\documentstyle[twoside, amssymb]{article}
% amssymb is used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil On the prescribed-period problem \hfil EJDE--1998/05}%
{EJDE--1998/05\hfil A.A. Zevin \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~05, pp. 1--9. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
On the prescribed-period problem for autonomous Hamiltonian systems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 58F14, 58F22.
\hfil\break\indent
{\em Key words and phrases:} Autonomous Hamiltonian system, periodic solutions,
\hfil\break\indent global analysis, prescribed-period problem.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted March 3, 1997. Published February 20, 1998.} }
\date{}
\author{A. A. Zevin}
\maketitle
\begin{abstract}
Asymptotically quadratic and subquadratic autonomous Hamiltonian
systems are considered. Lower bounds for the number of periodic solutions
with a prescribed minimal period are obtained. These bounds are expressed
in terms of the numbers of frequencies corresponding to the critical points
of the Hamiltonian. Results are based on a global analysis
of families of periodic solutions emanating from these points.
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\section{Introduction}
We consider the autonomous Hamiltonian system
$$
J\dot x=H_x (x), \qquad J=
\left[\begin {array} {cc} 0& -I_n\\I_n & 0\end{array}\right]\,, \eqno (1.1)
$$
where $x\in {\Bbb R}^{2n}$ and $I_n$ is the identity matrix of order $n$.
Here, the Hamiltonian $H(x)$ is assumed to be analytical.
The prescribed-period problem for system (1.1) consists of finding
conditions on $H(x)$ that guarantee the existence of periodic
solutions of a given period $ T$. During the recent years there appeared a
considerable amount of work devoted to this problem, see for example \cite{Ra}.
The most interesting results are those that establish existence
of solutions having minimal period $T$. For convex asymptotically
subquadratic and superquadratic Hamiltonians, the existence of
at least one such solution was proved by Clarke and Ekeland \cite{Cl},
Ambrosetti and Mancini \cite{Am}, Ekeland and Hofer \cite{Ek}, Girardi and
Matseu \cite{Gi}. For second-order Hamiltonian systems, with an even
potential function, some multiplicity results were obtained by Van
Groesen \cite{Va}. In these articles, the solution is obtained
as a minimum of a functional, through the use of variational techniques.
In this paper, we investigate the prescribed-period problem
with minimal period $T$. The results obtained are based on an examining
the behavior of families of periodic solutions emanating
from the equilibrium points. Note this such approach to nonlocal
analysis of autonomous Hamiltonian systems was previously utilized
in \cite{Z1,Z2}.
Now we present the main results.
Suppose there are positive constants $\alpha $ and $r$ such that for
$\| x\|>r$ ,
$$
-\alpha I_{2n}< H_{xx}(x)<\alpha I_{2n} \leqno (H)
$$
i.e., $-\alpha \|y\|^2<(H_{xx}(x)y,y)<\alpha \|y\|^2$
for $y\ne 0$, where
$\|y\|=(y,y)^{1/2}$ is the Euclidean norm of $y$.
We assume that $H(x)$ has a finite number of critical
points $x^p$, $p=1,\dots,k$ in ${\Bbb R}^{2n}$ ($H_x(x^p )=0$).
Also assume that they are non-degenerate ($\det H_{xx}(x^p) \ne 0$).
Let $\pm i\omega_r^p$ ($\omega_r^p >0$, $r=1,\dots,r_p \le n$) denote the
purely imaginary eigenvalues of the matrix $J^{-1} H_{xx}(x^p)$, and let
$$
l_r^p =i(Jx_r^p,x_r^p)\,, \eqno (1.2)
$$
where $x_r^p$ is the eigenvector associated with the eigenvalue $i\omega_r^p$ .
As known in \cite{Ya}, $l_r^p$ is real and $l_r^p\ne 0$ for simple
eigenvalues. The quantity $\omega_r^p$ is called a frequency of
{\em first kind}, or of {\em second kind}, if $l_r^p >0 $ or $l_r^p <0$,
respectively.
