\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 128, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/128\hfil Existence of periodic orbits]
{Existence of periodic orbits for vector fields \\ via Fuller index and
the averaging method}
\author[P. Benevieri, A. Gavioli, M. Villarini\hfil EJDE-2004/128\hfilneg]
{Pierluigi Benevieri, Andrea Gavioli, Massimo Villarini} % in alphabetical order
\address{Pierluigi Benevieri \hfill\break
Dipartimento di Matematica Applicata ``G. Sansone''\\
Universit\`a degli Studi di Firenze \\
Via S. Marta 3, 50139 Firenze, Italy}
\email{benevieri@dma.unifi.it}
\address{Andrea Gavioli \hfill\break
Dipartimento di Matematica Pura e Applicata ``G. Vitali'' \\
Universit\`a degli Studi di Modena e Reggio Emilia \\
Via Campi 213/b, 41100 Modena, Italy}
\email{gavioli@unimo.it}
\address{Massimo Villarini \hfill\break
Dipartimento di Matematica Pura e Applicata ``G. Vitali'' \\
Universit\`a degli Studi di Modena e Reggio Emilia \\
Via Campi 213/b, 41100 Modena, Italy}
\email{villarini@unimo.it}
\date{}
\thanks{Submitted September 3, 2003. Published November 3, 2004.}
\subjclass[2000]{34C25, 34C29, 34C40, 37C10, 57R25}
\keywords{Vector fields on manifolds; periodic orbits; Fuller index}
\begin{abstract}
We prove a generalization of a theorem proved by
Seifert and Fuller concerning the existence of periodic
orbits of vector fields via the averaging method.
Also we show applications of these results to Kepler motion
and to geodesic flows on spheres.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\section{Introduction}
Let $X_0$ be a smooth vector field on a closed manifold $M$; we
will refer to it as the {\it unperturbed} vector field. A smooth homotopy
$\varepsilon \mapsto X_\varepsilon $ will be called a {\it perturbation} of
$X_0$. A tipical situation when all the orbits of $X_0$ are closed leads to a
{\it fibration by circles} of $M$, generated by the $S^1$-action whose
infinitesimal generator is $X_0$.
Relevant examples are harmonic oscillators having the same frequency,
the geodesic flow on spheres and the regularized Kepler motion.
We are interested in the problem
of existence of periodic orbits for perturbed vector fields. Concerning
this problem two important results should be mentioned: the Seifert-Fuller
Theorem \cite{seifert, fuller1}, and the Reeb-Moser Theorem
\cite{reeb, moser}. In this article we clarify the relationships
between these two results. In particular, we will prove the former by proving a
generalization of the latter: in doing so, we realize an approach to the use of
perturbation theory to draw conclusions about the qualitative dynamics
proposed by Anosov in \cite[page 181]{anosov}.
In the next section we prove the
Seifert-Fuller Theorem via the averaging method for one-frequency
systems. It is based on a generalization of the Reeb-Moser Theorem, contained
in our Theorem \ref{locale}. In the last section, we give some examples and
applications: They concern the regularized Kepler motion and an
existence result of periodic orbits which can be considered as a
multidimensional version of the Poincar\'e-Bendixson Theorem. In the first
case we also prove the existence of periodic orbits for Hamiltonian
perturbations in the negative energy case, which generalizes an analogous one
given by Moser in \cite{moser} in the nondegenerate case.
\section{A proof of a Theorem by Seifert and Fuller via Averaging Method on a
Manifold}
In this section we give a new proof - and a slight generalization -
of a well-known result by Seifert \cite{seifert} and Fuller \cite{fuller1},
\cite{fuller2} concerning the existence of closed orbits for vector fields
arising from perturbations of a fibration by circles on a closed
manifold. Our approach is strongly motivated by Anosov's comments in
\cite{anosov}. Let us quote Anosov's own words in \cite{anosov}, page 181:
``We digress slightly to discuss a possible approach to the proof of this
theorem (of Seifert-Fuller, or of Seifert-Reeb as referred to by Anosov)
using perturbation theory, the description of what we refer to as
the Reeb-Moser Theorem about the existence of closed nondegenerate orbits
corresponding to nondegenerate singular points of the averaged vector field
follows.
But we do not, in fact, exclude cases in which the
equilibrium points [ of the averaged vector field ] are degenerate or even
non-isolated. Such cases could, of course, be investigated by perturbation
theory, but it is not clear {\it a priori} what result of such an investigation
would be and whether it would be possible to handle all the cases which arise
in a uniform way $\dots$ In summary, perturbation theory provides effective
computation procedure in a specific situation, but is less effective than
topological considerations in studying the qualitative behaviour in the general
case.''
Actually, our generalization of the Seifert-Fuller Theorem will
show how Anosov's approach to the use of perturbation theory for the
qualitative study of differential systems can be made effective even in the
degenerate cases.
To state the Seifert-Fuller Theorem we need a short introduction to the Fuller
index theory. We give a simplified version of it, well-suited for our goals,
and refer to \cite{fuller1}, \cite{fuller2} for a thoroughly discussion of the
theory and related results.
We point out that every mathematical object in this article is assumed smooth:
$C^2$-regularity would be enough.
Let $M$ be a $n$-dimensional closed (i.e. compact boundaryless) manifold and let
\[
X : M \to T M
\]
be a vector field. Consider an open set $\Omega \subseteq M$, bounded away from
the set $sing ( X ) $ of the singular points of $X$. Let $03$
there exist examples \cite{sullivan} of closed manifolds foliated by circles
having unbounded minimal periods (equiv. lenghts).
Then, the orbit space
\[
\widetilde M = M/S^1
\]
is a closed $(n-1)$-manifold, and
\[
i_F ( X_0 ; M \times ] \pi , 3 \pi [) = \chi ( \widetilde M ),
\]
where $\chi ( \widetilde M ) $ is the Euler characteristic of $ \widetilde M$.
