\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 275, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/275\hfil Oscillations with discontinuous energy]
{Oscillations with one degree of freedom and discontinuous energy}

\author[M. Frasson, M. Gadotti, S. Nicola, P. T\'aboas \hfil EJDE-2015/275\hfilneg]
{Miguel V. S. Frasson,  Marta C. Gadotti, \\
 Selma H. J. Nicola, Pl\'acido Z. T\'aboas}

\address{Miguel V. S. Frasson \newline
Departamento de Matem\'atica Aplicada e Estat\'istica,
ICMC-Universidade de S\~ao Paulo,
Avenida Trabalhador S\~ao-carlense 400,
13566-590 S\~ao Carlos SP, Brazil}
\email{frasson@icmc.usp.br}

\address{Marta C. Gadotti \newline
 Departamento de Matem\'atica,
IGCE -- Universidade Estadual Paulista,
Avenida 24A 1515,
13506-700 Rio Claro SP, Brazil}
\email{martacg@rc.unesp.br}

\address{Selma H. J. Nicola \newline
Departamento de Matem\'atica,\quad
Universidade Federal de S\~ao Carlos,
Rodovia Washington Luis, km 235 Norte,
13565-905 S\~ao Carlos SP, Brazil}
\email{selmaj@dm.ufscar.br}

\address{Pl\'acido Z. T\'aboas  \newline
Departamento de Matem\'atica Aplicada e Estat\'istica,
ICMC-Universidade de S\~ao Paulo,
Avenida Trabalhador S\~ao-carlense 400,
13566-590 S\~ao Carlos SP, Brazil}
\email{pztaboas@icmc.usp.br}

\thanks{Submitted September 30, 2015. Published October 23, 2015.}
\subjclass[2010]{34C25, 34D20, 37G15}
\keywords{Periodic solutions; discontinuous energy;
orbital stability; bifurcation}

\begin{abstract}
  In 1995 for a linear oscillator,  Myshkis imposed a constant impulse
  to the velocity, each moment the energy reaches a certain level. The
  main feature of the resulting system is that it defines a nonlinear
  discontinuous semigroup. In this note we study the orbital stability
  of a one-parameter family of periodic solutions and state the
  existence of a period-doubling bifurcation of such solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The solutions of the damped linear oscillator
\begin{equation}\label{eq:oscill-plain}
\ddot x+2\alpha\dot x+\omega^2x=0,\quad \omega>\alpha>0,
\end{equation}
are supposed to undergo a fixed instantaneous increase of velocity whenever 
they reach a certain level $E_0>0$ of energy. More precisely, the following 
condition is imposed
\[
\frac12(\dot x^2(t)+\omega^2x^2(t))=E_0\Rightarrow
\lim_{s\to{t+}}\dot x(s)=\dot x(t)+\sigma,\quad\sigma>0.
\]
This note concerns the resulting discontinuous dynamical system in the 
plane $x\dot x$. Motivated by a pioneering work by Myshkis \cite{myshkis-1}, 
we obtain the existence of  orbitally asymptotically stable \emph{simple}
 periodic solutions, i.e., solutions which have exactly one impulse in 
the period. We accomplish a period-doubling bifurcation for such solutions.

The main feature of the problem is to be autonomous; that is, besides
the involved equation being autonomous, the moments of impulses are not 
previously known. Therefore the solution operator of the whole system defines 
a discontinuous semigroup.

Specific references to the subject are Myshkis \cite{myshkis-3} 
and Samoilenko-Perestyuk \cite{samoilenko-perestyuk}.
 For a wider class of related poblems see 
\cite{gadotti-taboas, gadotti-taboas-2, gyori-3, halanay-wexler,
ladeira-nicola-taboas, laks-bainov-simeonov, myshkis, myshkis-2, 
myshkis-3, pandit-deo} and references therein.

Section 2 aims to build a context for the problem. 
In Section 3 we state elementary properties of positive simple periodic solutions. 
In Section 4 we prove the existence of orbitally unstable positive simple 
periodic solutions with small amplitude and of orbitally asymptotically stable 
with large amplitudes. Finally, in Section 5 we give a sufficient condition 
for a period-doubling bifurcation of such solutions.