Denote by $d^p$ the number of negative eigenvalues of the matrix
$J^{-1}H_{xx}(x^p)$, and by $n_1^p(T)$ and $n_2^p(T)$ the numbers
of the frequencies of first and second kind satisfying the
inequality $\omega_r^p >2\pi/T$.
Let
$$
m(T)= \left| \sum_{p=1}^k (-1)^{d^p}[n_1^p(T)-n_2^p(T)] \right|\,. \eqno (1.3)
$$
The following theorem gives a lower bound for the number of
$T$-periodic solutions.
\begin{theorem} For $T<2\pi/\alpha$, equation (1.1) has at least
$ m(T)$ periodic solutions with minimal period $T$.
\end{theorem}
The proof (given in Section 3) is based on the following arguments.
It appears that at least $ m(T)$ Lyapunov families of periodic solutions
$x_j(t,s)$
with the initial minimal periods $T_j(0)T$; therefore, $x_j(t,s)=T$
for some $s=s_j$.
\section{Preliminary remarks}
First let us recall some known facts relating to periodic
solutions of autonomous Hamiltonian systems.
For any solution $x(t)$ of (1.1), with
$$
H(x(t))\equiv h=\mbox{constant}\,, \eqno (2.1)
$$
and using this integral, equation (1.1) is reduced to a system of order
$2n-1$ depending on the parameter $h$. As a result, a periodic solution
$x(t)$ belongs to a one-parameter family.
The existence of such families in a neighbourhood of an
equilibrium position $x^p$ is established by the Lyapunov center theorem
\cite{Ly}. Namely, if for some $k$, the values $\omega_r^p$ satisfy the
condition
$$
\frac{\omega_r^p}{\omega_k^p}\ne m,\quad m=1,2,\dots;\quad
r=1,\dots,r_p,\,r\ne k, \eqno (2.2)
$$
then there exists a unique family of periodic solutions $x_k^p(t,s)=
x_k^p (t+T_k^p (s),s)$ such that $x_k^p(t,s)\to x^p$ and $T_k^p(s)\to
2\pi/\omega_k^p$
as $s\to 0$. Thus the $r_p$ families bifurcate from an
equilibrium point $x^p$.
Under continuation of such a family in the
parameter $ s$ branching can occur. It appears (J.Mallet-Paret and
J.Yorke \cite{Ma} that from the corresponding continuum of solutions one can
chose a one-parameter family of solutions (\lq\lq snake") $x_k^p(t,s)$
possessing the following properties.
The snake may terminate at an equilibrium point $x^q$;
in other words, different families emanating
from the points $x^p$ and $x^q$ may coalesce
(it is possible that $q=p$). Otherwise, the snake is continuable
to an arbitrary large value of the modulus
$$
M_k^p(s)=T_k^p(s)+| h_k^p(s)| +\max_t \| x_k^p(t,s)\| \eqno (2.3)
$$
where $h_k^p(s)=H(x_k^p(t,s))$. The minimal
period of the snake $T_k^p(s) $ is continuous but at most countable
number of points $s_r$
where it drops by a factor $q>1$ (at such a point, a family
with the minimal period $T_k^p(s_r)/q $ may branch off the snake). The left and
right
limits of $T_k^p(s)$ at $s_r$ are equal.
By scaling the time $t=\tau T$, equation (1.1) is reduced to
$$
Jx^{\prime}=TH_x(x), \qquad x^{\prime} =dx/d\tau\,, \eqno (2.4)
$$
so that 1-periodic solutions of (2.4) correspond to $T$--periodic
solutions of (1.1). The
variational equation associated with such a solution $x(\tau)$ is
$$
Jy^{\prime}=TA(\tau)y, \qquad A(\tau)=H_{xx}(x(\tau)). \eqno (2.5)
$$
Let $\rho_k, \; k=1,\dots,2n $ be the Floquet multipliers of (2.5) (the
eigenvalues
of the monodromy matrix $W(T)$ where $W(t)$ is the matrix of a fundamental
system of solutions satisfying the condition $W(0)=I_{2n}$).
