $H=1$. It follows immediately from (\ref{cross5}),(\ref{pertsc})
that the uniformity assumptions are fulfilled with $\alpha(h)=\ln h$, and $\beta(h)=O(h)$.

Thus, by Theorem~\ref{Thm2a}, there exist orbits bounded
from below by a solution of the equation
\begin{equation}\label{eqex}
h'(\nu)=h(\nu)\left(\max\{\hat v_a(\nu),\hat v_b(\nu)\}-\delta\right)
\end{equation}
where
\begin{equation}\label{vcnut}
{\hat v}_c(\nu)=\frac{1}{2T_c(1,\nu)} \int_0^{T_c(1,\nu)}
p_c(t;1,\nu) g'_{\nu}(q_c(t;1,\nu),\nu) p_c(t;1,\nu) dt.
\end{equation}

Let us estimate the solutions of (\ref{eqex}). Denote
$\hat T_c(\nu)\equiv T_c(1,\nu)$ (i.e. this is the period of the orbit $L_c$
of the frozen system in the energy level $H=1$). By the invariance of the frozen system
\begin{equation}\label{expfr}
H=\frac{1}{2}\;p\;g(q,\nu)\;p
\end{equation}
with respect to energy scaling, we find that
\[T_c(h,\nu)=\hat T_c(\nu)/\sqrt h.\]
Let us introduce the action variable (see Section~\ref{adi})
\[J_c(h,\nu)=\int_0^{T_c(h,\nu)} p_c(t;h,\nu) \dot q_c(t;h,\nu) dt.\]
As $\dot q=g(q,\nu)\;p$ in system (\ref{expfr}), this gives us the following explicit
formula for the action
\begin{equation}\label{Eq:Jc}
\begin{array}{l}\displaystyle
J_c(h,\nu)=\int_0^{T_c(h,\nu)} p_c(t;h,\nu)\; g(q_c(t;h,\nu),\nu)\; p_c(t;h,\nu) dt =\\
\\ \displaystyle \qquad\qquad\qquad
= 2h T_c(h,\nu)=2\sqrt h\hat T_c(\nu).
\end{array}
\end{equation}
Now recall that by general formula (\ref{actder}),
\begin{equation}\label{jdein}
\frac{\partial}{\partial\nu}\hat J_c(1,\nu)=-\hat v_c(\nu)\hat T_c(\nu)\,.
\end{equation}
Equations (\ref{Eq:Jc}) with $h=1$ and (\ref{jdein}) imply
that in the case of Hamiltonian (\ref{geoex}) there is
a closed formula which expresses $\hat v_c(\nu)$ in terms of $\hat T_c(\nu)$:
\begin{equation}\label{vcact}
\hat v_c(\nu)=-2\frac{d}{d\nu} \ln \hat T_c(\nu)\,.
\end{equation}

Plugging this in (\ref{eqex}) we find
\[\frac{d}{d\nu} \ln h(\nu)= - \frac{d}{d\nu} (\ln \hat T_a(\nu)+\ln\hat T_b(\nu))+
\left|\frac{d}{d\nu} (\ln \hat T_a(\nu)-\ln\hat T_b(\nu))\right|-\delta,\]
which gives us
\[\ln h(\nu)-\ln h_0 = \ln \hat T_a(0)\hat T_b(0)- \ln \hat T_a(\nu)\hat T_b(\nu)+
\int_0^\nu \left|\frac{d}{d\nu} \ln (\hat T_a(\nu)/\hat T_b(\nu))\right|d\nu-\delta\nu.\]
As we see, solutions of (\ref{eqex}) tend exponentially to infinity for all sufficiently
small $\delta$, provided
\begin{equation}\label{congop}
\liminf_{s\rightarrow+\infty} \frac{1}{s}
\int_0^s \left|\frac{d}{d\nu} \ln (\hat T_a(\nu)/\hat T_b(\nu))\right| d\nu >0,
\end{equation}
and the functions $\hat T_{a,b}$ are bounded away from zero and infinity uniformly
for all $\nu\geq0$

