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Differential and Integral Equations
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Volume 22, Numbers 9-10 (2009)}
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%%%%%%% {\ms {\bf Lakshmi Burra and Fabio Zanolin, }
%%%%%%% Chaotic dynamics in a simple class of Hamiltonian systems with
%%%%%%% applications to a pendulum with variable length
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\def\leftmark{\sc Lakshmi Burra and Fabio Zanolin}
\def\rightmark{\sc Chaos in a pendulum with variable length}
\begin{document}
\title{ Chaotic dynamics in a simple class of Hamiltonian systems with
applications to a pendulum with variable length }
\thanks{AMS Subject Classifications: 34C25, 37C25, 37D45.}
\date{}
\maketitle
\vspace{ -1\baselineskip}
{\small
\begin{center}
{\sc Lakshmi Burra and Fabio Zanolin} \\
Department of Mathematics and Computer Science\\
University of Udine,
via delle Scienze 206, 33100 Udine, Italy \\[10pt]
\dedicatory{Dedicated to Patrick Habets and Jean Mawhin}
\end{center} }
\numberwithin{equation}{section}
\allowdisplaybreaks
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]
\def\stretchx{\Bumpeq{\!\!\!\!\!\!\!\!{\longrightarrow}}}
\smallskip
\begin{quote}
\footnotesize {\bf Abstract.} We prove the existence of chaotic
dynamics in a simple Hamiltonian system of the form $\ddot{x} + q(t) f(x)
= 0,$ where $q(t)$ is a periodic function of constant sign. Applications
are given to a pendulum equation with variable length.
\end{quote}
\section{Introduction}\label{sec-1} The second-order nonlinear
differential equation
\begin{equation}\label{eq-1.1}
\ddot x + f(x) = 0,
\end{equation} can be considered as one of the most commonly studied
examples of a conservative system. It represents many different physical
phenomena which are described as a simple frictionless system of one
degree of freedom. Typical examples \cite{MeHa-92} range from the simple
pendulum equation (for $f(x)= a \sin(x)$), to the classical second-order
ODE generating elliptic functions (for
$f(x) = (1+k^2)x - 2k^2 x^3$). In both of the above mentioned examples,
the phase--plane portrait of the associated first-order planar
Hamiltonian system consists of a center bounded (from above and from
below) by two heteroclinic orbits. In the present paper we investigate
the case in which a perturbation is introduced in the form of a weight
function $q(t)$ which is periodic and of constant sign, which leads us to
the study of the equation
\begin{equation}\label{eq-1.2}
\ddot x + q(t) f(x) = 0.
\end{equation} For such kinds of nonlinear equations with weight,
different methods have been employed to study associated boundary-value
problems. See for example \cite{DeHa-06, MaWi-89}. It is our goal to
show, under certain conditions, the existence of chaotic dynamics in
equation $(\ref{eq-1.2}).$ In this context, we would like to mention
that some results on chaos have been obtained by other authors for some
special forms of $q(t)$ and $f(x).$ In particular, for pendulum type
equations, Hastings and McLeod in \cite{HaMc-93}, as well as Furi,
Martelli et al. in \cite{FuMaON-04}, using direct arguments proved
interesting results about the presence of chaos for the nonlinear
pendulum with variable length
\begin{equation}\label{eq-1.3}
\ddot x + (1 + r\sin(\omega t)) \sin(x) = 0,
\end{equation} which, as can be seen, is a special case of
$(\ref{eq-1.2}).$
In our paper we present a different approach, based on recent results
from the theory of topological horseshoes and linked twist maps. Such
results have been recently applied in \cite{BuZa->} to equation
$(\ref{eq-1.2})$ with $q(t)$ changing its sign and $f(x)$ a periodic
function. Here, we consider the case where $q(t)$ is of constant sign (for
example, we could choose $q(t) > 0,$ for all $t\in {\mathbb R}$), but we
do not assume any periodicity condition on $f(x).$ Actually, in order to
simplify our argument in the proofs, we choose $q(t)$ to be a stepwise
function.
To be more specific, we give now the hypotheses on $f$ and $q$ which will
be assumed throughout the paper.
Let $f: {\mathbb R} \to {\mathbb R}$ be a locally Lipschitz function
satisfying the following:
\begin{itemize}
\item[$(\textbf{H})\;$]{\em There exist $a,b$ with $a < 0 < b$ such that
$f(a) = f(0) = f(b) =0,$ and
$$f(x) < 0\;\; \forall \, x\in \, (a,0),\quad f(x) > 0\;\;
\forall \, x\in (0, b),\quad
\int_a^b f(s)\,ds =0.$$ }
\end{itemize} Let $q: {\mathbb R} \to {\mathbb R}$ be a $T$-periodic
function such that
\begin{equation}\label{eq-1.4} q(t):=\;
\left\{
\begin{array}{llll}
\; &A,&\quad\mbox{for }\; t\in [0,\tau_A) \\ &B,&\quad\mbox{for }\; t\in
[\tau_A,\tau_A + \tau_B ),
\end{array}
\right.
\end{equation} where $A, B > 0$ and $\tau_A,\tau_B > 0,$ with
$$\tau_A + \tau_B = T.$$ With this choice, the weight function is a
positive constant equal to $A$ on the time interval $[0,\tau_A).$ At
the time $t=\tau_A,$ it switches to $B$ and remains at this value for a
length of time
$\tau_B.$ At the time $T$ the function $q(t)$ switches again to $A,$
and so on, since it is a $T$-periodic function.
Within this framework, we consider the Poincar\'{e} map associated to the
system
\begin{equation}\label{eq-1.5}
\left\{
\begin{array}{llll} &\dot x = &y\\ &\dot y = - &q(t) f(x),
\end{array}
\right.
\end{equation} which is equivalent to $(\ref{eq-1.2})$ written in the
phase--plane. Indeed, we recall that the fundamental theory of ODEs
guarantees that, for every $t_0\in {\mathbb R}$ and $z_0 = (x_0,y_0)\in
{\mathbb R}^2,$ there exists a unique solution
$\zeta(\cdot)=\zeta(\cdot;t_0,z_0)$ of
$(\ref{eq-1.5}),$ satisfying the initial condition $x(t_0) = x_0,
y(t_0) = y_0,$ which is defined on a right maximal interval of
existence $[t_0,\omega) \,\subseteq [t_0,+\infty).$ The Poincar\'{e} map
for the $T$-periodic problem is defined as
$$\Phi: {\mathbb R}^2\supseteq \mbox{dom}(\Phi)\to {\mathbb R}^2,\quad
\Phi: z_0\mapsto \zeta(t_0+T;t_0,z_0),$$ that is, the function which maps
any initial point $z_0$ (at the time $t=t_0$) to the point after a period
$T,$ on the trajectory of $(\ref{eq-1.5})$ departing from $z_0.$
Usually, we take $t_0 = 0$ (this will be always assumed, unless otherwise
mentioned). The domain
$\mbox{dom}(\Phi)$ of the Poincar\'{e} map is an open subset of the plane
and $\Phi$ is a homeomorphism onto its image. Since we do not assume any
boundedness condition on $f,$
$\Phi$ is not necessarily defined on the whole plane. However, during the
course of the proofs of our results, we confine ourselves to a compact
invariant region of the strip $[a,b]\times {\mathbb R}$ on which the map
$\Phi$ is well defined.
With this background, we prove the following theorems.
\begin{theorem}\label{th-1.1} Let $f$ satisfy condition ${\bf{(H)}}$ and
let $q$ be as in $(\ref{eq-1.4})$ with
$ 0 < A < B.$ Then, there are constants
$\beta^* > 0$ and, for every
$m\geq 2,$
$\alpha^*_m >0,$ such that for all
\begin{equation}\label{eq-1.1as}
\tau_A > \alpha^*_m\,\quad\mbox{and }\; \tau_B > \beta^*,
\end{equation} the Poincar\'{e} map associated to $(\ref{eq-1.5})$
induces chaotic dynamics on $m$ symbols.