Therefore, if $\chi ( \widetilde M ) \not= 0$ and $\varepsilon $ is
sufficiently small, each vector field $X_\varepsilon$ of a given smooth
homotopy has at least one closed orbit.
\end{theorem}
\noindent\textbf{Remark.} In general, the Fuller index is a
rational number, while, in the cases when the statement of Theorem
\ref{fuller} applies, it is always an integer. This is a consequence of the
fact that in the situation considered in the above theorem only closed orbits
with minimal period in $] \pi , 3 \pi [ $ are detected.
To present our proof - and the promised slight generalization - of the
above theorem, we need to introduce the basic elements of averaging method on a
manifold, mainly focusing on the one-frequency case as treated by Moser in
\cite{moser}. We will refer to
\[
X_0 : M \to TM
\]
as the {\it unperturbed vector field}, and to the $X_\varepsilon $'s of a
smooth homotopy as the {\it perturbations} of $X_0$. We will use the notation
\[
X_{\varepsilon } = X_0 + \varepsilon P + O ( {\varepsilon }^2) .
\]
The {\it averaged vector field} of $X_\varepsilon $ on $M$,
$\underline X_ \varepsilon : M \to TM$,
is defined as
\begin{equation} \label{average1}
\underline X_ \varepsilon = { 1 \over { 2 \pi }} \int_0^{ 2 \pi } ( \phi_0
^t )_* X_{\varepsilon }dt,
\end{equation}
where $( \phi_0 ^t )_*
X_{\varepsilon}=d\phi_0 ^{-t}X_{\varepsilon } \circ \phi_0 ^t$ and
$\phi_0 ^t$ denotes the flow of $X_0$. The main
property of $ \underbar X_ \varepsilon $ is that
\[
( \phi_0 ^t )_* \underbar X_ \varepsilon = \underbar X_ \varepsilon
\]
or equivalently that
$[X_0 , \underbar X_ \varepsilon] \equiv 0 $.
As a straightforward consequence we get that, if
\[
p: M \to \widetilde M = M/S^1
\]
($p_*$ denotes the Fr\'echet derivative of $p$) is the projection of the
$S^1$-bundle, then
\[
\overline X_\varepsilon = p_* \underbar X_ \varepsilon
\]
is a well-defined vector field
\[
\overline X_\varepsilon : \widetilde M \to T \widetilde M.
\]
We still call it {\it averaged vector field} on $\widetilde M$.
Recalling that $X_\varepsilon=X_0+\varepsilon P + O(\varepsilon^2)$, and using
the above formula \eqref{average1}, we obtain
\[
\underline X_\varepsilon=X_0+\varepsilon\underline P + O(\varepsilon^2),
\]
where $\underline P$ is the averaged vector field of $P$ (on $M$), defined
as
\[
\underline P= \frac{1}{2\pi}\int_0^{2\pi}(\phi_0^t)_* P \,dt.
\]
Therefore, $\overline X_\varepsilon= \varepsilon \overline P+O(\varepsilon^2)$,
where $\overline P=p_*\underline P$ is the averaged vector field on
$\widetilde M$.
In local
trivializing coordinates of the bundle $ p: M \to \widetilde M = M/S^1 $,
corresponding to straightening coordinates of $X_0$, by using the ``action
coordinate'' $I$ to parametrize $\widetilde M$ and
the ``angular coordinate'' $\theta $, with $\theta = \theta \, mod \, 2 \pi $,
to
parametrize $S^1$, we obtain
\begin{gather*}
X_0: \begin{cases} \dot I =0 \\ \dot \theta =1, \end{cases}
\\
X_\varepsilon:\begin{cases}
\dot I = \varepsilon g(I,\theta,\varepsilon)\\
\dot \theta = 1+\varepsilon f(I,\theta,\varepsilon), \end{cases}
\\
\underline X_\varepsilon: \begin{cases}
\dot I = \varepsilon G(I)+O(\varepsilon^2)\\
\dot \theta = 1+\varepsilon f(I,\theta,\varepsilon), \end{cases}\\
\overline X_\varepsilon: \begin{cases}
\dot I = \varepsilon G(I)+O(\varepsilon^2). \end{cases}
\end{gather*}
It is easy to check that $G(I)$ is the expression of $\overline P$ in local
trivializing coordinates, that is,
\[
G(I)= \frac{1}{2\pi}\int_0^{2\pi}g(I,\theta,0) d\theta.
\]
The geometric meaning of the vector field $\overline P$, or
equivalently of $G(I)$, is given by the following argument,
essentially due to Moser \cite{moser}. The use of local trivializing
coordinates allows us to locally identify the bundle $p:M\to
\widetilde M$ with the product $U\times S^1$, where $U$ is open in
$\widetilde M$. On the other hand, $U$ can be viewed as an $(m-1)$-dimensional
submanifold of $M$, represented in local coordinates as
$\{(I,0),|I|0$ small enough.
For a sufficiently small $\overline \varepsilon>0$, consider a cross
section $\Sigma$ of $X_\varepsilon$, $|\varepsilon|<\overline \varepsilon$,
that is an $(m-1)$-dimensional submanifold of $M$, contained in $U$, which is
transverse in each of its points to $X_\varepsilon$.
In addition, consider the one parameter family of
Poincar\'e maps
\[
F = F_\varepsilon :A \times (-\overline \varepsilon,\overline
\varepsilon)\to \Sigma,
\]
where $A$ is an open subset of $\Sigma$. The
existence of closed orbits of $X_\varepsilon$, for
$|\varepsilon|<\overline \varepsilon$, with initial data $(I,0)$,
$I\in A$, and minimal period close to $2 \pi$, is then reduced to the existence
of $I=I(\varepsilon)$ such that $F
(I(\varepsilon),\varepsilon)=I(\varepsilon)$.