Thus, we arrive at the following result.

\begin{Theorem}\label{Thm4}
If the periods $\hat T_a(\nu)$ and $\hat T_b(\nu)$ of the periodic orbits $L_{a,b}$
of the frozen system in the energy level $H=1$ are bounded away from zero 
and infinity for all $\nu\geq0$, and if condition (\ref{congop}) is satisfied, then
Hamiltonian system (\ref{geoex})
has orbits for which $H$ exponentially grows to infinity.
\end{Theorem}

Note that similar to the previous section, if $g(q,t)$ is periodic or 
quasi-periodic in time, condition (\ref{congop}) is equivalent to
\begin{equation}\label{qpero}
\hat T_a(\nu)/\hat T_b(\nu)\neq {\rm const}.
\end{equation}


\subsection{Time-dependent billiard-like potentials}

Another example: Let $D\in {\mathbb R}^m$, $m\geq 2$, be a bounded region whose boundary is
composed of a finite number of strictly concave (when looking from inside of $D$)
smooth $(m-1)$-dimensional manifolds which intersect by non-zero angles.
Let $V_0(q)$ ($q\in D$) be a positive function such that
$$V_0(q)\rightarrow+\infty \;\mbox{ as }\;\; q\rightarrow \partial D.$$
Consider the Hamiltonian system
\begin{equation}\label{bils}
H=\sum_{i=1}^m \frac{p_i^2}{2} + V_0(q) + V_1(q,t),
\end{equation}
where $V_1(q,\nu)$ is uniformly bounded, along with the first and second derivatives,
for all $q\in D$ and  $\nu\in {\mathbb R}$. By scaling time, energy and momenta by the rule
(\ref{scalg}) with $s=\varepsilon^{-2}$, this system transforms into
\begin{equation}\label{bilh}
H=\sum_{i=1}^m \frac{p_i^2}{2} + \varepsilon^2 V_0(q)+\varepsilon^2 V_1(q,\varepsilon t).
\end{equation}
Thus, at large energies, it is a slow perturbation of the singular Hamiltonian
$$H=H_b=\sum_{i=1}^m \frac{p_i^2}{2} +
\left\{\begin{array}{ll} 0 & \mbox{at }\; q\in D,\\
+\infty & \mbox{at }\; q\not\in D.\end{array}\right.$$

This defines a billiard in $D$: inertial motion inside $D$ and reflection
at the boundary. In order to ensure the actual closeness of (\ref{bilh}) to
a billiard with the standard reflection law (``the angle of reflection equals
the angle of incidence'') we need certain assumptions. Namely,
let $S\subset\partial D$ be the set of ``corner'' points, i.e. those where $\partial D$ is not smooth
(these are the points where the smooth boundary components intersect).
Assume that there exists an open neighbourhood $U$ of $\partial D\backslash S$
such that
\begin{equation}\label{barpat}
V_0(q)=W(Q(q))
\end{equation}
for all $q\in U$. Here $Q(q)$ is the so-called pattern
function: it is at least $C^2$-smooth function defined for all $q\in U$,
its first derivative $\partial Q/\partial q$ does not vanish in $U$,
and the smooth boundary components of $D$ are given by the equation $Q=0$. The
function $Q$ thus defines the shape of the billiard region $D$. The function $W$
defines the growth of the potential as the boundary is approached. We assume that
$W'\neq 0$ for all small $Q$ and that its inverse function $W^{-1}$ satisfies
\begin{equation}\label{smref}
W^{-1}(hV)_{\displaystyle\stackrel{\longrightarrow}{_{C^2}}} 0 \; \mbox{ as }\;
h\rightarrow+\infty,
\end{equation}
on any interval $V\geq C >0$. Roughly speaking, by representing the potential in
form (\ref{barpat}) we achieve that its gradient (``the reaction force'') is, in the
limit $h=+\infty$, normal to the billiard boundary, which is an obvious necessary
condition for the validity of the standard reflection law; condition (\ref{smref})
ensures the $C^1$-closeness of Poincar\'e map for the smooth flow (\ref{bilh})
at large $h$ to the Poincar\'e map for the billiard flow, outside the set of singular
trajectories, i.e. those which hit $S$ or which are tangent to a smooth component of the billiard boundary at some  point.