\end{theorem} We claim that our theorem is stable with respect to small
perturbations. In fact, we have the following.
\begin{theorem}\label{th-1.2} Let $f$ satisfy condition ${\bf{(H)}}$ and
let $q$ be as in $(\ref{eq-1.4})$ with
$ 0 < A < B. $ Then, there are constants
$\beta^* > 0$ and, for every
$m\geq 2,$
$\alpha^*_m >0,$ such that for all
$
\tau_A > \alpha^*_m$ and $ \tau_B > \beta^*,
$ there is $\varepsilon > 0$ $($depending on $f,q,\tau_{A},\tau_{B} )$ such
that the Poincar\'{e} map associated to
$$\ddot x + c \dot x + r(t) f(x) =0$$ $($where $c\in{\mathbb R}$ is a
constant and $r(t)$ is a locally integrable $T$-periodic function$)$
induces chaotic dynamics on $m$ symbols, provided that
$$|c| < \varepsilon, \quad \int_0^T |r(t) - q(t)|\,dt < \varepsilon.$$
\end{theorem}
We would like to make the observation that in Theorem \ref{th-1.2} the
weight $r(t)$ need not necessarily be a stepwise function (like $q$) and
it may be as smooth as we like.
For the sake of simplicity in the exposition, we have confined ourselves
to the study of a weight function $q(t)$ having only one jump in
$[0,T).$ The same method and argument of proof can be adapted (at the
expense of a more cumbersome treatment) to the case in which $q(t)$ has
more than one jump in $[0,T).$
The conclusions in Theorem \ref{th-1.1} and Theorem \ref{th-1.2} hold
good if $0 < B < A,$ provided that we suitably reverse the roles of
$\alpha^*_m$ and $\beta^*$ to some new constants $\alpha^*$ and
$\beta^*_m.$
Before starting with the proof of Theorem \ref{th-1.1} and its variant
Theorem \ref{th-1.2}, we have to explain the sense in which we mean ``
chaos '' and we need also to present the topological tools we use to
demonstrate chaotic dynamics. This is done in the next section.
\section{Topological tools}\label{sec-2} The definition of chaotic
dynamics considered in our work for a map $\psi$ corresponds to the
concept of chaos in the coin-tossing sense, as explained by Smale in
\cite{Sm-98}. We find $m\geq 2$ pairwise disjoint compact sets that we
label with
$m$ symbols and prove that given any itinerary of $m$-symbols there is
some point which, under the iterates of $\psi,$ follows the same
itinerary. For example, in the case of $m=2,$ we have two disjoint
compact sets ${\mathcal K}_0$ and ${\mathcal K}_1$ such that if we fix
any two-sided sequence of symbols $(s_i)_{i\in{\mathbb Z}}$ with $s_i\in
\{0,1\},$ then there exists at least one point $w\in \mbox{dom}(\psi)$
such that $\psi^{i}(w)$ belongs to ${\mathcal K}_0$ or to ${\mathcal K}_1$
according as the $i$-th symbol is $s_i=0$ or $s_i=1.$ Our definition puts
special emphasis on periodic points which arise from periodic sequences
of symbols. This is of special importance for the applications to ODEs,
where the map $\psi$ is the Poincar\'{e} map $\Phi$ (defined above) and
periodic points correspond to subharmonic solutions. Formally, our
definition can be stated as the following.
\begin{definition}\label{def-1.1} Let $X$ be a metric space, let $\psi:
X\supseteq D_{\psi} \to X$ be a map and ${\mathcal D}\subseteq D_{\psi}$
a nonempty set. Assume also that $m\geq 2$ is an integer. We say that
\textit{$\psi$ induces chaotic dynamics on $m$ symbols in the set
${\mathcal D}$} if there exist $m$ nonempty pairwise disjoint compact sets
$ {\mathcal K}_0, {\mathcal K}_1,\dots, {\mathcal K}_{m-1}\,\subseteq
{\mathcal D} $ such that, for each two--sided sequence $(s_i)_{i\in
{\mathbb Z}} \in \Sigma_m:= \{0,\dots,m-1\}^{\mathbb Z},$ there exists a
corresponding sequence $(w_i)_{i\in {\mathbb Z}}\in {\mathcal D}^{\mathbb
Z}$ with
\begin{equation}\label{eq-1.ct1} w_i \,\in\, {\mathcal K}_{s_i}\,\quad
\mbox{ and }\; w_{i+1} = \psi(w_i),\;\; \forall\, i\in {\mathbb Z},
\end{equation} and, whenever $(s_i)_{i\in {\mathbb Z}}$ is a $k$-periodic
sequence of $m$ symbols $($that is,
$s_{i+k} = s_i$ for all $i\in {\mathbb Z} )$ for some $k\geq 1,$ there
exists a $k$-periodic sequence of points $(w_i)_{i\in {\mathbb Z}}\in
{\mathcal D}^{\mathbb Z}$ satisfying $(\ref{eq-1.ct1}).$
\end{definition}
Definition \ref{def-1.1} agrees with other ones considered in the
literature about chaotic dynamics for ODEs with periodic coefficients
(see \cite{CaDaPa-02, SrWo-97, WoZg-00}). We notice that if the map
$\psi$ fulfills Definition \ref{def-1.1} and is \textit{continuous and
injective on}
$${\mathcal K}:=\bigcup_{i=0}^{m-1}{\mathcal K}_i
\subseteq {\mathcal D}$$ (as in the case of a Poincar\'{e} map), then
there exists a nonempty compact set
$ \Lambda\subseteq {\mathcal K}$ which is invariant for $\psi$ and such
that $\psi|_{\Lambda}$ is semi-conjugate, by a continuous surjection $g,$
to the two-sided Bernoulli shift
$\sigma: \Sigma_m\to \Sigma_m$ on
$m$ symbols, $\sigma: (s_i)_{i\in{\mathbb Z}}\mapsto
(s_{i+1})_{i\in{\mathbb Z}}.$ Moreover, the subset of $\Lambda$ made by
the periodic points of $\psi$ is dense in
$\Lambda$ and the counterimage by $g$ of any periodic sequence in
$\Sigma_m$ contains a periodic point of $\psi.$ Recall also that the
semiconjugation to the Bernoulli shift is one of the typical requirements
for chaotic dynamics as it implies a positive topological entropy for the
map
$\psi|_{\Lambda}.$
In order to produce chaos as put forth in Definition \ref{def-1.1}, we
employ a topological method that we name ``stretching along the paths''
which is based on some results developed in \cite{PaZa-04a,PaZa-04b}
and further refined in \cite{PaZa-08}.
Let $X$ be a metric space. By a \textit{path} in $X$ we mean a continuous
mapping $\gamma: {\mathbb R}\supseteq [a,b]\to X.$ We set
${\bar{\gamma}}:= \gamma([a,b]).$ Also, without loss of generality, we
usually take
$[a,b]=[0,1].$ By a \textit{sub--path} we mean the restriction of
$\gamma$ to a compact subinterval of its domain. An \textit{arc} is the
homeomorphic image of the compact interval $[0,1].$ A subset ${\mathcal
R}\subseteq X$ is called a \textit{generalized rectangle} if there is a
homeomorphism $\eta$ of
$[0,1]^2 \subseteq {\mathbb R}^2$ onto
${\mathcal R} \subseteq X$. Given $\eta$ we define the \textit{contour
of} ${\mathcal R}$ as
$$\vartheta{\mathcal R}:= \eta(\partial [0,1]^2).$$ Notice that the set
denoted by $\vartheta{\mathcal R}$ is not necessarily the boundary of
${\mathcal R}$ in the usual sense (for instance, if ${\mathcal R}=X$ the
boundary is empty). In our applications, however,
$X\equiv{\mathbb R}^2$ and hence $\vartheta \equiv \partial.$ We also
define an \textit{oriented rectangle} in $X,$ as a pair
$ {\widetilde{\mathcal R}}:= ({\mathcal R},{\mathcal R}^-), $ where
$ {\mathcal R}^-:={\mathcal R}^-_l \cup {\mathcal R}^-_r $ and
${\mathcal R}^-_l, {\mathcal R}^-_r \subseteq \vartheta
\mathcal R$ are two disjoint arcs. The indices ``$l$'' and ``$r$'' for
the two components ${\mathcal R}^-_l$ and ${\mathcal R}^-_r$ of $\mathcal
R^-$ stand for ``left'' and ``right'' respectively. In the same manner,
we can define a ``base'' ${\mathcal R}^+_{b}$ and a ``top'' ${\mathcal
R}^+_{t}$ components contained in
$\vartheta{\mathcal R}.$ Formally, we define ${\mathcal R}^+$ as the
closure of the set $\vartheta{\mathcal R}\setminus {\mathcal R}^-$ and
then we decompose it as
$ {\mathcal R}^+:= {\mathcal R}^+_{b} \cup {\mathcal R}^+_{t},$ with
${\mathcal R}^+_{b}$ and ${\mathcal R}^+_{t}$ two compact disjoint arcs.