\section{Object of study and basic facts}

By the time scaling $\tau=\omega t$ and the change of variables 
$\xi(\tau)=(\omega/\sqrt{2E_0})x(\tau/\omega)$ 
Equation \eqref{eq:oscill-plain} is written as
$\xi''+2a\xi'+\xi=0$, where $'=d/d\tau$, $a=\alpha/\omega\in(0,1)$ and
the locus of level $E_0$ of energy is taken to the circle $S:\xi^2+{\xi'}^2=1$
in the plane $\xi\xi'$. Retrieving the original notation and formulating
the problem in the $x\dot x$ plane we obtain
\begin{equation}\label{eq:system}
\begin{gathered}
\dot x=y,\\
\dot y=-x-2ay
\end{gathered}
\end{equation}
with the impulsive condition
\begin{equation}
(x(t),y(t))\in S\,\Rightarrow\, (x(t+),y(t+))=(x(t),y(t)+v).\label{eq:jump}
\end{equation}

Solutions of \eqref{eq:system} will be denoted by $z$ and $z(\cdot;t_0,z_0)$,
if $z(t_0;t_0,z_0)=z_0$, or briefly $z(\cdot;z_0)=z(\cdot;0,z_0)$.
As the eigenvalues of \eqref{eq:system} are $-a\pm\delta i$, with 
$\delta=\sqrt{1-a^2}>0$, the origin is a stable focus and the energy 
decreases strictly along nontrivial solutions, since
\begin{equation}\label{eq:dissipacao}
\dot E(z(t))= -2a(y(t))^2, \quad t\in\mathbb{R}.
\end{equation}

Let $a=\sin b$, $b\in(0,\pi/2)$, so that $\delta=\cos b$. 
If $\bar z(\cdot)=z(\cdot;(0,-1))$,
\begin{equation}\label{eq:spannersolution}
\bar z(t) = -\delta^{-1} e^{-at} \bigl(\sin \delta t, \cos (\delta t +b)\bigr),
\quad t\in\mathbb{R}.
\end{equation}
As $\bar z(\cdot)$ crosses the $y$ axis at $(0,-\sigma)=(0,-e^{-2a\pi/\delta})$, 
completing a lap around the origin, if $\gamma= \bar z(\mathbb{R})$, the family 
$\{\mu\gamma\}_{\mu\in(\sigma,1]}$ describes all nontrivial orbits 
of \eqref{eq:system}. That is, the general nontrivial solution is
\begin{equation*}\label{eq:generalsolution}
z(\cdot)=\mu\bar z(\cdot+\tau),\quad \tau\in\mathbb{R},\quad\sigma<\mu\leq 1.
\end{equation*}



\begin{definition}\label{solsia} \rm
 A solution of \eqref{eq:system}, \eqref{eq:jump} through $b_0\in\mathbb{R}^2$ at $t=t_0$
 is a function $\phi:[t_0,\infty)\to\mathbb{R}^2$ such that $\phi(t_0)=b_0$ and
\begin{enumerate}
\item $\phi(t-)=\phi(t)$, for all $t\in(t_0,\infty)$;
\item $\phi\in C^1$ and satisfies \eqref{eq:system} in  
 $(t,t+\epsilon_t)$, for all $t\in[t_0,\infty)$ and some $\epsilon_t>0$.
\item $\phi$ is continuous in $t$ if $\phi(t)\in\mathbb{R}^2\setminus S$ and 
$\phi(t+)=\phi(t)+(0,v)$ if $\phi(t)\in S$.
\end{enumerate}
\end{definition}

\begin{remark}\rm
\begin{enumerate}
\item $\phi$ is denoted by $\phi(\cdot;t_0,b_0)$ or $\phi(\cdot;b_0)$ if $t_0=0$.

\item A function $\psi:(\tau,\infty)\to\mathbb{R}^2$ is solution of 
 \eqref{eq:system}, \eqref{eq:jump} in $(\tau,\infty)$ if 
 $\psi\bigl|_{[\,t_0,\infty)}= \phi(\cdot;t_0,\psi(t_0))\bigr.$,
 for any $t_0\in(\tau,\infty)$.