As the boundary components are strictly concave, the billiard in $D$ is dispersing.
This implies \cite{Sin} the hyperbolicity of the billiard flow (outside the set of singular trajectories); 
moreover, periodic orbits are dense in the phase space \cite{KaS,BSC}. We call a billiard
orbit regular, if it stays bounded away from the singularities, i.e. from the
set of points in the phase space which correspond to corner or to a tangency to the
billiard boundary.

\begin{Theorem}\label{Thm:billiard}
Let $L_a: q=q_a(t)$ and $L_b: q=q_b(t)$ be two regular
periodic trajectories of the billiard in $D$, corresponding to kinetic energy
equal to $1$. Denote
\begin{equation}\label{brvcb}
\bar V_c(\nu)=\frac{1}{T_c}\int_0^{T_c} V_1(q_c(t),\nu) dt,
\end{equation}
where $T_c$ ($c=a,b$) is the period of $L_c$. Assume that condition (\ref{cong}) (or
condition (\ref{qper}) in the case of periodic or quasiperiodic dependence
of $V$ of $t$) is fulfilled. Then system (\ref{bils}) has orbits for which $H$
linearly grows to infinity.
\end{Theorem}

\noindent
{\em Proof.} It follows from \cite{Sin,SC} that for any two regular periodic orbits $L_a$ and 
$L_b$ in the strictly dispersing billiard there exists a pair of transverse heteroclinic orbits 
$\Gamma_{ab}$ and $\Gamma_{ba}$, which are also regular. Take a sufficiently small neighbourhood 
of $L_a\cup L_b\cup\Gamma_{ab}\cup\Gamma_{ba}$ in the intersection of the phase space with 
the level $H_b=1$. The hyperbolic set $\Lambda$ of the orbits that stay in this neighbourhood
consists of regular orbits only; as a whole, $\Lambda$ stays bounded away from
the singularity. One can take two small cross-sections, $\Sigma_a$ and $\Sigma_b$,
to the orbits $L_a$ and $L_b$ in the intersection of the phase space with the level $H_b=1$, 
such that every orbit of $\Lambda$ returns to $\Sigma_a\cup\Sigma_b$
at a finite time; the corresponding Poincar\'e maps are smooth, as the orbits of
$\Lambda$ undergo only regular collisions with the billiard boundary. We have the same 
picture in every other energy level because of the invariance of the billiard flow
with respect to energy scaling.