Since $F$ is smooth, it can be expanded as
\[
F (I,\varepsilon)=I+\varepsilon \frac{\partial}{\partial
\varepsilon} F (I,0)+O(\varepsilon^2),
\]
where the equality $F(I,0)=I$ follows from the $2\pi$-periodicity of $X_0$.
Observe that $O(\varepsilon^2)$ is uniform with respect to $I$, if $|I|0$ and a neighborhood $U$ of $I=0$ in
$\widetilde M$, such that for every $\varepsilon$, $|\varepsilon|<\overline
\varepsilon$, there exists at least one closed orbit $\gamma_{\varepsilon}$ of
$X_\varepsilon$, corresponding to the initial datum $(I,0)$, $I \in U$, and
such that $\gamma_{\varepsilon} \mapsto \{ I=0 \}$ (Hausdorff topology).
\item[(iii)] (Hale \cite{hale}) Assume that $0 \in U \subseteq {\mathbb R}^{n-1}$
is a hyperbolic singular point of $\overline P $. Then the closed
orbit $\gamma_{\varepsilon}$ is hyperbolic, hence isolated among the closed
orbits of $X_\varepsilon$ having periods in $] \pi , 3 \pi [$.
\item[(iv)] (Fuller \cite{fuller2}) In the same assumptions of the previous
statement there exists a small tubular neighborhood $\widetilde
\gamma_{\varepsilon}$ of $\gamma_{\varepsilon}$ in $M$ such that
\[
i_F (X_\varepsilon ; \widetilde \gamma_{\varepsilon} \times ] \pi , 3 \pi [)
= \mathop{\rm sign}(- \varepsilon )^n i_{P-H} (\overline P ; 0 ),
\]
where $i_{P-H} (\overline P ; 0 )$ is the Poincar\'e-Hopf index of $\overline
P$ at $0$.
\end{itemize} \end{theorem}
\begin{proof} From the assumptions of statement i) and from the basic
relationship
between the Poincar\'e map and the averaged vector field we get
\[
F (I(\varepsilon_n ),\varepsilon_n) - I ( \varepsilon_n ) = \varepsilon_n (
\frac{\partial}{\partial \varepsilon} F (I( \varepsilon_n),\varepsilon_n) + O
(\varepsilon_n) ) = 0
\]
for a sequence of nonzero $\varepsilon_n \to 0$ and consequently for $
I(\varepsilon_n) \to 0$. This is clearly impossible if
\[
\overline P (0) = \frac{1}{2\pi}\frac{\partial}{\partial \varepsilon} F
(0,0)\not= 0 ,
\]
then i) follows.
To prove the second statement we must prove the existence of
$\varepsilon \mapsto I(\varepsilon ) $, $I(0)=0$, such that
\[
F (I(\varepsilon ),\varepsilon) - I ( \varepsilon ) = 0
\]
or equivalently, just expanding the Poincar\'e map with respect to the
parameter $\varepsilon$, we must prove the existence of nontrivial solutions of
\[
\varepsilon ( \frac{1}{2\pi}\frac{\partial}{\partial \varepsilon} F
(I,\varepsilon) + O (\varepsilon) ) = 0 .
\]
Of course, this is a straightforward consequence of the Implicit Function
Theorem, of the basic equality
\[
\frac{1}{2\pi}\frac{\partial}{\partial \varepsilon} F (I,0)=G(I)
\]
and of the hypothesis that $G(I)$ has a nondegenerate zero at $0$.
The proofs of both the statements iii) and iv) follow from the following
argument. Let $\lambda_j (I, \varepsilon )$, $ \mu_j (I)$,
$j=1, \dots , \dim \widetilde M $ be respectively the eigenvalues of
\[
\frac{\partial}{\partial I} F (I,\varepsilon)
\]
and of
\[
\frac{\partial}{\partial
I} \overline P (I).
\]
Let us remark that, even if $X_0 : M \to TM $ does not admit a global
section, i.e. a one-codimensional closed submanifold, diffeomorphic to
$\widetilde M $, which is everywhere transverse to $X_0$, the functions
$\lambda_j (I, \varepsilon )$, $j=1, \dots , n = \dim \widetilde M $ are
well-defined, if the multiplicity of the eigenvalues is considered. In fact, we
can choose an one-codimensional distribution $\mathcal D $ of small disks on $M$,
everywhere transverse to $X_0$, and we can compute the relative local first
return maps: the eigenvalues $\lambda_j (I, \varepsilon )$ turn out to be
independent of $\mathcal D $. Moreover, let us observe that all the conclusions
about the computations of the various indices are not affected by a small
smooth homotopy of $P$ still keeping $I=0$ as a hyperbolic singular point of
$\overline P$, having eigenvalues of the linearization at $0$ which are all
distincts: hence we will suppose this is the situation we are dealing with.
Then, again as a straightforward consequence of the basic equality
\[
F (I(\varepsilon ),\varepsilon) = I ( \varepsilon ) + 2 \pi \varepsilon
\overline P (I) + O(\varepsilon^2 ),
\]
we get
\[
\frac{\partial}{\partial
I} F (I (\varepsilon ),\varepsilon) = E + 2 \pi \frac{\partial}{\partial
I} G (I) \varepsilon + O ( \varepsilon^2 )
\]
and so finally, using the fact that the $\mu_j (I)$'s are all distinct,
we get the equality
\[
\lambda_j (I, \varepsilon ) = 1 + 2 \pi \varepsilon \mu_j (I) + O (
\varepsilon^2 )
\]
from which both statements iii) and iv) easily follow.