Since (2.4) is autonomous, (2.5) has always a multiplier $\rho = 1$
corresponding to the periodic solution
$y_1(\tau) = x^{\prime}(\tau)$. In view of integral (2.1), the
multiplicity of this multiplier equals two \cite{Po}. Hereafter, we assume
$\rho_1=\rho_2=1$. If the corresponding elementary divisors of the matrix
$W(T)$ are simple, there exists one more $1$-periodic solution $y_2(\tau)$.
In the case of non-simple divisors the second solution is of the form
$y_2(\tau) =f_2(\tau)+\tau y_1(\tau)$ where $f_2(\tau+1)=f_2(\tau)$.
Substituting this solution in (2.5), we find that $f_2(\tau)$
satisfies the equation
$$
Jf_2^{\prime}=TA(\tau)f_2 - Jy_1(\tau)\,. \eqno (2.6)
$$
If $\rho_k\ne 1$ for $k>2$, the corresponding invariant subspaces of
the matrix $W(T)$ are $J$-orthogonal \cite{Ya}, i.e.,
$$
(Jy_r(0),y_q(0))=0 \qquad \mbox{for } r\le 2,\; q>2\,. \eqno (2.7)
$$
If $\rho_k = 1$ for some $k > 2$, relation (2.7) is reached by an
appropriate
choice of the vectors associated with the multiplier $\rho = 1$.
Setting $r=1$ in (2.7), and taking into account that $Jy_1=H_x$ and that
$(Jy_1,y_1)=0$, we find
$$
(H_x(x(0)), y_q(0))=0 \qquad\mbox{for } q\ne2. \eqno (2.8)
$$
As $H_x(x(0))\ne0$, from (2.8) it follows
$$
(H_x(x(0)), y_2(0))\ne0. \eqno (2.9)
$$
Fixing the energy $h$ and one of the coordinates (so that the trajectory of
$x(\tau)$ is transversal to the corresponding $(2n-2)$--dimensional disc
$B$),
we obtain the Poincare map $G(v,h)$ with $G(v_0,h)=v_0$
where $v_0$ corresponds to the solution
$x(\tau)$.
The eigenvalues of the matrix of partial derivatives $G_v(v_0,h)$
are the multipliers $\rho_3,\dots,\rho_{2n}$ of equation (2.5).
Let $x(t,s)$ be a snake, then
$$
v_0(s)=G(v_0(s),h(s)). \eqno (2.10)
$$
Differentiating (2.10), we obtain
$$
D(s)v_{0s}(s) = h_s(s)G_{h}(v_0(s),h(s)) \eqno (2.11)
$$
where $D(s)=[I_{2n-2}- G_v(v_0(s),h(s))]$, $h_s(s) = dh(s)/ds$, \newline
$G_{h}(v_0,h) = \partial G(v_0,h)/\partial h $, and $v_{0s}(s)=dv_{0}(s)/ds$.
As is seen from (2.11), if $h_s(s_k) = 0$, then $U(s_k) = \det D(s_k) = 0$
(and, therefore,
$\rho_q(s_k) = 1$ for some $q>2$ \cite{Ma}).
Generically, $h(s)$ is a Morse function
($h_{ss}(s_k)\ne 0 $ for $h_{s}(s_k)= 0 $ \cite{Au}) and $v_{0s}(s_k)\ne 0$, so
the functions
$h_s(s)$ and $U(s)$ change their signs at the same points $s_k$.
As is mentioned above, the frequencies $\omega _k^p$ are classified
into
that of first or second kind depending on the sign of $l_k^p$
(this complies with the Krein's classification \cite{Kr} of the corresponding
multipliers $ \rho_k^p=\exp(i\omega_k^p T)$ of the linearized system).