According to \cite{TRK1,TRK2}, under conditions (\ref{barpat}),(\ref{smref}), at small
$\varepsilon$ a finite-time flow map of the smooth system (\ref{bils}) near
a regular orbit of the billiard  flow is close, along with the first derivatives,
to the corresponding map for the billiard flow (while only an autonomous case was considered
in the quoted papers, the results and proofs do not change for our case where
a bounded and slow non-autonomous term $\varepsilon^2 V_1$ is added to the Hamiltonian).
Therefore, for any compact interval of the energy values,
the Poincar\'e maps defined on the cross-section $\Sigma_a\cup\Sigma_b$
by the smooth system (\ref{bils}) is close to the Poincar\'e map of the billiard flow. Now, applying
the scaling transformation (\ref{scalg}) exactly as we did it in the proof of
Theorem \ref{Thm3}, we find that the uniformity assumptions [UA1] and [UA2] are
fulfilled with $\alpha(h)=h$ and the Poincar\'e map can be written in the form (\ref{crossinf}).
The equation for $\bar h$ in the last line of (\ref{crossinf}) is
found from the relation
\begin{equation}\label{dlthb}
\bar h= h+\varepsilon^3 \int \frac{\partial}{\partial \nu} V_1(q(t),\varepsilon t) dt
\end{equation}
(see (\ref{bilh})) where the integral is taken over the corresponding
orbit of the smooth system. As the orbits of the smooth system are close to the orbits
of the billiard after an appropriate time-parametrization \cite{TRK2}, the integral
in (\ref{dlthb}) tends, as $\varepsilon\rightarrow 0$, to the integral over
the limit billiard orbit, i.e. it remains uniformly bounded along with
the first derivatives with respect to initial conditions for any compact set of energy values.
Now, by using the scaling transformation (\ref{scalg}) with
$s=h$ we find that the integral in (\ref{dlthb}) behaves as $O(h^{-1/2})$, which
gives the equation (\ref{crossinf}) for $\bar h$ indeed. After formula (\ref{crossinf}) 
is established for the Poincar\'e map, the rest of the proof follows exactly like in Theorem \ref{Thm3}.~\qed

Note that like in Theorem~\ref{Thm3}, the conditions on the billiard domain $D$ may be relaxed. We do
not need the billiard to be dispersing; it is enough to have a locally-maximal,
uniformly-hyperbolic, transitive, compact, invariant set $\Lambda$ composed of
regular orbits. Theorem~\ref{Thm:billiard} holds true,
provided the regular orbits $L_a$ and $L_b$ belong to $\Lambda$.

\subsection{Nonautonomous perturbation of a homogeneous potential}
In the last example, we consider the Hamiltonian system
\begin{equation}\label{polh}
H=T(p) + V_0(q) + V_1(q,t) + V_2(q,t),
\end{equation}
where $T$ is a quadratic polynomial of momenta $p\in R^m$ ($m\geq 2$),
$V_0$ is a degree $d\geq 3$ homogeneous polynomial of the coordinates $q\in {\mathbb R}^m$,
$V_1$ is a degree $d-1$ homogeneous polynomial of $q$, and $V_2$ is a polynomial of
$q$ of degree less than $d-1$; the coefficients of $V_1$ and $V_2$ are smooth
functions of time, uniformly bounded, along with the first derivative, for all $t$.

By scaling time, energy, momenta and coordinates by the rule
\begin{equation}\label{scalh}
t\rightarrow t/s^{1/2-1/d}, \quad H\rightarrow H s, \quad
p\rightarrow p \sqrt{s}, \quad q\rightarrow q s^{1/d}
\end{equation}
with $s=\varepsilon^{-2d/(d-2)}$ this system transforms into
\begin{equation}\label{polhh}
H=T(p) + V_0(q)+ \varepsilon^{\frac{2}{d-2}} V_1(q,\varepsilon t) + O(\varepsilon^{\frac{4}{d-2}}).
\end{equation}
It is a small and slow
perturbation of the homogeneous Hamiltonian
\begin{equation}\label{homp}
H=T(p) + V_0(q).
\end{equation}
This system is invariant with respect to the scaling (\ref{scalh}), hence its behaviour
is the same in every energy level. Assume that system (\ref{homp}) has
a locally-maximal, uniformly-hyperbolic, compact, transitive, invariant set $\Lambda$
in the energy level $H=1$. Take any two periodic trajectories
$L_a: \{q=q_a(t), p=p_a(t)\}$ and $L_b: \{q=q_b(t), p=p_b(t)\}$ from $\Lambda$. Denote
\begin{equation}\label{brvch}
\bar V_c(\nu)=\frac{1}{T_c}\int_0^{T_c} V_1(q_c(t),\nu) dt\qquad (c=a,b).
\end{equation}