Suppose now that $\chi: X \supseteq D_{\chi}\to X$ is a map
(not necessarily continuous on its whole domain $D_{\chi}$) and let
${\widetilde{\mathcal A}}:= ({\mathcal A},{\mathcal A}^-)$ and
${\widetilde{\mathcal B}}:= ({\mathcal B},{\mathcal B}^-)$ be oriented
rectangles in the metric space $X.$ Let also ${\mathcal H}\subseteq
{\mathcal A}\cap D_{\chi}$ be a compact set.
\begin{definition}\label{def-1.2} We say that \textit{$({\mathcal
H},\chi)$ stretches ${\widetilde{\mathcal A}}$ to ${\widetilde{\mathcal
B}}$ along the paths} and write
$ ({\mathcal H},\chi): {\widetilde{\mathcal A}} \stretchx
{\widetilde{\mathcal B}}, $ if the following conditions hold:
\begin{itemize}
\item{} $\chi$ is continuous on ${\mathcal H};$
\item{} for every path $\gamma: [0,1]\to {\mathcal A}$ such that
$\gamma(0)\in {\mathcal A}^-_l$ and $\gamma(1)\in {\mathcal A}^-_r$ (or
$\gamma(0)\in {\mathcal A}^-_r$ and $\gamma(1)\in {\mathcal A}^-_l$),
there exists a subinterval $[t',t'']\subseteq [0,1]$ such that
$ \gamma(t)\in {\mathcal H},\quad \chi(\gamma(t))\in {\mathcal B},$
for all $t\in [t',t'']$ and, moreover, $\chi(\gamma(t'))$ and
$\chi(\gamma(t''))$ belong to different components of
${\mathcal B}^-.$
\end{itemize} In the special case in which ${\mathcal H} = {\mathcal
A}\subseteq D_{\chi},$ we simply write
$ \chi: {\widetilde{\mathcal A}} \stretchx {\widetilde{\mathcal B}}. $
\end{definition}
The stretching property is related to the theory of \textit{topological
horseshoes}
\cite{BuWe-95,KeYo-01,ZgGi-04} and therefore provides us the proper tool
for the proof of chaotic dynamics without the necessity of checking
conditions of hyperbolicity as required in the classical applications of
Smale's theory (see \cite{Mo-73}).
To prove the main results of the paper, we use the following Theorem
\ref{th-1.main}. For notational convenience, in what follows, we have
designated the basic set of $m$-symbols as $\{1,\dots,m\}$ instead of
$\{0,\dots,m-1\}.$
\begin{theorem}[\cite{PaZa-08}]\label{th-1.main} Let $\psi_r\,:X\supseteq
D_{\psi_r}\to X$ and $\psi_s\,:X\supseteq D_{\psi_s}\to X$ be given maps.
Let
${\widetilde{\mathcal M}} := ({\mathcal M},{\mathcal M}^-)$ and
${\widetilde{\mathcal N}} := ({\mathcal N},{\mathcal N}^-)$ be oriented
rectangles in $X$ with ${\mathcal M}\subseteq D_{\psi_r}$ and ${\mathcal
N}\subseteq D_{\psi_s}.$ Suppose that the following conditions are
satisfied:
\begin{itemize}
\item[${\bf{(H_r)}}$] There exists $m\geq 2$ pairwise disjoint
compact sets
${\mathcal K}_1, \dots,{\mathcal K}_m\subseteq {\mathcal M}$ such that
$ ({\mathcal K}_i,\psi_r): {\widetilde{\mathcal M}}
\stretchx {\widetilde{\mathcal N}},\;\;\forall\, i=1,\dots,m; $
\item[${\bf{(H_s)}}$]
$\psi_s\,: {\widetilde{\mathcal N}} \stretchx {\widetilde{\mathcal M}}.$
\end{itemize} Then the map $\psi:=\psi_s\circ \psi_r$ induces chaotic
dynamics on $m$ symbols in the set
$${\mathcal K}:= \bigcup_{i=1}^m {\mathcal K}'_i,
\quad\mbox{with } {\mathcal K}'_i:= {\mathcal K}_i\cap
{\psi_r}^{-1}({\mathcal N}).$$ Moreover, for each sequence of $m$ symbols
$\textbf{s}=(s_n)_n\in \{1,\dots,m\}^{\mathbb N},$ there exists a compact
connected set ${\mathcal C}_{\textbf{s}}\subseteq {\mathcal K}_{s_0}$ with
$ {\mathcal C}_{\textbf{s}}\cap {\mathcal M}^+_{b}\not=\emptyset,\quad
{\mathcal C}_{\textbf{s}}\cap {\mathcal M}^+_{t}\not=\emptyset
$ and such that, for every $w\in {\mathcal C}_{\textbf{s}}$ there exists
a sequence $(y_n)_{n}$ with $y_0 = w$ and
$$y_n \in {\mathcal K}_{s_n},\quad \psi(y_n) = y_{n+1},\;\forall\,
n\geq 0.$$
\end{theorem} See \cite{PaZa-08} for a proof. In the applications of
Theorem \ref{th-1.main} to the ODE models considered in the present
paper, we take $X\equiv{\mathbb R}^2$ and the maps $\psi_r,\psi_s$ to
be the Poincar\'{e} maps associated with some planar systems for
different time intervals.
\section{A graphical description}\label{sec-3} Just to have a graphical
feel of the problem, we now consider a specific example of Eq.
$(\ref{eq-1.2})$ choosing
$$f(x) = \sin(\pi x).$$ In this case, condition $(\textbf{H})$ is
satisfied with $a=-1$ and $b=1.$ For this particular example we choose a
periodic function $f(x),$ but we would like to stress that this is not a
condition which is required for our results. Indeed, we confine our
attention only to the strip $[-1,1]\times{\mathbb R}.$
The weight function $q(t)$ is as in $(\ref{eq-1.4})$ with $0 < A < B.$
The differential system $(\ref{eq-1.5})$ is given by the superposition of
the two systems
\begin{equation}\label{eq-3.1} ({I}_{A}):\quad \left\{
\begin{array}{llllll}
\dot x& = &y \\ \dot y &= - &A \,\sin(\pi
x)\end{array}\right.\hspace{2cm} \mbox{for}\,\,t \,\in \, [0,\tau_A ) ,
\end{equation} and
\begin{equation}\label{eq-3.2} ({I}_{B}):\quad \left\{
\begin{array}{llll}
\dot x& = &y \\ \dot y &= -& B\,\sin(\pi
x)\end{array}\right.\hspace{2cm} \mbox{for}\,\,t \,\in \,
[\tau_A,\tau_A+\tau_B).
\end{equation}
The global dynamics is as follows: Along the interval $[0,\tau_A]$ we
consider the trajectories of the system
$({I}_{A})$ and at time $t= \tau_A$ we switch to the system $({I}_{B})$.
Without loss of generality we can assume that $[0,\tau_B] \equiv
[\tau_A,\tau_A+\tau_B]$
(the systems being autonomous and hence invariant for time shift).
Some trajectories of $({I}_{A})\; and \;({I}_{B})$ are shown in Figure
\ref{fig:1}.
\begin{figure}[htp]
\begin{center}
\subfigure[The first system.]
{\label{fig:1a}
\includegraphics[scale=0.5]{level1}
}
\qquad \subfigure[The second system.]
{\label{fig:1b}
\includegraphics[scale=0.5]{level2}
}
\end{center}
\vskip -5pt
\caption{Some energy level lines of the two systems
$(I_{A})\, \& \, (I_{B})\,$ for $A=2\pi,$ $B=5\pi,$ with the equilibrium
points $(0,0)$ and $(\pm 1,0),$ marked.}\label{fig:1}\end{figure}
A superimposition of the phase portraits of the two systems shows the
presence of linked annuli from which we construct the oriented rectangles
explained earlier. Such oriented rectangles are obtained by intersecting
a region included between a pair of level lines of one system with a
region between a pair of level lines of the other. The intersection of
these two sets gives rise to two regions $\mathcal{P}$ and $\mathcal{Q}$
with no interior points in common as shown in Figure \ref{fig:2}.