\item The solution $\phi(\cdot;t_0,b_0)$ is unique, but in general there
is no uniqueness for backward continuations. If $|b_0|\geq 1$,
$\phi(\cdot;t_0,b_0)$ has a continuation to $(-\infty,\infty)$. If $|b_0|<1$,
in general a maximal interval of existence to the left is bounded below.
\end{enumerate}
\end{remark}


\section{Positive simple solutions} \label{sec: pss}

For the dynamics of \eqref{eq:system}, \eqref{eq:jump} the only relevant 
solutions are $\phi(\cdot;b)$ with $|b|\geq 1$, as they are the only that
eventually undergo impulses. There is no loss of generality in taking $|b|=1$ 
and we do so. We denote by $\mathfrak C$ the class of such solutions.

\begin{definition} \rm
Let $\phi(\cdot;b)$, $|b|=1$, be a periodic solution of
\eqref{eq:system}, \eqref{eq:jump} with minimal period $\omega>0$. 
The point $\phi(0;b)$ is called \emph{vertex} of $\gamma=\phi(\cdot;b)$.
We say that $\phi(0;b)$ is simple if it has a unique impulse in
$[0,\omega)$. If $\phi(\cdot;b)=(x(\cdot),y(\cdot))$, it is positive when
$x(t)>0$ for all $t$.
\end{definition}

We close this section by setting some standing notations. 
A number $\beta$, identified to any $\beta'\equiv\beta\mod2\pi$,
indicates a point $(\cos\beta,\sin\beta)\in S$ or its arc length coordinate in $S$. 
The context will clarify the meaning in each case. For $\beta\in S$ we 
denote $\phi_\beta=\phi(\cdot;\beta)$ and, if $|\beta+(0,v)|>1$,
we set $t_1=t_1(\beta)>0$ such that $\phi_\beta(t_1)\in S$ and 
$\phi_\beta(t)\notin S$ for $0<t<t_1$.

\begin{definition} \rm
If $D=\{\beta\in S\mid |\beta+(0,v)|>1\}$, we define the return map
 $\Phi_v:D\to S$ by $\Phi_v(\beta)=\phi_\beta(t_1(\beta))$ for all $\beta\in D$.
\end{definition}

Clearly, if $\beta^*\in D$ is a fixed point of $\Phi_v$, $\phi_{\beta^*}$ 
is a simple periodic solution whose period is $t_1(\beta^*)$ and $\beta^*$ 
is the vertex of the simple cycle $\phi_{\beta^*}(\mathbb{R})$.
 If $\beta^*$ is an attractor fixed point, $\phi_{\beta^*}$ is 
orbitally asymptotically stable and, if it is repelling, $\phi_{\beta^*}$ 
is orbitally unstable. Here the orbital stability must be in the sense of 
conditional stability relative to the class $\mathfrak C$, see \cite{lefschetz}, 
since if $\phi=\phi(\cdot;b)$, $|b|=1$, there are points $b'$ inside
$S$ arbitrarily close to $b$ and therefore $\phi(t;b')\to(0,0)$, as $t\to\infty$.

If $\beta\in S$, let $s_\beta$ be the vertical line $s_\beta\!: x=\cos\beta$ 
and $t_\beta>0$ such that $z(-t_\beta;\beta)= (\cos\beta,y_\beta)\in s_\beta$
and $z(t;\beta)\notin s_\beta$ for $-t_\beta<t<0$.
We set $v_\beta=y_\beta-\sin\beta$, so that $\phi_\beta$ is a positive 
simple periodic solution of \eqref{eq:system}, \eqref{eq:jump},
 $v_\beta>0$. We denote by $\alpha=\alpha_\beta$ the polar angle of 
$z(-t_\beta;\beta)$, according to Figure \ref{fig:positsimpcycle}.