By the scaling invariance, we obtain that a pair of saddle periodic orbits $L_{a,b}(h)$
exists in every energy level with $h>0$; since the orbits belong to
a transitive hyperbolic set, they are connected by transverse heteroclinic
orbits $\Gamma_{ab}$ and $\Gamma_{ba}$.
As the frozen system for (\ref{polhh}) is close to (\ref{homp}) (recall that $d>2$),
the former also possesses a heteroclinic cycle in every energy level.
By applying scaling transformation (\ref{scalh}) with $s=h$,
one can immediately see that the change of the energy along
an orbit of system (\ref{polhh}) for one round near $L_c(h)$ is given by
\begin{equation}\label{hch}
\bar h - h = \varepsilon^{\frac{d}{d-2}} h^{1/2}(\int_0^{T_c} V_1(q_c(t),\nu) dt+\dots),
\end{equation}
while the return time to the cross-section behaves as
\begin{equation}\label{nuch1}
\bar\nu-\nu=\varepsilon h^{-\frac{d-2}{2d}} (T_c+\dots),
\end{equation}
where the dots stand for the terms which tend to zero as the distance to $L_c(h)$
diminishes and $\varepsilon\rightarrow0$.

Now, like we did it in the examples
above, by using the scaling transformation (\ref{scalh}) one can check that
the Poincar\'e map satisfies the uniformity assumptions [UA1] and [UA2] with
$$\alpha(h)=h^{\frac{1}{d}}$$ and
$$\beta=O(\varepsilon^{\frac{2}{d-2}} h^{1-\frac{1}{d}}).$$
This, along with equations (\ref{hch}),(\ref{nuch1})
implies that system (\ref{polhh}) has, for any $\delta>0$, orbits
for which the energy grows not slower than the solution of the equation
\begin{equation}\label{eqgrh}
h'(\nu)=\varepsilon^{\frac{2}{d-2}} h^{1-\frac{1}{d}}
(\max\{\frac{d}{d\nu} \bar V_a(\nu), \frac{d}{d\nu} \bar V_b(\nu)\}-\delta)
\end{equation}
(see Theorem~\ref{Thm2a}). By scaling energy back in order to return to the
original system (\ref{polh}): $h\rightarrow h \varepsilon^{2d/(d-2)}$, we rewrite this
equation as
$$h'(\nu)=h^{1-\frac{1}{d}}
(\max\{\frac{d}{d\nu} \bar V_a(\nu), \frac{d}{d\nu} \bar V_b(\nu)\}-\delta).$$
This is solved as
\begin{eqnarray*}
\lefteqn{h(\nu)^{1/d}-h(0)^{1/d}=}\\
&=&\frac{1}{2d} \left( \bar V_a(\nu) + \bar V_b(\nu)-
\bar V_a(0) - \bar V_b(0) + \int_0^\nu \left|\frac{d}{d\nu}(\bar V_a-\bar V_b)\right| d\nu -
2\delta\;\nu\right).
\end{eqnarray*}
Thus, we arrive at the following result.

\begin{Theorem}\label{Thm:homo}
If\/ {\rm (\ref{cong})} is fulfilled, then system\/ {\rm (\ref{polh})} has
orbits for which $H$ grows to infinity as $t^d$.
\end{Theorem}

Like in Theorems~\ref{Thm3} and~\ref{Thm:billiard} above, in the case of periodic or
quasiperiodic dependence
of $V_1$ of $t$, condition (\ref{cong}) reduces to (\ref{qper}).

\medskip

As we see, every time we have a chaotic Hamiltonian system
which is invariant with respect to a scaling of energy, its non-autonomous
perturbation creates orbits of growing to infinity energy, provided very non-restrictive
conditions of type (\ref{cong}), (\ref{qper}) or (\ref{qpero}) are fulfilled. The rate of the
energy growth with time depends on how the perturbation term rescales, and
is determined by solving the corresponding equation (\ref{acc}).


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