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=1]{2regions}
\end{center}\caption{The
linked annuli determining the two regions $\mathcal{P}$ (the upper) and
$\mathcal{Q}$ (the lower).} \label{fig:2}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[The upper region
$\mathcal{P}.$]
{
\label{fig:3a}
\includegraphics[scale=0.5]{upper}
}\qquad
\subfigure[The lower region
$\mathcal{Q}.$]{\label{fig:3b}\includegraphics[scale=0.5]{lower}}
\end{center}
\vskip -5pt
\caption{The two regions $\mathcal{P}$ and $\mathcal{Q}$ determined by
the two systems of level lines are then oriented, by suitably
choosing the $[\cdot]^-$ -sets.}
\label{fig:3}
\end{figure}
Following the theory developed above, we take an arbitrary path $\gamma$
in the region $\mathcal{P},$ from
$\mathcal{P}^{-}_{l}$ to $\mathcal{P}^{-}_{r}$, that is,
$$\gamma : [0,1] \rightarrow \mathcal{P},\quad\mbox{with } \; \gamma(0)
\in
\mathcal{P}^{-}_{l} \; \mbox{and}\; \gamma(1) \in
\mathcal{P}^{-}_{r}$$ (see Figure \ref{fig:4}), and we consider its
deformation in the phase--plane under the action of the Poincar\'{e} map.
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.5]{combined}
\end{center}
\caption{The path $\gamma$ in the region $\mathcal{P}.$}
\label{fig:4}
\end{figure}
Let $\psi_A $ be the Poincar\'{e} map associated to the system
$({I}_{A})$ for the time interval $[0,\tau_A]$ and $\psi_B$ be the
Poincar\'{e} map associated to the system $({I}_{B})$ for the time
interval $[0,\tau_B].$ We observe that during the evolution of the path
$\gamma$ under the influence of the Poincar\'{e} map $\psi_A$ the points
of ${\mathcal P}^-_{r},$ which lie on the heteroclinic orbit of
$({I}_{A})$ \footnote{that is, the orbit in the upper half plane
connecting $(-1,0)$ to
$(1,0)$ (see first figure in Figure \ref{fig:1}).}, do not move beyond
the point $(1,0)$ in the phase--plane, because $(1,0)$ is an equilibrium
point of the system $({I}_{A}).$ Hence the corresponding angular
coordinates remain between $0$ and $\pi/2$.
On the other hand, the points of ${\mathcal P}^-_{l},$ which lie on a
periodic orbit of $({I}_{A}),$ wind around the origin
$(0,0),$ in the phase--plane. The corresponding number of turns around
the origin increases with $\tau_{A}.$
The curve $\gamma$ goes from $\gamma(0)$ to $\gamma(1)$ in the region
$\mathcal{P},$ with $\gamma(1)$ on the heteroclinic phase--curve and
$\gamma(0)$ on a closed periodic phase--curve (see Figure \ref{fig:4}). The
Figures \ref{fig:5a} and \ref{fig:5b} show the deformation of $\gamma$
with time. Under the action of the Poincar\'{e} map $\psi_{A},$ the
point $\gamma(0)$ winds around the origin while $\gamma(1)$ does not move
beyond the point
$(1,0)$ (independent of the length of the time interval). Thus, the points
on $\gamma$ towards $\gamma(0)$ wind faster than points towards
$\gamma(1),$ causing the curve to undergo a stretching. In any case the
points of $\psi_A(\gamma(s))$ for $s\in [0,1]$ remain in the invariant
region between the two level lines of system
$I_{A}$ of Figure \ref{fig:1a}, as can be seen in the next Figures
\ref{fig:5a} and \ref{fig:5b}.
The resulting image of $\gamma$ is that of a spiral if $\tau_A$ is
sufficiently large. After a certain time this spiral executes at least
two crossings of the region $\mathcal{Q}$ from
$\mathcal{Q}^{-}_{l} $ to $\mathcal{Q}^{-}_{r},$ satisfying the condition
${\bf{(H_r)}}$ with the choice $\psi_r\equiv \psi_A$ and
${\mathcal M}\equiv {\mathcal P},$ ${\mathcal N}\equiv {\mathcal Q}$.
\begin{figure}[htp]
\begin{center}
\subfigure[$\gamma$ after a short interval of time
($t=0.2$).]{
\label{fig:5a}
\includegraphics[scale=0.5]{curve02}}\qquad
\subfigure[$\gamma$ after a longer interval of time
($t=5.5$).]{
\label{fig:5b}
\includegraphics[scale=0.5]{secondcross}}
\end{center}
\vskip -5pt
\caption{The stretching of $\gamma$ with time.}
\label{fig:5}
\end{figure}
In order that the condition ${\bf{(H_s)}}$ be satisfied for
$\psi_s\equiv \psi_B\,$, we repeat the same argument starting from a path
in ${\mathcal Q}$ joining the two components of ${\mathcal Q}^-.$ In this
case, we just need to show that the image of the path by the Poincar\'{e}
map $\psi_B$ crosses the region
${\mathcal P}$ once. Hence, a curve in ${\mathcal Q}$ is now considered
and we look for its deformation under the influence of the second system.
Compared to the previous case (i.e., system
$I_A$) now the underlying geometry is different. The two level lines
considered for system $I_{B}$ are both periodic (recall Figure
\ref{fig:1b}). But, again the curve undergoes a stretching since the
points on the outer trajectory move slower than the points on the inner
orbit. Therefore, after sufficient time has elapsed, there is a
deformation of the curve and the required crossing of the region
$\mathcal {P}$ is achieved. Figure
\ref{fig:6} shows the crossing.
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{finalstretch}
\end{center}
\caption{The path in ${\mathcal Q}$ under the mapping $\psi_B$
crosses the region $\mathcal{P}$ after a sufficiently long time
($t=12$).}
\label{fig:6}
\end{figure}
As a final comment we note that from this numerical example it can be
seen that the lower bounds $\alpha^*_m < \tau_A$ and $\beta^* < \tau_B$
are not necessarily large.
Having visualized our argument in the proof for the particular case $f(x)
= \sin(\pi x),$ we proceed with the analysis for a general $f(x),$ which
also justifies the graphical ``proof'' described by the above images.
\section{Technical preliminaries}\label{sec-4} In this section, we show
how the topological lemmas stated in Section \ref{sec-2} are applied to
the Poincar\'{e} map $\Phi$ associated to system $(\ref{eq-1.5}).$ Here
and henceforth, we assume that $f: {\mathbb R}\to {\mathbb R}$ is a
locally Lipschitz function satisfying condition $(\textbf{H})$ and $q:
{\mathbb R}\to {\mathbb R}$ is a $T$-periodic function as in
$(\ref{eq-1.4}),$ with
$$\tau_A + \tau_B = T \quad\mbox{and }\; 0 < A < B.$$ Due to the special
form of the weight function $q(t),$ it is convenient to study the
auxiliary autonomous Hamiltonian system
$$ (J_{\mu})\qquad\left\{
\begin{array}{llll} &\dot x = &y\\ &\dot y = - &\mu f(x),
\end{array}
\right.
$$ with $\mu > 0$ treated as a parameter. If we denote by
$\Phi_{\mu}$ the Poincar\'{e} map for system $(J_{\mu}),$ we find the
relation
\begin{equation}\label{eq-4.1}
\Phi = \Phi_{B}\circ\Phi_{A}.