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1} % art-fgnt-figure0
\end{center}
\caption{Positive simple cycle.} \label{fig:positsimpcycle}
\end{figure}

\begin{remark} \rm
For any $v\in(0,e^{a\pi/\delta}+1)$, there exists exactly one positive
 simple cycle of \eqref{eq:system}, \eqref{eq:jump} since
$\beta\in(-\pi/2,0)\mapsto v_\beta\in (0,\,e^{a\pi/\delta}+1)$ is a 
continuous bijection.
\end{remark}

\section{Orbital stability}

Now we show that, for some $\zeta>0$, the solution $\phi_\beta$ 
of \eqref{eq:system}, \eqref{eq:jump} is orbitally unstable if 
$\beta\in(-\zeta,0)$ and orbitally asymptotically stable if 
$\beta\in(-\pi/2,-\pi/2+\zeta)$.


\begin{lemma}\label{lem:technicallemma}
$v_\beta = -2\beta + o(\beta)$ as $\beta\to0-$.
\end{lemma}

\begin{proof} 
Let $\beta\in(-\pi/2,0)$. System \eqref{eq:system} in polar coordinates,
\begin{gather*}
\dot r=-(2a\sin^2\theta)r,\\
\dot\theta=-(1+a\sin2\theta),
\end{gather*}
yields
\begin{equation}\label{eq:cp}
r'=\bigl(2a\sin^2\theta/(1+a\sin2\theta)\bigr)r,\quad( '=d/d\theta).
\end{equation}
and a parametrization of $\phi_\beta$ is
\begin{equation}\label{eq:A(theta)}
r_\beta(\theta)=e^{A_\beta(\theta)}
=\exp\Big[2a\int_\beta^\theta\frac{\sin^2s} {1+a\sin2s}\,ds\Big],\quad
\theta\in\mathbb{R}. 
\end{equation}
As the integrand in \eqref{eq:A(theta)} will be a regular participant,
 we introduce the notation
\[
q_a(s)=\frac{\sin^2s} {1+a\sin2s}.
\]
For any small $\epsilon>0$ such that $\alpha=-(1+\epsilon)\beta<\pi/2$, 
the inequality
\[
A_\beta(\theta)\leq -\frac{2a(2+\epsilon)(1+\epsilon)^2}{1-a}\beta^3,\quad
\theta\in[\beta,-(1+\epsilon)\beta],
\]
yields
\[
r_\beta(-(1+\epsilon)\beta)=e^{A_\beta(-(1+\epsilon)\beta)}=1+O(\beta^3)\quad\text{as } \beta\to{0-}.
\]

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % art-fgnt-figure1
\end{center}
\caption{$v_\beta=-2\beta+o(\beta)$ as $\beta\to0-$.}\label{fig:orderofv_beta}
\end{figure}

If $r^\epsilon=|p_\epsilon|$, $p_\epsilon$ being the intersection of the 
half lines $s_1:\theta=-(1+\epsilon)\beta$ and $s_2:r(\theta)\cos\theta=\cos\beta$, 
$\theta\in(0,\pi/2)$, the similarity of the triangles $mnO$ and 
$p_\epsilon qO$ seen in Figure \ref{fig:orderofv_beta} yields
\[
r^\epsilon=\frac{\cos\beta}{\cos(1+\epsilon)\beta}
=1+ \frac{(2+\epsilon)\epsilon}{2!}\beta^2+O(\beta^4) \quad\text{as }\beta\to{0-}.
\]
For $|\beta|$ small enough, the estimates above imply 
$r_\beta(-(1+\epsilon)\beta)<r^\epsilon$, so that 
$y_\beta/\cos\beta<-\tan(1+\epsilon)\beta$ and
\[
1<-\frac{y_\beta}{\sin\beta}<\frac{\tan(1+\epsilon)\beta}{\tan\beta}.
\]
Taking limits as $\beta\to0-$,
\[
1\leq \liminf_{\beta\to0-}-\frac{y_\beta}{\sin\beta}\leq
\limsup_{\beta\to0-}-\frac{y_\beta}{\sin\beta}\leq  1+\epsilon,
\]
so that $\lim_{\beta\to0-}y_\beta/\sin\beta=-1$. 
Therefore $y_\beta=-\beta+o(\beta)$ and  hence $v_\beta=-2\beta+o(\beta)$, 
as $\beta\to0-$.
\end{proof}

The theorem below in what concerns orbital instability is a result 
by Myshkis \cite{myshkis-1}. We give an alternative approach to extend it.

\begin{theorem}\label{theo:unst-st} 
There is a number $\zeta>0$ such that if
$\beta\in(-\zeta,0)$, the simple periodic solution $\phi_\beta$ of 
\eqref{eq:system}, \eqref{eq:jump} is orbitally unstable and 
if $\beta\in(-\pi/2,-\pi/2+\zeta)$, $\phi_\beta$ is orbitally asymptotically stable.
\end{theorem}

\begin{proof} 
Let $\beta\in(-\pi/2,0)$ and $\epsilon_1\ne0$ so that 
$\beta+\epsilon_1=\beta_1\in(-\pi/2,0)$. We take $|\epsilon_1|$ smaller 
if necessary to assure the existence of 
$\Phi_{v_\beta}(\beta_1)=\beta+\epsilon_2\in(-\pi/2,0)$, as it is seen 
in Figure \ref{fig:epsilon xi} for the case $\epsilon_1<0$.