In fact, from the definition of the Fuller index for hyperbolic periodic
orbits, see \cite{fuller1}, we have that
\[ i_F (X_\varepsilon ; \widetilde
\gamma_{\varepsilon} \times ] \pi , 3 \pi [) = (-1)^\sigma
\]
where $\sigma$ is
the number of eigenvalues of the monodromy operator $\frac{\partial}{\partial
I} F (I (\varepsilon ),\varepsilon)$ in $ ] 1 , + \infty [ $. Therefore
$\sigma$ is equal, in the case $\varepsilon > 0$, to the number of the
$\mu_j$'s which are real and greater than zero, or, from the fact that the
system is real, to the number of the $\mu_j$'s having positive real parts: this
conclude the proof in the case when $\varepsilon$ is positive; the case of
negative
$\varepsilon$ is analogous.
\end{proof}
We can now state and prove the main result of this section: it is an extension
of statement ii) of the above thorem (Reeb-Moser Theorem) to the degenerate
case. Let us consider the one-parameter family of vector fields on $M$
\[
X_\varepsilon =X_0 + \varepsilon P + O( \varepsilon^2 ).
\]
We are going to show how the topological properties of the Frechet derivative
of $\varepsilon \mapsto X_\varepsilon $ - namely the vector field $P$ -
determine the existence of closed orbits of $X_\varepsilon$, with $\varepsilon$
sufficiently small. A generalization of this approach will be considered in
the remark at the end of this section.
Let $\widetilde A$ be an open subset of $\widetilde M$, whose boundary is a
boundaryless $(m-2)$-dimensional manifold. We recall that, in this case, the
{\it index} of
the averaged vector field $ \overline P : \widetilde M \to T \widetilde M $
in $\widetilde A$ is well defined if $sing \overline P \cap \partial \widetilde
A = \emptyset $, where $sing \overline P $ is the set of singular points of
$\overline P$. We have the equality
\[
\mathop{\rm ind}( \overline P , \widetilde A ) = \deg(\frac{\overline P}{\Vert \overline P
\Vert}, \partial \widetilde A),
\]
where $\deg(\frac{\overline P}{\Vert \overline P \Vert},
\partial \widetilde A)$ stands for the ordinary Brouwer degree.
\begin{theorem}\label{locale}
Let $\widetilde A $ be an open subset of $\widetilde M$ with $\partial
\widetilde
A$ a compact boundaryless manifold. Suppose $sing \overline P \cap \partial
\widetilde A = \emptyset $.
Let $ p: M \to \widetilde M $ be the bundle projection map and $A =
p^{-1} ( \widetilde A ) \subseteq M $. Then, there exists $\overline \varepsilon
> 0 $ such that, for every $\varepsilon$, $ \vert \varepsilon \vert < \overline
\varepsilon $, the set $ A \times ] \pi , 3 \pi [ $ is admissible for
$X_\varepsilon$ and
\begin{equation} \label{21}
i_F (X_\varepsilon ;A \times ] \pi , 3 \pi [)
= \mathop{\rm sign}(- \varepsilon )^n \mathop{\rm ind} ( \overline P , \widetilde A ),
\end{equation}
where $n= \dim \widetilde M $.
\end{theorem}
Before giving the proof of this theorem we deduce as a corollary Theorem
\ref{fuller}
\[
i_F (X_\varepsilon ;M \times ] \pi , 3 \pi [) = \chi ( \widetilde M ) .
\]
\begin{proof}[Proof of Theorem \ref{fuller}]
We just need to use the above theorem
and the additive property of the index, together with the well-known equality
of the global index of a vector field on a closed manifold and the Euler
characteristic of the manifold itself. The presence of the factor $sign(-
\varepsilon )^n$ in the formula \eqref{21} is obviously immaterial
when the global situation is considered. This is clear if
$\dim \widetilde M$ is even and this is a consequence of the fact that $\chi (
\widetilde M ) = 0$ in the case when $\dim \widetilde M$ is odd .
\end{proof}
\begin{proof}[Proof of Theorem \ref{locale}]
We carry on the proof in the case $\widetilde A$ is completely contained in one
local chart of $\widetilde M$ and referred to local coordinates $I$ as
$\widetilde A = B_R (0) = \{ I \in \widetilde M : \vert I \vert < R \} $. The
general case is completely analogous and can be reduced to the above situation
by choosing local trivializing coordinates $ (I,\theta)$ on the bundle,
decomposing $\widetilde A $ in local charts and patching the various parts of
it together, taking into account the additivity property of the index.
Let us observe that in our situation, we have
\[
A =B_R (0)\times S^1.
\]
Moreover, as we are working in local coordinates, we will refer to $\overline P
$ as $ G(I)$. The assumption that $G(I) \not= 0$, for $ I \in \partial B_R (0)
$, and statement i) of Theorem \ref{main theorem} imply that for $\vert
\varepsilon \vert < \overline \varepsilon $, $\overline \varepsilon $
sufficiently small, $X_\varepsilon$ has no closed orbits with periods in $] \pi
, 3 \pi [$ passing through points of $\partial A $. This proves that $A$ is
admissible for the $X_\varepsilon$'s.
In the following part of the proof we suppose $\varepsilon$ to be fixed and
sufficiently small, according to the above specified request, and we compute
$i_F (X_\varepsilon ;A \times ] \pi , 3 \pi [)$. Let
\[
\rho : M \to \mathbb R ^+
\]
be a smooth bump function, such that $\rho
\equiv 0 $ in $ M - p^{-1} ( B_R (0) )= M - A $, while $ \rho \equiv 1 $ in $
p^{-1} ( B_{\mu R} (0) ) $ where $\mu $ is sufficiently small in order that
$G(I) \not= 0 $ for $\mu R \leq \vert I \vert \leq R$. The bump function $\rho$
allows to localize a smooth homotopy $\lambda \mapsto X_{\varepsilon , \lambda
}$ in the local chart containing $B_R (0)\times S^1 $. Therefore, we just need
to define $X_{\varepsilon , \lambda }$ in local coordinates $(I,\theta)$. Let
$n= \dim M$ and $V \in {\mathbb R}^{n-1} - \{ 0 \} $, and let us define such (local)
homotopy as
\[
\lambda \mapsto X_{\varepsilon , \lambda } (I,\theta) = X_\varepsilon
(I,\theta) + \lambda \rho (I,\theta) V.