\end{equation} The plan of the proof is to show that Theorem
\ref{th-1.main} applies for $\Phi$ with $\psi_{r} =\Phi_{A}$ and $\psi_{s}
=\Phi_{B}$ for a suitable choice of oriented rectangles. As described in
the visual examples of Section \ref{sec-3}, such oriented rectangles will
be obtained by the intersection of invariant regions between the level lines
of the systems $(J_{A})$ and
$(J_{B}).$ To this end, we first study the energy level lines associated
to $(J_{\mu}),$ focusing our attention to the strip
$ {\mathcal S}:= [a,b]\times{\mathbb R}, $ where, according to hypothesis
$(\textbf{H}),$
$$a < 0 < b,\quad f(a) = f(0) = f(b) =0.$$ Since $(J_{\mu})$ is a
conservative system having as first integral the energy function
$$E(x,y) = E_{\mu}(x,y):= \frac{1}{2} y^2 + \mu F(x),
\quad \mbox{with } \; F(x):= \int_0^x f(s)\,ds,$$ the trajectories of
$(J_{\mu})$ lie on energy level lines of constant energy. We denote by
$ \Gamma^{e}= \Gamma^{e}_{\mu} := \{(x,y)\in {\mathcal S}:\, E_{\mu}(x,y)
= e\} $ the part of the level line of energy $e$ which is contained in
${\mathcal S}.$ We also introduce the critical energy (or the energy of
the separatrices)
$$e_{\rm crit} = e_{\mu, \rm crit}:= \mu F(b).$$ Recall that
$$f(x) < 0\;\; \forall \, x\in (a,0),\quad f(x) > 0\;\;
\forall \, x\in (0,b ),\quad
\int_a^b f(s)\,ds =0,$$ and, hence, $F$ is strictly decreasing on $[a,0]$
and strictly
increasing on $[0,b],$ with $e_{\rm crit}= \mu F(a) = \mu F(b) > 0.$
Now, for every $\mu > 0,$ we define the set
$${\mathcal W}= {\mathcal W}_{\mu}:= \{(x,y)\in {\mathcal S}:\,
E_{\mu}(x,y) \leq e_{\mu, \rm crit}\},$$ which is a compact subset of
${\mathcal S}$ which is contained in the rectangle
$[a,b]\times[-d_{\mu},d_{\mu}],$ where
$ d_{\mu}:= \sqrt{\mu}\,\sqrt{2 F(b)} $ is the maximal height of
$\Gamma\,^{ e_{\rm crit}}_{\mu}.$
The boundary of ${\mathcal W}$ can be represented as
$$\partial{\mathcal W}_{\mu} = \{(x,y)\in {\mathcal S}:\, E_{\mu}(x,y) =
e_{\mu, \rm crit}\} = \Gamma\,^{ e_{\rm crit}}_{\mu} = {\mathcal
O}^+_{\mu} \cup {\mathcal O}^-_{\mu} \cup\{(a,0), (b,0)\},$$ where
${\mathcal O}^+_{\mu}$ and ${\mathcal O}^-_{\mu}$ are the heteroclinic
orbits
$${\mathcal O}^+_{\mu}:=\{(x,y)\in{\mathcal S}: y> 0,\, E_{\mu}(x,y) =
e_{\mu, \rm crit}\},$$ and
$${\mathcal O}^-_{\mu}:=\{(x,y)\in{\mathcal S}: y< 0,\, E_{\mu}(x,y) =
e_{\mu, \rm crit}\}.$$ The trajectory ${\mathcal O}^+$ connects $(a,0)$
to $(b,0)$
in the upper half of the phase--plane and ${\mathcal O}^-$ goes from
$(b,0)$ to $(a,0)$ in the lower half of the phase--plane (see, as an
example, the external
trajectories in Figure \ref{fig:1a}).
For every fixed value $e\in {\mathbb R},$ with
$ 0 < e < e_{\mu, \rm crit}, $ the level line $\Gamma^{e}$ is a
periodic orbit of system
$(J_{\mu})$ (see, as an example, the internal trajectory in Figure
\ref{fig:1a} and the two orbits in Figure \ref{fig:1b}). We denote by
${\mathcal T}_{\mu}(e)$ the fundamental period of
$\Gamma^{e}_{\mu}.$ Such minimal period can be computed by the
time-mapping formula \cite[Chapter V.1]{Ha}
$${\mathcal T}_{\mu}(e):= 2 \left( T^+_{\mu}(e) + T^-_{\mu}(e)\right),$$
with
$$T^+_{\mu}(e):=\frac{1}{\sqrt{\mu}}\,\int_{0}^{x^+(e)}
\frac{ds}{\sqrt{2( F(x^+(e)) - F(s))}},$$ and
$$T^-_{\mu}(e):=\frac{1}{\sqrt{\mu}}\,\int_{x^-(e)}^{0}
\frac{ds}{\sqrt{2( F(x^-(e)) - F(s))}},$$ where
$ a < x^-(e) < 0 < x^+(e) < b,$ $ F(x^-(e)) = F(x^+(e)) =
\frac{e}{\mu}. $ The number $T^+_{\mu}(e)$ is equal to the time needed
for a solution of $(J_{\mu})$ to make a quarter lap in the clockwise
sense from the $y$-axis to the $x$-axis along the level line $\Gamma^{e}$
in the first quadrant. It coincides with the time needed for the same
solution to make a quarter lap in the clockwise sense from the $x$-axis
to the $y$-axis in the fourth quadrant. Similar considerations can be
made for $T^-_{\mu}(e).$
For any fixed $\mu > 0,$ the ``period'' time--mapping
$$ ( 0,e_{\mu,\rm crit} ) \ni e\mapsto {\mathcal
T}_{\mu}(e)\in ( 0,+\infty)$$ is continuous and such that ${\mathcal
T}_{\mu}(e)\to +\infty$ as $e\to e_{\mu,\rm crit}.$ Without further
assumptions on $f$ we cannot guarantee that ${\mathcal T}_{\mu}(\cdot)$
is monotone increasing (as in the case of the classical pendulum
equation). For instance, we could have that
${\mathcal T}_{\mu}(e)\to +\infty$ as $e\to 0^+.$ However, in any case,
we have that there exists $\delta \in\, (0,e_{\mu,\rm crit})$ such that
for every $\varepsilon >0,$ there is
$\nu\in \, (e_{\mu,\rm crit}-\varepsilon, e_{\mu,\rm crit})$ such that
\begin{equation}\label{eq-nu} {\mathcal T}_{\mu}(e) < {\mathcal
T}_{\mu}(\nu),\quad\forall\, e\in \, [\delta,\nu ).
\end{equation}
If we introduce polar coordinates into the system $(J_{\mu})$ with pole
at $(0,0),$ any solution $(x(t),y(t))$ of equation $(J_{\mu})$ with
initial point in ${\mathcal W}_{\mu}\setminus\{(0,0)\}$ can be written as
$$x(t) = \rho(t)\cos\theta(t),\;\; y(t) = \rho(t)\sin\theta(t),$$ and we
have that
$$-\dot\theta(t) = \frac{y(t)^2 + \mu f(x(t))\,x(t)}{x(t)^2 + y(t)^2}.$$
We denote by $\theta(\cdot,z)$ the angular coordinate associated to the
solution
$(x(\cdot,z),$ $y(\cdot,z))$ of $(J_{\mu})$ with $(x(0),y(0))=z\in
{\mathcal W}_{\mu}\setminus\{(0,0)\}.$ Accordingly, we have
$$\theta(0,z) - \theta(t,z) = \int_0^t \frac{y(s,z)^2 + \mu
f(x(s,z))\,x(s)}{x(s,z)^2 + y(s,z)^2}\,ds.$$ We observe that, for $t\geq
0$ and
$ e= E(z)\in ( 0,e_{\mu, \rm crit} ), $ it holds that
$\theta(0,z) - \theta(t,z)= 2 k \pi$ (with $k$ a nonnegative integer) if
and only if
$t = k {\mathcal T}_{\mu}(e).$ More precisely, we have that
$$\theta(t,z) \lesseqgtr \theta(0,z) - 2 k \pi\quad\mbox{if and only if
}\; t \gtreqless k {\mathcal T}_{\mu}(e).
$$ We now fix a level
$ e_0 \in ( 0,e_{\mu, \rm crit} ) $ and focus our attention on the region
in the strip ${\mathcal S}$ between
$\Gamma^{e_0}$ and $\Gamma^{e}_{\mu, \rm crit},$ and define the set
$${\mathcal W}_{e_0}={\mathcal W_{e_0,\mu}}:=\{(x,y)\in {\mathcal S}: e_0
\leq E_{\mu}(x,y) \leq e_{\mu, \rm crit}\}.$$ Notice that the set
${\mathcal W_{e_0,\mu}}$ is invariant; that is, every solution
$(x(t),y(t))$ of equation $(J_{\mu})$ with initial point $(x(0),y(0))
\! \in \!