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3} % art-fgnt-figure2
\end{center}
\caption{$\beta+\epsilon_2= \Phi_{v_\beta}(\beta+\epsilon_1)$.}
\label{fig:epsilon xi}
\end{figure}

Firstly we notice that $\epsilon_1$ and $\sigma$ are related by the equation
\[
\frac{v_\beta+\sin(\beta+\epsilon_1)}{\cos(\beta+\epsilon_1)}=\tan(\alpha+\sigma),
\]
therefore, the implicit function theorem about 
$(\epsilon_1,\sigma)=(0,0)$ yields
\begin{equation}\label{eq:dsigmadepsilon}
\sigma=\frac{v_\beta\sin\beta+1}{|b_\beta|^2}\,\epsilon_1+o(\epsilon_1),
\end{equation}
as $\epsilon_1\to0$. By \eqref{eq:A(theta)}, 
if $b_1=\beta_1+(0,v_\beta)$, $\epsilon_2$ must satisfy
\[
|b_1|\exp\Big[2a\int_{\alpha+\sigma}^{\beta+\epsilon_2}  q_a(s)\,ds\Big]=1.
\]
As $|b_1|=\sqrt{(v_\beta+\sin(\beta+\epsilon_1))^2+\cos^2(\beta+\epsilon_1)}$, 
we have
\[
\bigl(v_\beta^2+2v_\beta\sin(\beta+\epsilon_1)+1\bigr)
\exp\Big[4a\int_{\alpha+\sigma(\epsilon_1)}^{\beta+\epsilon_2}   q_a(s)\,ds\Big]=1
\]
and the implicit function theorem leads to
\begin{equation}\label{eq:key}
\epsilon_2=\frac1{q_a(\beta)|b_\beta|^2}
\big[q_a(\alpha)(1+v_\beta\sin\beta)- \frac{v_\beta\cos\beta}{2a}\big] 
\epsilon_1 +o(\epsilon_1),
\end{equation}
as $\epsilon_1\to0$. Let
\begin{equation}\label{eq:F(beta)}
F(\beta)=\frac1{q_a(\beta)|b_\beta|^2}
\big[q_a(\alpha)(1+v_\beta\sin\beta)- \frac{v_\beta\cos\beta}{2a}\big],
\end{equation}
so that $F(\beta)<0$ and \eqref{eq:key} is 
$\epsilon_2=F(\beta)\epsilon_1+o(\epsilon_1)$, as $\epsilon_1\to0$, 
for short. Since 
$\lim_{\beta\to-\pi/2}|b_\beta|= \lim_{\beta\to-\pi/2}-(1+v_\beta\sin\beta)
=e^{a\pi/\delta}$,
\begin{equation}\label{eq:Fbeta<1}
|F(\beta)|\to e^{-a\pi/\delta}<1,\quad\text{ as }\beta\to-\pi/2.
\end{equation}

On the other hand, we have $|\sin\beta|<|\sin\alpha|<y_\beta$, 
see Figure \ref{fig:orderofv_beta}, so that by Lemma \ref{lem:technicallemma}, 
$q_a(\alpha)/q_a(\beta)\to1$ and $v_\beta=O(\beta)$, as $\beta\to0$, 
therefore recalling that $q_a(\beta)=O(\beta^2)$ as $\beta\to0$,
\begin{equation}\label{eq:Fbetatoinf}
|F(\beta)|\to\infty\quad\text{ as }\beta\to0.
\end{equation}
For some $\zeta>0$, Eqs. \eqref{eq:Fbeta<1} and \eqref{eq:Fbetatoinf} 
imply that $|F(\beta)|<1$ if  $\beta\in(-\pi/2,-\pi/2+\zeta)$ and 
$|F(\beta)|>1$ if $\beta\in(-\zeta,0)$. In other words, any  
$\beta\in (-\pi/2,-\pi/2+\zeta)$ is an attractor fixed point of the return
 map $\Phi_{v_\beta}$ and any $\beta\in(-\zeta,0)$ is a repelling 
fixed point of $\Phi_{v_\beta}$.   
 \end{proof}