\]
Of course, $A \times ] \pi , 3 \pi [$ is still admissible for $X_{\varepsilon ,
\lambda }$, for sufficiently small $\lambda$. Let $\lambda $ be one of such
sufficiently small values: a straightforward application of Sard's Theorem
implies that for almost any choice of $V$, the averaged vector field $\overline
X_{\varepsilon , \lambda }$ has only hyperbolic singular points in $B_R (0)$.
{}From the basic results of degree theory we have the following chain of
equalities
\begin{align*}
\mathop{\rm ind}( \overline P , B_R (0) )
&= \deg({{\overline P}\over {\Vert \overline P \Vert}},\partial B_R (0)) \\
&= \deg({{\overline X_\varepsilon}\over {\Vert\overline X_\varepsilon \Vert}},
\partial B_R (0)) \\
&=\deg( {{\overline X_{\varepsilon , \lambda}}\over
{\Vert \overline X_{\varepsilon, \lambda} \Vert}},\partial B_R (0)) \\
&= \sum_{I_j \in \mathop{\rm sing}(\overline X_{\varepsilon , \lambda }) } i_{P-H} ( \overline
X_{\varepsilon , \lambda } ; I_j ) .
\end{align*}
On the other hand, statement iv) of Theorem \ref{main theorem} and the
homotopy invariance of the Fuller index give
\begin{align*}
\sum_{I_j \in \mathop{\rm sing}(\overline X_{\varepsilon , \lambda }) } i_{P-H} ( \overline
X_{\varepsilon , \lambda } ; I_j )
&= \mathop{\rm sign}(- \varepsilon )^n i_F
(X_{\varepsilon , \lambda };p^{-1} (B_R (0) \times ] \pi , 3 \pi [)\\
&=\mathop{\rm sign}(- \varepsilon )^n i_F (X_\varepsilon ;p^{-1} B_R (0) \times ]
\pi , 3\pi [)
\end{align*}
and finally
\[
\mathop{\rm sign}(- \varepsilon )^n i_F (X_\varepsilon ;p^{-1} B_R(0)
\times ] \pi , 3 \pi [) = \mathop{\rm ind}( \overline P , B_R (0) ).
\]
\end{proof}
\noindent\textbf{Remark.} It is easy to see that, as a consequence of Theorem \ref{locale},
a closed orbit of $X_\varepsilon $ with initial datum $(I,0)$, $I \in
\widetilde A$, exists whenever $\mathop{\rm ind}( X_\varepsilon , p^{-1} ( \widetilde A ) )
\not= 0 $.
\smallskip
\noindent\textbf{Remark.} One could try to use the Fuller index approach to
investigate the existence of periodic orbits of minimal period greater
than $ 2 \pi $.
Actually, these trajectories do not exist. More precisely we can state the
following property:
\begin{quote} for any given number $T > 2 \pi $, there exists $\overline
\varepsilon $ such that for every $0 < \varepsilon < \overline \varepsilon $
the vector field $X_\varepsilon$ has no periodic orbits having minimal periods
in $] 2 \pi , T[ $.\end{quote}
The proof of this claim is an obvious consequence of the fact that the
eigenvalues $\lambda_j (I, \varepsilon ) $ defined in the proof of Theorem
\ref{main theorem} verify
$ \lambda_j (i, \varepsilon ) = 1 + O(\varepsilon )$.
\smallskip
\noindent\textbf{Remark.} Theorem \ref{locale} easily generalizes to the case when
$\overline P \equiv 0$ on $\widetilde M$. Let
\begin{gather*}
X_\varepsilon = X_0 + \varepsilon P + \cdots + \varepsilon^k P^{(k)} + O
(\varepsilon^{k+1} )\,,\\
\underline P^{(k)} = \frac{1}{2 \pi} \int_{0}^{2 \pi} (\phi_0^t)_*
P^{(k)}dt\,,\\
\overline P^{(k)} = p_* \underline P^{(k)}.
\end{gather*}
Also suppose that
$\overline P \equiv \cdots \equiv \overline P^{(k - 1)} \equiv 0$
while $ \overline P^{(k)} \not= 0 $.
Then the same arguments leading to Theorem \ref{locale} give
\[
i_F (X_\varepsilon ;A \times ] \pi , 3 \pi [)
= \mathop{\rm sign}(- \varepsilon )^{kn}
\mathop{\rm ind}( \overline P^{(k)} , \widetilde A ).
\]
\section{Examples and applications}
This final part of the article contains some applications of the results
contained in the previous section. Specifically, we give applications of the
``degenerate version of the Reeb-Moser Theorem'', namely of Theorem
\ref{locale}, as well as applications of the classical Seifert-Fuller Theorem.
In our opinion they have some interest, originality and relationship with the
present article. This section is divided in two subsections, labeled by a latin
letter and a short title.
\subsection*{Hamiltonian degenerate perturbations of the Kepler motion}
Let
\[
H_0 (p,q) = \frac{1}{2} \vert p \vert ^2 - \frac{1}{\vert q \vert }
\]
be a Kepler Hamiltonan, $q = (q_1,\dots,q_n)$, $p = ( p_1 , \dots , p_n)$.
In the case $n=2$ $H_0$ is the Hamiltonian of the Newtonian gravitational field
describing a two-body system. Let
\[
H_\varepsilon (p,q, \varepsilon ) = H_0 (p,q) + \varepsilon K (p,q,
\varepsilon )
\]
be a perturbed Hamiltonian, where $K (p,q, \varepsilon ) $ is smooth and
satisfies a smoothness assumption also as a function
\[
K ( \vert p \vert ^2 q - (2 p \cdot q) p ,\frac{p}{\vert p \vert ^2
}\varepsilon)
\]
near $ \vert q \vert = 2$, $p=0$, $\varepsilon =0$. Such a smoothness condition could be verified following
an analogous case presented in \cite[Section 5, p. 628]{moser}.