{\mathcal W_{e_0,\mu}},$ remains in ${\mathcal W_{e_0,\mu}}$ for all~$t.$
We also introduce the sets
$${\mathcal U}:=\{(x,y)\in {\mathcal S}: e_0 \leq E(x,y) \leq e_{\mu, \rm
crit},\;\, x\geq 0, \, y\geq 0\},$$
$${\mathcal V}:=\{(x,y)\in {\mathcal S}: e_0 \leq E(x,y) \leq e_{\mu, \rm
crit}, \;\, x\geq 0, \, y\leq 0\}.$$
\begin{figure}
\begin{center}
\includegraphics[scale=1.]{UV1}
\end{center}
\caption{The regions ${\mathcal U}$ and ${\mathcal V}$
for $f(x) = \sin(\pi x).$}
\label{fig:7}
\end{figure}
We are interested in those solutions with initial points in
${\mathcal U},$ for which we assume
$ \theta(0,z)\in [0,\pi/2].$ Note that, for $e=e_{\mu, \rm crit},$ the
solution lies on the heteroclinic orbit ${\mathcal O}^+$ and therefore
$$0 \leq x(0,z) \leq x(t,z) \leq b,\;\; y(0,z) \geq y(t,z) \geq 0,\quad
\forall \, t\geq 0.$$ Hence,
$ \theta(t,z) \geq 0,$ for all $t\geq 0.$ On the other hand, for
$e=e_0,$ the solution lies on the periodic orbit $\Gamma^{e_0}$ and
therefore
$$\theta(t,z) \leq \theta(0,z) -2\pi\, \Big \lfloor
\frac{t}{{\mathcal T}_{\mu}(e_0)} \Big \rfloor.$$ Hence,
$$\theta(t,z) \leq -\frac{3\pi}{2} - 2 \Big ( \Big \lfloor
\frac{t}{{\mathcal T}_{\mu}(e_0)} \Big \rfloor -1 \Big )\pi
< -\frac{\pi}{2} - 2 \Big ( \Big \lfloor
\frac{t}{{\mathcal T}_{\mu}(e_0)} \Big \rfloor -1 \Big )\pi.
$$ For each positive integer $i,$ we define the compact set
\begin{equation}\label{eq-3.1ht} {\mathcal H}_i(t):= \{z\in {\mathcal U}:
\, \theta(t,z)\in [-\pi/2 - 2(i-1)\pi,-2(i-1)\pi]\,\}.
\end{equation} For $i\geq 1$ and $t > 0$ fixed, we have that
$z\in {\mathcal H}_i(t)$ if and only if the solution of $(\ref{eq-3.1})$
with
$(x(0),y(0))=z\in{\mathcal U}$ performs
$i-1$ turns in the clockwise sense
around the origin and ends in the region
${\mathcal V}$ at the time $t.$ By definition and the well-defined nature
of the angular function, we have that
$${\mathcal H}_i(t)\cap {\mathcal H}_j(t) =\emptyset,\quad \mbox{for }\;
i\not=j.$$
If, for some $m\geq 2,$
\begin{equation}\label{eq-tA} t\geq m{\mathcal T}_{\mu}(e_0),
\end{equation} then
\begin{equation}\label{eq-4.2}
\theta(t,z) < -\frac{\pi}{2} - 2(m-1)\pi,
\end{equation} and the sets ${\mathcal H}_1,\dots {\mathcal H}_m$ are
all nonempty.
As a next step, we fix two levels
$ e_1, e_2 \in (0,e_{\mu, \rm crit} ),$ with $ e_1 < e_2, $
we consider the region in the strip ${\mathcal S}$ between
$\Gamma^{e_1}$ and $\Gamma^{e_2}$ and define the set
$${\mathcal W}_{e_1}^{{}\,e_2}={{\mathcal W}_{e_1,}^{{}
\,e_2}}_{\mu}:=\{(x,y)\in {\mathcal S}: e_1 \leq E_{\mu}(x,y) \leq
e_2\}.$$ The set ${\mathcal W}_{e_1}^{{}\,e_2}$ is invariant for
the flow associated to
$(J_{\mu}).$ We also introduce the sets
$$\Omega:=\{(x,y)\in {\mathcal S}: e_1 \leq E(x,y) \leq e_2,\;\, x\geq
0, \, y\geq 0\},$$
$$\Xi:=\{(x,y)\in {\mathcal S}: e_1\leq E(x,y) \leq e_2,
\;\, x\geq 0, \, y\leq 0\}.$$
\begin{figure}
\begin{center}
\includegraphics[scale=1.]{ox1}
\end{center}
\caption{The regions $\Omega$ and $\Xi$
for $f(x) = \sin(\pi x).$}
\label{fig:8}
\end{figure}
We are interested in those solutions with initial points in $\Xi,$ for
which we assume
$ \theta(0,z)\in [-\pi/2,0].$ For every $z\in \Xi$ the solution of
$(J_{\mu})$ starting at $z,$ for $t=0,$ lies on the closed orbit
$\Gamma^{e}\in {\mathcal W}_{e_1}^{{}\,e_2},$ for
$e = E_{\mu}(z)\in [e_1,e_2]$ and is periodic of period ${\mathcal
T}_{\mu}(e).$ Therefore, as observed before,
$$\theta(0,z) -2\pi \Big \lceil
\frac{t}{{\mathcal T}_{\mu}(e)} \Big \rceil\leq
\theta(t,z) \leq \theta(0,z) -2\pi \Big \lfloor
\frac{t}{{\mathcal T}_{\mu}(e)} \Big \rfloor.$$ Suppose now that $z_1$
and $z_2$ are two arbitrarily chosen points in $\Xi$ with
$z_1\in \Gamma^{e_1}$ and $z_2\in \Gamma^{e_2}.$
We also assume that
$ {\mathcal T}_{\mu}(e_1) < {\mathcal T}_{\mu}(e_2).
$ We claim that, for every $t>0$ sufficiently large,
there exists $k\in {\mathbb Z}$ such that
\begin{equation}\label{eq-kpi} [\theta(t,z_1),\theta(t,z_2)]\supseteq
[2k\pi, \frac{\pi}{2} + 2k\pi];
\end{equation} that is, the angular displacement between
$\theta(t,z_1)$ and $\theta(t,z_2)$ covers the angular extent of the
region $\Omega.$
In order to be sure that equation $(\ref{eq-kpi})$
holds good we need to have that
\begin{equation}\label{eq-kpi1}
\theta(t,z_2) - \theta(t,z_1) \geq \frac{\pi}{2} + 2\pi.
\end{equation} Now, we have the following inequalities:
$$\theta(t,z_2)\geq \theta(0,z_2) - 2\pi \Big \lceil
\frac{t}{{\mathcal T}_{\mu}(e_2)} \Big \rceil> \theta(0,z_2) - 2\pi\,
\frac{t}{{\mathcal T}_{\mu}(e_2)} - 2\pi,$$ and
$$
\theta(t,z_1) \leq \theta(0,z_1) -2\pi \Big \lfloor
\frac{t}{{\mathcal T}_{\mu}(e_1)} \Big \rfloor < \theta(0,z_1) -2\pi\,
\frac{t}{{\mathcal T}_{\mu}(e_1)} + 2\pi.$$ Hence,
\begin{eqnarray*}
\theta(t,z_2) - \theta(t,z_1) &>& \theta(0,z_2) -
\theta(0,z_1) + 2\pi t \Big (\frac{1}{{\mathcal T}_{\mu}(e_1)} -
\frac{1}{{\mathcal T}_{\mu}(e_2)} \Big ) - 4\pi\\ &\geq& 2\pi t
\frac{{\mathcal T}_{\mu}(e_2) - {\mathcal T}_{\mu}(e_1)} {{\mathcal
T}_{\mu}(e_1){\mathcal T}_{\mu}(e_2)} - 4\pi - \frac{\pi}{2}.
\end{eqnarray*} Thus, we can conclude that $(\ref{eq-kpi1})$ holds if
\begin{equation}\label{eq-tB} t\geq \frac 72 \,\frac{{\mathcal
T}_{\mu}(e_1){\mathcal T}_{\mu}(e_2)} {{\mathcal T}_{\mu}(e_2) -
{\mathcal T}_{\mu}(e_1)}.
\end{equation} This completes the proof of our claim.
\section{Proof of the main results} In this section, we prove Theorem
\ref{th-1.1}. The proof of Theorem \ref{th-1.2} follows by a repetition
of the same argument of Theorem \ref{th-1.1} and observing that the
corresponding estimates are still true under small perturbations (see
\cite{PaZa-08} for a similar observation). Hence, it is omitted.
\subsection{Construction of the oriented rectangles} Choosing two
different values for the parameter $\mu$,
say $\mu = A$ and $\mu =B,$ we obtain two different differential
systems of $(J_{\mu}).$
Then the two systems could be written as
$$ (J_{A})\qquad\left\{
\begin{array}{llll} &\dot x = &y\\ &\dot y = - &A f(x)
\end{array}
\right.
\;,
\qquad (J_{B})\qquad\left\{
\begin{array}{llll} &\dot x = &y\\ &\dot y = - &B f(x)
\end{array}
\right.