\section{Period doubling bifurcation} \label{sec: per-doub}

Solutions $\phi_\beta$ of \eqref{eq:system}, \eqref{eq:jump} 
change from stable to unstable when $\beta$ varies over $(-\pi/2,0)$ 
from left to the right. Therefore it is natural to expect a bifurcation 
in between. In this section we apply the theorem below \cite[Theorem 12.7]{devaney} 
to confirm that this indeed occurs at least for small dampings.

\begin{theorem}[Period doubling bifurcation]\label{theo:devaney}
Let $\{f_\lambda\}$ a one-parameter family of real functions and suppose that
  \begin{enumerate}
  \item $f_\lambda(0)=0$ for all $\lambda$ in an interval about $\lambda_0$;
  \item $f_{\lambda_0}'(0)=-1$;
  \item $\displaystyle
    \frac{\partial(f_\lambda^2)'}{\partial\lambda}\Big|_{\lambda=\lambda_0}(0)\neq
    0$.
  \end{enumerate}
 Then there is an interval $I$ about $0$ and a function $p:I\to\mathbb{R}$
  such that
  \[
  f_{p(x)}(x)\neq x \quad\text{and}\quad f^2_{p(x)}(x)=x.
  \]
\end{theorem}

By the proof of Theorem \ref{theo:unst-st} there is a 
$\beta^*_a \in (-\pi/2,0)$, $0<a<1$, such that $F(\beta^*_a)=-1$. 
Now we show that such a $\beta^*_a$ is a period doubling bifurcation point 
of the family of periodic solutions $\phi_{\beta}$, $-\pi/2 < \beta <0$, 
at least if $a$ is small enough.

\begin{theorem}\label{theo:perdoub}
If $a\in(0,1)$ is sufficiently small, then any $\beta^*_a\in(-\pi/2,0)$ 
such that $F(\beta^*_a)=-1$ is a period doubling bifurcation point for the 
family $\phi_\beta$, $-\pi/2<\beta<0$.
\end{theorem}

\begin{proof}
Let us follow  \eqref{eq:key} to define the family of functions 
$f_\beta$, $-\pi/2<\beta<0$, in such a way that
\[
\epsilon_2=f_\beta(\epsilon_1)= F(\beta)\epsilon_1+o(\epsilon_1),
\]
as $\epsilon_1\to0$. Condition $(1)$ of Theorem \ref{theo:devaney}, 
$f_\beta(0)=0$ for all $\beta\in(-\pi/2,0)$, is immediate and, if $'$ denotes
 for a moment $d/d\epsilon_1$, Condition $(2)$, $f'_{\beta^*_a}(0)=F(\beta^*_a)=-1$, 
follows from the definition of $\beta^*_a$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth]{fig4} %art-fgnt-figure3
\end{center}
\caption{$\beta_1=\Phi^2_v(\beta_1)\ne\Phi_v(\beta_1)=\beta_2$.}
\label{fig:doubleperiod}
\end{figure}