Under these conditions the
Hamiltonian motion on a negative energy level, say on $\{ H_\varepsilon (p,q,
\varepsilon ) = - \frac{1}{2} \}$, can be embedded, as a flow, after a
reparametrization of the independent variable and a smooth change of
coordinates, in a Hamiltonian $\varepsilon$-perturbation of the geodesic flow
$X_0$ on $S^n$ (with respect to the standard metric on the $n$-sphere ). This is the
so called {\it regularization of the perturbed Kepler motion}, see \cite{moser}
for details. Let
\[
X_\varepsilon : T_1 S^n \to T(T_1 S^n )
\]
be the corresponding one-parameter family of vector fields on the unitary
tangent bundle of the $n$-sphere realizing the perturbation of the geodesic
vector field
\[
X_0 : T_1 S^n \to T(T_1 S^n ) .
\]
In \cite{moser} Moser proved, as a consequence of statement ii) in Theorem
\ref{main theorem}, the following theorem of existence of periodic orbits for
the perturbed geodesic flow on spheres, or equivalently for the perturbed
Kepler motion.
\begin{theorem}[\cite{moser}]\label{keplero1}
Let $\overline X_\varepsilon $ be the averaged vector field with respect to the
unperturbed geodesic flow $ X_0$ on a sphere.
If, for $\varepsilon $ sufficiently small, $\overline X_\varepsilon $ has
a nondegenerate singular point, then $X_\varepsilon $ has a
(nondegenerate) periodic orbit.
\end{theorem}
Moreover, let $H_\varepsilon $ be the Hamiltonian of
$\overline X_\varepsilon $ and consider the regularization of the perturbed
Kepler motion on negative energy manifolds. For every $\varepsilon$
sufficiently small such that a nondegenerate singular point of the averaged
vector field arising from the regularization exists,
it has at least one closed orbit.
Actually such ``closed'' orbit could be a collision orbit. We
will not consider this question here.
Our Theorem \ref{locale} permits to drop the (particularly heavy in the
Hamiltonian case) non-degeneracy assumption in Theorem \ref{keplero1}.
\begin{theorem} \label{keplero2}
The same conclusion as in the previous theorem, regarding the existence of
closed orbits for the perturbation $X_\varepsilon $ of the geodesic flow on
spheres, holds if $\overline X_\varepsilon = \varepsilon \overline P
+O(\varepsilon^2) $ has a degenerate zero with nonzero index or more generally
if there exists a ball $B_R$ in the orbit space of $X_0$ such that
\[
\mathop{\rm ind}( \overline P ,\partial B_R ) \not= 0 .
\]
Actually, if $\varepsilon $ is sufficiently small, the perturbed geodesic
vector field $X_\varepsilon $ has always at least one closed geodesic.
An analogous conclusion holds for the perturbed Kepler motion.
\end{theorem}
\begin{proof}
The first part of the theorem is a straightforward application of Theorem
\ref{locale}, while the second one is a consequence of the Seifert-Fuller
Theorem. In both cases we just need to reduce the dynamical situation to a
geometric model well-suited for application of the one-frequency averaging
method. We will do that referring to the application of the Seifert-Fuller
Theorem, the rest of the proof being completely analogous. The geodesic vector
field $ X_0 : T_1 S^n \to T( T_1 S^n ) $ defines a fibration by circles
\[
T_1 S^n \to G_{2,n+1},
\]
where $ G_{2,n+1} $ is the Grassmannian manifold of oriented $2$-planes in
$\mathbb R^{n+1} $, obtained after identification of a great circle in $S^n$ by
the $2$-plane through the origin containing it. We apply now the Seifert-Fuller
Theorem
\[
i_F ( X_0; T_1 S^n \times ] \pi , 3 \pi [ ) = \chi ( G_{2,n+1}) \not= 0.
\]
In fact a straightforward computation of the Betti numbers of $G_{2,n+1}$, based for instance on the cell structure of the Grassmann manifolds as exposed in \cite{MS}, together with the definition of the Euler characteristic as the alternating sum of the Betti numbers, leads to
\[
\chi (G_{2,n+1}) = \begin{cases}
n+1 & \text{if } n \text{ is odd}\\
n & \text{if } n \text{ is even}.
\end{cases}
\]
\end{proof}
\noindent\textbf{Remark.}
It is probably worthwhile mentioning that the above result and
the approach used for its proof are not unrelated with the deeply studied
problem of existence of closed geodesics after perturbation of the standard
metric on $S^n$. We stress the fact that in the above theorem we considered {\it
arbitrary} perturbations, and not only perturbations arising from a perturbation
of the standard metric on the sphere. For such particular perturbations not
only existence but also multiplicity results are known, obtained through a
variational approach (see \cite{geodetiche}).
\subsection*{A Poincar\'e-Bendixson type existence theorem of periodic orbits}
This second application of the ideas related to the Fuller index
approach in the averaging method for one-frequency systems deals with a
situation which is frequently present in mechanics.
Let $J \subseteq \mathbb R$ be
an interval and $M$ be a closed manifold: the dynamic variable $h$
parametrizing $J$ is called the {\it energy} of the {\it unperturbed dynamical
system}
\[
X_0: J \times M \to \mathbb R \times TM .