\;.
$$ In order to construct the oriented rectangles, we proceed as follows:
$\bullet$ We start with the critical level line
$\Gamma\,^{e_{A, \rm crit}}$ for system $(J_A).$ Recall that it crosses
the $y$-axis at the points $(0,\pm d_A),$ with
$ d_{A}= \sqrt{A}\,\sqrt{2 F(b)}. $
$\bullet$ Next, we fix an energy level $e_1$ for system $(J_B),$ with
$ e_1 \in ( A F(b), B F(b) ) $ and consider the level line
$\Gamma^{e_1}_{B}$ of system $(J_B).$ Since $e_1 < e_{B,\rm crit} = B
F(b)$ we have that $\Gamma^{e_1}_{B}$ is a closed curve (a periodic orbit
for system $J_{B}$). It crosses the $y$-axis at the points $(0,\pm\sqrt{2
e_1})$ with $\sqrt{2 e_1} > d_{A}.$
$\bullet$ We take an energy line $\Gamma^{e_2}_{B}$ of system $(J_B)$
with $e_2$ satisfying
$$e_1 < e_2 < e_{B, \rm crit},\quad \mbox{with }\; {\mathcal
T}_{B}(e_1) < {\mathcal T}_{B}(e_2).$$ Such a choice is always possible
provided that $e_2$ is sufficiently close to
$e_{B,\rm crit}$ (as already explained in $(\ref{eq-nu})).$ Observe that
$\Gamma^{e_2}_{B}$ crosses the positive $x$-axis at a point
$(\tilde{x},0),$ with $\tilde{x}\in\, (0,b)$ and $F(\tilde{x}) = e_2/B.$
By construction, we have that
$ {\frac{A}{B} e_2 < \frac{A}{B}\,e_{B, \rm crit} =e_{A, \rm
crit}.}$
$\bullet$ Lastly, we fix, for system $(J_A),$ a level line
$\Gamma^{e_0}_{A}$ with $e_0$ satisfying
$ \frac{A}{B} e_2 \leq e_0 < e_{A, \rm crit}. $ In other words, we
choose a level line $\Gamma^{e_0}_{A}$ which intersects the positive
$x$-axis at a point $(\hat{x},0),$ with $\tilde{x}\leq \hat{x} < b.$ For
example, we could take $e_0$ so that $\Gamma^{e_0}_{A}$ and
$\Gamma^{e_2}_{B}$ intersect at the same point on the positive $x$-axis
(as in Figure \ref{fig:2}).
With the level lines chosen as explained, we consider the regions
${\mathcal W}_{e_0} = {\mathcal W}_{e_0,A}$ and ${\mathcal
W}_{e_1}^{\,e_2} ={\mathcal W}_{e_1,B}^{\,e_2}$
and take their intersection. From the sets obtained by such an
intersection we call the sets
${\mathcal P}:={\mathcal U}\cap \Omega$ and ${\mathcal Q}:={\mathcal
V}\cap \Xi$, as in Figure \ref{fig:2}. More precisely,
\begin{eqnarray*} {\mathcal P}&=& {\mathcal W}_{e_0} \cap {\mathcal
W}_{e_1}^{{}\,e_2}\,\cap \{(x,y):\, x\geq 0,\, y\geq 0\}\\ &=& \{(x,y) :
e_0 \leq E_{A}(x,y)\leq e_{A, \rm crit},\; e_1 \leq E_{B}(x,y)\leq
e_2,\; x\geq 0,\; y\geq 0 \},\\ {\mathcal Q}&=& {\mathcal W}_{e_0} \cap
{\mathcal W}_{e_1}^{{}\,e_2}\,\cap \{(x,y):\, x\geq 0,\, y\leq 0\} =
\{(x,y): (x,-y)\in {\mathcal P}\},
\end{eqnarray*} are generalized rectangles that we are going to
orient. Indeed, we set
$${\mathcal P}^-_{l}:= {\mathcal P}\cap \Gamma\,^{e_0}_A,\quad{\mathcal
P}^-_{r}:= {\mathcal P}\cap {\mathcal O}^+,\quad {\mathcal P}^-:=
{\mathcal P}^-_{l} \cup{\mathcal P}^-_{r},$$ to form the oriented
rectangle ${\widetilde{\mathcal P}}= ({\mathcal P},{\mathcal P}^-).$
Similarly, we define ${\widetilde{\mathcal Q}}= ({\mathcal Q},{\mathcal
Q}^-),$ by choosing
$ {\mathcal Q}^-_{l}:= {\mathcal Q}\cap \Gamma\,^{e_1}_B,\quad{\mathcal
Q}^-_{r}:= {\mathcal Q}\cap \Gamma\,^{e_2}_B,\quad {\mathcal Q}^-:=
{\mathcal Q}^-_{l} \cup{\mathcal Q}^-_{r}. $ A quick check for the
above statement can be made as follows: For every $\lambda\in [e_1,e_2]$
and $\sigma\in [e_0,e_{A, \rm crit}],$ we look at the intersection
between $\Gamma^{\lambda}_B$ and $\Gamma^{\sigma}_A$ in the interior of
the strip ${\mathcal S}.$ This corresponds to solving the system of
equations
\begin{equation}\label{eq-5.0} E_B(x,y) = \lambda,\quad E_A(x,y) =
\sigma,\qquad \mbox{for } \; a < x < b.
\end{equation} Such a system yields the equation
\begin{equation}\label{eq-5.1} F(x) = \frac{\lambda -\sigma}{B-A},\quad
x\in \, (a,b),
\end{equation} which has two solutions if and only if
\begin{equation}\label{eq-5.2} 0 < \frac{\lambda -\sigma}{B-A} < F(b) =
F(a).
\end{equation} The left inequality in $(\ref{eq-5.2})$ will hold provided
$\lambda > \sigma$ for each $\lambda\in [e_1,e_2]$ and $\sigma\in
[e_0,e_{A, \rm crit}].$ Thus, we are led to verify that $e_1 > e_{A, \rm
crit} = A F(b),$ which is true by our choice of $e_1.$ The right
inequality in $(\ref{eq-5.2})$ will hold provided
$\lambda - \sigma < (B-A)F(b)$ for each $\lambda\in [e_1,e_2]$ and
$\sigma\in [e_0,e_{A, \rm crit}].$ Thus, we are led to verify that $e_2 -
e_0 < (B-A)F(b),$ which is true because $e_2 - e_0 \leq e_2 - (A/B) e_2 =
(B-A)e_2/B < (B-A)\,e_{B, \rm crit}/B = (B-A)F(b).$
\noindent Thus we can conclude that $(\ref{eq-5.1})$ has a unique
positive solution given by
$x=F_{\rm right}^{-1}(\frac{\lambda -\sigma}{B-A}),$ where $F_{\rm
right}^{-1}$ is the inverse of the strictly increasing smooth function
$F$ restricted to $(0,b)$ with range onto $ ( 0,F(b) ).$ Having found
$x\in \,(0,b)$ which is the solution of $(\ref{eq-5.1}),$ we have to plug
it in any of the two equations of $(\ref{eq-5.0})$ and look for $y.$ This
gives rise to
$$\frac{1}{2} y^2 + B F(x) = \lambda, \quad F(x) = \frac{\lambda
-\sigma}{B-A},$$ which has real solutions provided $B(\lambda - \sigma)
\leq (B-A)\lambda;$ that is
$B\sigma \geq A\lambda,$ for each $\lambda\in [e_1,e_2]$ and $\sigma\in
[e_0,e_{A, \rm crit}].$ Thus, we are led to verify that $B e_0 \geq A e_2,$
which is true by our choice of $e_0.$
\noindent In conclusion, we can define the set ${\mathcal P}$ and,
respectively, ${\mathcal Q},$ by the homeomorphisms:
$$h_{\pm}: [e_1,e_2]\times [e_0,e_{A, \rm crit}]\to {\mathcal
P},{\mathcal Q}$$
$$h(\lambda,\sigma):= \Big (F_{\rm right}^{-1} \Big (\frac{\lambda
-\sigma}{B-A} \Big ),
\pm\sqrt{2\, \frac{\sigma B - \lambda A}{B-A}} \Big ).
$$
\subsection{Proof of Theorem \ref{th-1.1}}\label{subsec-5.2} Set
\begin{equation}\label{eq-5.ab}
\alpha^*_m:=m {\mathcal T}_{{A}}\,(e_0),\quad\mbox{for }
m\geq 2\qquad \mbox{and }\;
\beta^*:=\frac 72 \,\frac{{\mathcal T}_{B}(e_1) {\mathcal T}_{B}(e_2)}{{\mathcal
T}_{B}(e_2) - {\mathcal T}_{B}(e_1)}
\end{equation} (see equation $(\ref{eq-tA})$ and $(\ref{eq-tB})$).