Now it remains to show that
\[
\big[\frac{\partial(f_\beta^2)'}{\partial\beta}\big]_{\beta=\beta^*_a}(0)
= \frac{\partial}{\partial \beta}\big[\bigl(F(\beta)\bigr)^2
\big]_{\beta=\beta^*_a}\ne0
\]
for $a$ small enough. Retaking the notation ${\,}'=d{\,}/d\beta$ this is
equivalent to $F'(\beta^*_a)\ne0$, since $F(\beta^*_a)\ne0$. We note
that if $\beta=\beta_a^*$,
\[
q_a(\beta)|b_\beta|^2= \frac{v_\beta\cos\beta}{2a}+ q_a(\alpha)(-v_\beta\sin\beta-1);
\]
therefore,
\begin{equation}\label{eq:Fprime}
\begin{aligned}
F'(\beta^*_a)
&=\Big[\frac1{q_a(\beta)|b_\beta|^2}
\Big(\frac{v_\beta\cos\beta}{2a}+ q_a(\alpha)(-v_\beta\sin\beta-1)\Big)
\Big]'_{\beta=\beta^*_a}\\
&\quad \frac1{q_a(\beta^*_a)|b_{\beta^*_a}|^2}
\Big[q'_a(\beta)|b_\beta|^2+2q_a(\beta)|b_\beta||b_\beta|'
+ \frac{v'_\beta\cos\beta-v_\beta\sin\beta}{2a}\\
&\quad + q_a'(\alpha)\alpha'(-v_\beta\sin\beta-1)
 +q_a(\alpha)(-v_\beta'\sin\beta -v_\beta\cos\beta)\Big]_{\beta=\beta^*_a}.
\end{aligned}
\end{equation}
It suffices to show that the term in the brackets in the right side of 
\eqref{eq:Fprime} is nonzero.

Equation \eqref{eq:A(theta)} implies 
$|b_\beta|=\exp\bigl[2a\int_\beta^\alpha q_a(s)ds\bigr]\to1$ as $a\to0$, 
uniformly in $\beta\in(-\pi/2,0)$. This yields 
$y_\beta\to-\sin\beta$ and $\alpha\to-\beta$, as $a\to0$, uniformly 
in $\beta\in (-\pi/2,0)$. Moreover, the implicit function theorem applied 
to the equation
\[
\exp\Big[2a\int_\beta^\alpha q_a(s)ds\Big]\cos\alpha=\cos\beta,
\]
leads to
\[
\alpha'(\beta)=\frac{\sin\beta(1+a\sin2\alpha)}{y_\beta(1+a\sin2\beta)}.
\]
Thus $\alpha'\to-1$ as $a\to0$, uniformly in $\beta\in(-\pi/2,0)$. 
Finally, we note that the following limits, taken as $a\to0$, are uniform 
in $\beta\in(-\pi/2,0)$:
\begin{gather*}
\lim q_a(\beta)=\sin^2\beta,\\
\lim q'_a(\beta)=\sin2\beta,\\
\lim v_\beta=-2\sin\beta,\\
\lim v'_\beta=-2\cos\beta,\\
\lim |b_\beta|'=0.
\end{gather*}
Therefore, the limit, as $a\to0$, of the term in the brackets in the right 
side of \eqref{eq:Fprime} is
\begin{equation}\label{eq:nonzero}
\sin2\beta+\lim_{a\to0}\frac{v'_\beta\cos\beta-v_\beta\sin\beta}{2a}
-\frac{\sin4\beta}2.
\end{equation}
Since $\lim_{a\to0}(v'_\beta\cos\beta-v_\beta\sin\beta)=-2\cos2\beta$,
in order to assure the expression \eqref{eq:nonzero} is nonzero, $\beta^*_a$ 
must be bounded away from $-\pi/4$ for $a$ small enough.
According to \eqref{eq:F(beta)} $\lim_{a\to0}-F(-\pi/4)=\infty$;
therefore, for some $\eta>0$, $\beta^*_a\notin(-\pi/4-\eta,-\pi/4+\eta)$. 
That is, $F'(\beta^*_a)\ne0$ for $a\in(0,1)$ sufficiently small.
\end{proof}

Figure \ref{fig:doubleperiod} shows a typical positive periodic orbit 
emanating from $\beta^*_a$.



\subsection*{Final remarks}
Smallness of $a$ is a request of our proof of Theorem \ref{theo:perdoub}, 
possibly this hypothesis can be weakened or even discarded.

 The larger is the coefficient $a \in (0,1)$, the larger is the region 
of stability in $(-\pi/2,0)$. In fact, by \eqref{eq:A(theta)},
 $r_{-\pi/2}(\pi)=e^{a\pi/\delta} \to \infty$ as $a \to 1$. 
Therefore, for any fixed $\beta \in (-\pi/2,0)$, one has 
$|b_{\beta}| \to \infty$ as $a \to 1$, so that the number $\epsilon_{2}$ 
in \eqref{eq:key} satisfies $\epsilon_{2} \to 0$, as $a \to 1$.

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\end{document}