\]
Such a vector field verifies:
\begin{itemize}
\item[(i)] $h$ is a first integral for $X_0$,
\item[(ii)] ${X_0}_{\vert \{ h=c \} } : M \to TM$ generates a fibration by
circles (with $c$-depending minimal periods $T_c $)
\[
p(c) : M \to \widetilde M,
\]
\item[(iii)] for $c_1 , c_2 \in J $ the fibrations $ p(c_j) : M
\to \widetilde M $, $j=1,2$, are isomorphic,
\item[(iv)] the Fuller
index of the fibration by circles satisfies
\[
i_F ( {X_0}_{\vert \{ h= c \} };
M \times ] \frac{1}{2} T_c , \frac{3}{2} T_c [ ) \not= 0 .
\]
\end{itemize}
In the sequel we
will always refer to the natural splitting of the tangent bundle $ T(J \times M
) = \mathbb R \times TM $ and the analogous $ T(J \times \widetilde M ) =
\mathbb R
\times T \widetilde M $ defined by the bundle map. Therefore, the averaged
vector field
\[
\overline P : J \times \widetilde M \to \mathbb R \times T
\widetilde M
\]
is canonically decomposed as
$\overline P = (\overline P^h , \overline P^{\widetilde M} )$.
s\smallskip
\noindent\textbf{Example.}
{\it Harmonic oscillators with the same frequency}.
Let $x \in \mathbb R^4 $ and
\[
\dot x = X_0 (x) = A x,
\]
where
$A = I_1 \oplus I_2$ and
\[
I_1 = I_2 = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}.
\]
Here $J= \mathbb R^+$, $M=S^3$, $\widetilde M = S^2 $ and the fibration $ p: S^3
\to S^2 $, which is the same for every energy level, is the Hopf fibration,
with $i_F ( X_0 ; S^3 \times ] \pi , 2 \pi [\, ) = 2 $. Of course, this
situation generalizes in an obvious way to the case of $n$ harmonic oscillators
with rationally dependent frequencies.
\smallskip
\noindent\textbf{Example.}
{\it Geodesic flow on spheres}. Here
\[
X_0 : T S^n \to T (T S^n )
\]
is the geodesic vector field, $n \geq 2$, with respect to the usual metric on
$S^n$. The tangent bundle is fibered through the level manifold of the kinetic
energy first integral as
\[
T S^n =( S^n \times \{ 0 \} )\cup (\cup_{h>0} T_h S^n).
\]
The exceptional $0$-fiber is diffeomorphic to $S^n$, while all the other
fibers are diffeomorphic to $T_1 S^n $. Therefore every energy interval $J =
(h_1 , h_2 ) \subseteq \mathbb R^+ $ defines a phase space for (the restriction
of) $X_0$ such that
\[
X_0 : \cup_{h_1 < h < h_2 } ( \{ h \} \times T_h S^n ) \to \cup_{h_1 < h <
h_2 } ( \{ h \} \times T(T_h S^n ) ),
\]
where $p(h) : T_h S^n \to G_{2 , n+1 }$
are isomorphic bundles for $ h_1 < h < h_2 $. Finally, we recall that
\[
i_F ( {X_0}_{\vert \{ h=c \} } ; T_h S^n \times ]
\frac{\pi }{c} , \frac{3 \pi }{c}[ ) = \chi ( G_{2 , n+1 } ) \not= 0 .
\]
These examples justify our attention to the perturbations
\[
X_\varepsilon : J \times M \to \mathbb R \times TM,
\]
where $X_\varepsilon = X_0 + \varepsilon P + O( \varepsilon^2 ) $ and $X_0$
satisfies the above listed properties i)-iv). It is easy to see that in general
$X_\varepsilon $ has no closed orbits: in the next theorem we will give some
relevant hypotheses implying the existence of periodic orbits.
\begin{theorem} \label{bendy}
Let $h_1$, $h_2 \in J$ be two energy levels, $h_1 < h_2 $, such that
\[
\overline P^h (h_1 , I ) \overline P^h (h_2 , I ) < 0
\]
for every $I \in \widetilde M$. Assume in addition that $\chi (\widetilde M)\neq 0$. Then, for every sufficiently small
$\varepsilon$, there exists a closed orbit of $X_\varepsilon $ having minimal
period between $\frac{1}{2} T( h_1 )$ and $\frac{3}{2} T( h_2 )$ and
corresponding to an initial datum $q \in J \times M $ such that $ h_1 < h ( q )
< h_2 $.
\end{theorem}
\begin{proof}
We prove the theorem in the case $\pi < T(h_1) < T(h_2) < 3 \pi$. This situation is the
general one, up to a reparametrization of the independent variable and of the energy $h$, not
affecting our geometric conclusions concerning the existence of a periodic
orbit. We will apply the Fuller index theory to the set
\[
\Omega = ( \,] h_1 , h_2 [ \times M ) \times ] \pi , 3 \pi [\,.
\]
Let us remark that, as $M$ is boundaryless,
\[
\partial \Omega = \{h_1 \} \times M \times ] \pi , 3 \pi [ \cup \{h_2 \} \times M \times ] \pi , 3 \pi [ \cup ]h_1 , h_2 [ \times M \times \{ \pi \} \cup ]h_1 , h_2 [ \times M \times \{3 \pi \}.
\]
The assumption
\[
\overline P^h (h_1 , I ) \overline P^h (h_2 , I ) < 0
\]
together with statement i) of Theorem \ref{main theorem} permits to conclude that $ \Omega $ is admissible for the
perturbation if $\varepsilon $ is sufficiently small, as we will always
suppose for the rest of the proof. Therefore, to conclude the proof we keep
$\varepsilon$ fixed and we prove that $ i_F (X_\varepsilon ; \Omega)
\not= 0 $. Let us define
\[
Y : ] h_1 , h_2 [ \times M \to {\mathbb R} \times TM
\]
through its components with respect to the canonical splitting as
\begin{gather*}
Y^M (h,q) = {X_0}_{\vert \{ h \} \times M },\\
Y^h (h,q) = \begin{cases}
h - \frac{h_1 + h_2 }{2} &\text{ if }\overline P^h (h_1 , I ) <0,\\
\frac{h_1 + h_2 }{2} -h &\text{ if } \overline P^h (h_1 , I ) >0.