With these choices we are now in a position to prove Theorem \ref{th-1.1}
using Theorem \ref{th-1.main}, with the following correspondences for
sets and maps:
$${\widetilde{\mathcal M}}:= {\widetilde{\mathcal P}}, \quad
{\widetilde{\mathcal N}}:= {\widetilde{\mathcal Q}},
$$
$${\mathcal K}_i:= {\mathcal H}_i(\tau_A)\cap {\mathcal P},
\quad\mbox{ for }\; i=1,\dots,m,$$
$$\psi_r:=\Phi_A,\;\;\psi_s:=\Phi_B,$$ so that the abstract map $\psi$
inducing chaotic dynamics on ${\mathcal M}$ is the Poincar\'{e} map
$\Phi$ restricted to the domain ${\mathcal P}.$ For the remainder of the
proof, we assume condition $(\ref{eq-1.1as}).$
For $q(t)$ defined as in $(\ref{eq-1.4}),$ the Poincar\'{e} map $\Phi:
z_0\mapsto \zeta(T;0,z_0)$ for system $(\ref{eq-1.5})$ splits as in
$(\ref{eq-4.1}).$ Firstly, we observe that ${\mathcal P}\subseteq
{\mathcal W}_A\subseteq \mbox{\rm dom}(\Phi_A)$ and
${\mathcal Q}\subseteq {\mathcal W}_B\subseteq\mbox{\rm dom}(\Phi_B).$
$\blacklozenge\;$ We prove ${\bf{(H_r)}}\,$ of Theorem \ref{th-1.main}.
Let $\gamma = \gamma(s): [0,1]\to {\mathcal P}$ be a continuous curve
such that $\gamma(0) \in {\mathcal P}_{l}^-$ and $\gamma(1) \in {\mathcal
P}_{r}^-.$ Since ${\mathcal P}\subseteq {\mathcal U},$ we know that
$ \theta(0,\gamma(s))\in [0,\pi/2],$ for all $s\in [0,1]. $ Observe
also that
$ E_A(\gamma(0)) = e_0$ and $ E_A(\gamma(1)) = e_{A, \rm
crit}. $ Hence, recalling also $(\ref{eq-4.2}),$ for
$ t = \tau_A > \alpha^*_m,$ we find
$$\theta(t,\gamma(0)) < - \frac{\pi}{2} - 2(m-1)\pi
\quad\mbox{and }\, \theta(t,\gamma(1))\geq 0.$$ Fix an $i\in
\{1,\dots,m\}.$ By the continuity of the map $s\mapsto
\Phi_A(t,\gamma(s)),$ there exist $[s_{1,i},s_{2,i}]\subseteq [0,1]$ such
that
$$\theta(t,\gamma(s_{1,i}))= -\pi/2 - 2(i-1)\pi,\;\;
\theta(t,\gamma(s_{2,i}))= -2(i-1)\pi,$$ and
\begin{equation}\label{eq-5.1ht}
\theta(t,\gamma(s))\in [-\pi/2 - 2(i-1)\pi,-2(i-1)\pi],\;\forall\, s\in
[s_{1,i},s_{2,i}].
\end{equation} In terms of the Poincar\'{e} map $\Phi_A,$ this means
that
$$\Phi_A(t,\gamma(s_{1,i}))\in {\mathcal V}\cap\{(x,y): x=0\},\;\;
\Phi_A(t,\gamma(s_{2,i}))\in {\mathcal V}\cap\{(x,y): y=0\}$$ and
moreover, comparing $(\ref{eq-5.1ht})$ with the definition of ${\mathcal
H}_i(t)$ in $(\ref{eq-3.1ht}),$
$ \gamma(s)\in {\mathcal H}_i(t),$ for all $s\in [s_{1,i},s_{2,i}]. $
Hence, by the continuity of the map $s\mapsto \Phi_A(t,\gamma(s)),$ there
exists $[s'_i,s''_i]\subseteq [s_{1,i},s_{2,i}]$ such that
$$\Phi_A(t,\gamma(s'_i))\in {\mathcal Q}_{l}^-,\;\;
\Phi_A(t,\gamma(s''_i))\in {\mathcal Q}_{r}^-\quad\mbox{and }\,
\Phi_A(t,\gamma(s))\in {\mathcal Q},\;\forall\, s\in [s'_i,s''_i].$$ In
addition we have
$ \gamma(s)\in {\mathcal K}_i = {\mathcal H}_i(t)\cap {\mathcal
P},$ for all $s\in [s'_i,s''_i].$ In this manner, according to
Definition \ref{def-1.2}, we have proved that
$ ({\mathcal K}_i,\Phi_A):{\widetilde{\mathcal P}}\stretchx
{\widetilde{\mathcal Q}},$ for all $i=1,\dots,m $ verifying condition
${\bf{(H_r)}}.$
$\blacklozenge\;$ Next, we prove ${\bf{(H_s)}}\,$ of Theorem
\ref{th-1.main}.
Let $\gamma = \gamma(s): [0,1]\to {\mathcal Q}$ be a continuous curve
such that $\gamma(0) \in {\mathcal Q}_{l}^-$ and $\gamma(1) \in {\mathcal
Q}_{r}^-.$ Since ${\mathcal Q}\subseteq \Xi,$ we know that
$ \theta(0,\gamma(s))\in [-\pi/2,0],$ for all $s\in [0,1]. $ Observe
also that
$ E_B(\gamma(0)) = e_1$ and $ E_B(\gamma(1)) = e_2.$
Hence, for
$ t = \tau_B > \beta^*, $ we find, according to $(\ref{eq-kpi}),$ that
there exists $k\in {\mathbb Z}$ such that
$$ [\theta(t,\gamma(0)),\theta(t,\gamma(1))]\supseteq [2k\pi,
\frac{\pi}{2} + 2k\pi];
$$ that is, the angular extent of
$[\theta(t,\gamma(0)),\theta(t,\gamma(1))]$ covers the angular extent of
the region $\Omega.$ By the continuity of the map $s\mapsto
\Phi_B(t,\gamma(s)),$ there exist $[s_{1},s_{2}]\subseteq [0,1]$ such that
$$\theta(t,\gamma(s_{1}))= 2k\pi,\;\; \theta(t,\gamma(s_{2,i}))=
\frac{\pi}{2} + 2k\pi,$$ and
$$
\theta(t,\gamma(s))\in [2k\pi,\frac{\pi}{2} + 2k\pi],\;\forall\, s\in
[s_{1},s_{2}].
$$ In terms of the Poincar\'{e} map $\Phi_B,$ this means that
$$\Phi_B(t,\gamma(s_{1}))\in \Omega\cap\{(x,y): y=0\},\;\;
\Phi_A(t,\gamma(s_{2}))\in \Omega\cap\{(x,y): x=0\},$$ and
$ \Phi_B(t,\gamma(s))\in \Omega$ for all $s\in [s_1,s_2].$ Hence,
by the continuity of the map $s\mapsto \Phi_B(t,\gamma(s)),$ there exists
$[s',s'']\subseteq [s_{1},s_{2}]$ such that
$$\Phi_B(t,\gamma(s'))\in {\mathcal P}_{l}^-,\;\;
\Phi_B(t,\gamma(s''))\in {\mathcal P}_{r}^-\quad\mbox{and }\,
\Phi_B(t,\gamma(s))\in {\mathcal P},\;\forall\, s\in [s',s''].$$ In this
manner, according to Definition \ref{def-1.2}, we have proved that
$ \Phi_B:{\widetilde{\mathcal Q}}\stretchx {\widetilde{\mathcal P}} $
satisfies condition ${\bf{(H_s)}}.$
This completes the proof of Theorem \ref{th-1.1}, showing the existence
of chaotic dynamics in the set ${\mathcal P}.$
\qed
\smallskip
As a final remark we would like to mention that the method of proof
applied here to the weighted equation $(\ref{eq-1.2})$ could also be
applied to some periodically forced equations of the type
$$\ddot{x} + f(x) = p(t),$$ with $f$ satisfying condition $(\textbf{H})$
(at least for some special forms of $p(t)$). This in turn would lead to
some new applications of chaos to forced pendulum type equations (see
\cite{Ma-04} for a comprehensive study on this topic).
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\end{document}