\end{cases}
\end{gather*}
It is clear that $\{(h,q):h - \frac{h_1 + h_2 }{2} = 0 \} \simeq M $ is an invariant
manifold for $Y$, fibered by circles by the $Y$-action, while no other point in
$] h_1 , h_2 [ \times M $ can be the initial datum for a periodic orbit of $Y$.
It is also easy to see that $ \Omega$ is admissible for $Y$ and that
- by an obvious homotopic perturbation - we get
\[
i_F ( Y ; \Omega ) = \chi ( \widetilde M ) \not= 0 .
\]
Therefore, the statement will be proved if we construct a smooth homotopy
between $Y$ and $X_\varepsilon $ still having $ \Omega $ as an
admissible set. Let
\[
\lambda \mapsto ( 1 - \lambda ) X_{\varepsilon } + \lambda Y = Z_{\varepsilon , \lambda }
\]
connecting $X_\varepsilon $ to $Y $. To prove the admissibility of $\Omega $ for $ \lambda \mapsto Z_{\varepsilon , \lambda } $, we observe that
\[
\overline Z_{\varepsilon , \lambda } = \varepsilon (1 - \lambda ) \overline P + \varepsilon
\lambda \overline Y
\]
and that $ \overline Y^{\widetilde M } = 0 $, while $\overline Y^h = h -
\frac{h_1 + h_2 }{2} $.
Now, let us suppose that $\overline P^h (h_1 , I ) < 0
$ and $ \overline P^h (h_2 , I ) > 0 $ for $I \in \widetilde M$, the opposite
situation being analogous.
Then
\begin{gather*}
\overline Z_{\varepsilon , \lambda }^{\widetilde M} = \varepsilon (1 - \lambda )
\overline P^{\widetilde M}\,,\\
\overline Z_{\varepsilon , \lambda }^h = \varepsilon (1 - \lambda ) \overline P^h
+ \lambda \overline Y^h
\end{gather*}
and therefore for $\varepsilon \not= 0$,
\[
\overline Z_{\varepsilon , \lambda }^h (h_1 , I)
\overline Z_{\varepsilon , \lambda }^h (h_2 , I) <0\, .
\]
Arguing as in Theorem \ref{main theorem}, statement (i), the above
inequality implies that no periodic orbit of $Z_{\varepsilon , \lambda }$
can intersect
\[
\{h_1 \} \times M \times ] \pi , 3 \pi [ \cup \{h_2 \} \times M \times ] \pi , 3 \pi [.
\]
Therefore, to prove that $\partial \Omega $ is admissible for
$\lambda \mapsto Z_{\varepsilon , \lambda }$ we must prove that no
periodic orbit of $Z_{\varepsilon , \lambda }$ intersects
$]h_1 , h_2 [ \times M \times \{ \pi \} \cup ]h_1 , h_2 [ \times M
\times \{3 \pi \}$
or, which is the same, that no periodic orbit of
$Z_{\varepsilon , \lambda }$ in $]h_1 , h_2 [ \times M$ has period
$\pi$ or $ 3 \pi $. Let $Z_{0 , \lambda }= (1- \lambda ) X_0 + \lambda Y$,
$0 \leq \lambda \leq 1$. For every $\lambda \in [0,1]$ these vector fields
has no periodic orbits of period $\pi $ or $ 3 \pi $. More precisely, if
\[
\phi_{0,\lambda }^t (h, I) = (\phi_{0,\lambda ;h}^t (h, I),
\phi_{0,\lambda ;I}^t (h, I))
\]
is the flow of $Z_{0 , \lambda }$, then, as
\[
Z_{0 , \lambda }^M (h , I) = X_{0 \vert \{h \} \times M },
\]
there exists $\delta >0 $ such that, for every
$(h,I) \in ]h_1 , h_2 [ \times M $,
\begin{gather*}
d(\phi_{0,\lambda ;I}^{\pi } (h , I) , I ) > 2 \delta\,, \\
d(\phi_{0,\lambda ;I}^{ 3\pi } (h , I) , I ) > 2 \delta\,,
\end{gather*}
where $d (\cdot,\cdot)$ is a distance defined by a Riemann metric on $M$.
Then, if
\[
\phi_{\varepsilon,\lambda}^t (h,I)=(\phi_{\varepsilon ,\lambda ;h}^t (h, I),
\phi_{\varepsilon ,\lambda ;I}^t (h, I))
\]
is the flow of $Z_{\varepsilon , \lambda }$ and if $\varepsilon $ is
sufficiently small, from the continuous dependence of the solutions from
parameters one has that for every $(h,I) \in [h_1 , h_2 ]\times M $
\begin{gather*}
d(\phi_{\varepsilon ,\lambda ;I}^{\pi } (h , I) , I ) > \delta\,,\\
d(\phi_{\varepsilon ,\lambda ;I}^{ 3\pi } (h , I) , I ) > 2 \delta
\end{gather*}
and therefore the vector fields $Z_{\varepsilon , \lambda }$ have no
periodic orbits with periods $\pi $ or $ 3 \pi $ in $\in [h_1 , h_2 ]\times M $.
This concludes the proof.
\end{proof}
\noindent\textbf{Remark.}
An application of the above theorem to the geodesic flow on
spheres (second example above) gives an existence result for closed geodesics.
In some sense, the above theorem is a theorem of Poincar\'e-Bendixson type,
too. In fact, the $\omega $-invariance of a region of the (multidimensional)
phase space, together with topological hypotheses on the averaged vector
field, {\it including that it always points
either inward or outward in the 2 boundary components}, imply the existence
of a closed orbit.
\subsection*{Aknowledgments}
The third author (Massimo Villarini) would like to thank
F. Podest\`a for his useful discussions.
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\end{document}