If $ H_{xx}(x^p) > 0$
or $H_{xx}(x^p )<0$ (i.e., $x^p$ is a minimum or a maximum of $H(x)$) all
eigenvalues of the matrix $J^{-1} H_{xx}(x^p )$ are purely imaginary \cite{Ya}.
Taking into account that $J^{-1} H_{xx}(x^p )x_k^p =i\omega_k^p x_k^p$ ,
we find
$$
l_k^p =i(Jx_k^p,x_k^p)=(H_{xx}(x^p)x_k^p,x_k^p)/\omega _k^p,
$$
so $l_k^p>0 $ or $l_k^p <0 $ and, therefore, all frequencies $\omega _k^p,\,
k=1,\dots,n$ are
of first or second kind, correspondingly. If the matrix $H_{xx}(x^p)$
is indefinite, there exist frequencies of each kind. Note that bilateral
bounds for their numbers expressed in the numbers of positive and
negative eigenvalues of the matrix $H_{xx}(x^p )$ are obtained in \cite{Z3}.
\section{Proof of main Theorem}
First let us establish some preliminary results. Let $x_k^p(t,s)$
be a Lyapunov family ($x_k^p (t,s)\to x^p$, $T_k^p(s)\to2\pi /\omega_k^p$
as $s\to 0$). The following lemma describes the behavior of the corresponding
energy $h_k^p (s)$ in the vicinity of $x^p$ depending on the kind of
the frequency $\omega_k^p$.
\begin{lemma} For small s, the function $h_k^p(s)$ increases or
decreases with $s$ if the frequency
$\omega_k^p$ is of first or second kind, respectively.
\end{lemma}
\paragraph{Proof}. Setting $x=x_k^p(t,s)$ in (2.1) and differentiating it
with respect to $s$, we obtain
$$
h_{ks}^p(s)=dh_{k}^p(s)/ds=(H_x(x_k^p(t,s)), x_{ks}^p(t,s))= (J\dot
x_{k}^p(t,s),x_{ks}^p(t,s)). \eqno (3.1)
$$
For small $s$, one can assume $s=| h-H(x^p)|$, then \cite{Ly}
$$
x_k^p(t,s)=s^{1/2}x_k^p \exp(i\omega _k^p t) + O(s,t) \eqno (3.2)
$$
where $O(s,t)/s^{1/2} \to 0 $ as $s \to 0$.
From (3.1) and (3.2) we obtain
$$
h_{ks}^p(0)=1/2i\omega_k^p(Jx_k^p,x_k^p)=1/2\omega _k^p l_k^p. \eqno (3.3)
$$
Thus, the sign of $h_{ks}^p(0)$ coincides with that of $l_k^p$.
The lemma is proved. \hfill$\fbox{\ }$\smallskip
Let $h(s)$ and $T(s)$ be the energy and period of a one-parameter family
$x(t,s)$.
\begin{lemma} If $h_s(s_k)=0$ , then $T_s(s_k)=0$.\end{lemma}
\paragraph{Proof}
Setting $x=x(\tau, s)$ in (2.4) and differentiating
it with respect to $s$, we obtain
$$
Jx^{\prime}_s=TA(\tau,s)x_s + T_sH_x(x(\tau,s)). \eqno (3.4)
$$
Let $T_s(s)\ne 0$. Taking into account that
$$H_x(x(\tau,s))= Jx^{\prime} (\tau,s)/T=Jy_1(\tau,s)/T\,,$$
we find that the function $- x_s(\tau,s)T(s)T_s^{-1}(s)$ satisfies
(2.6). Therefore, $x_s(\tau,s)$ may be represented in the form
$$
x_s(\tau,s) =-T_s(s)T^{-1}(s) [y_2(\tau,s)-\tau y_1(\tau,s)]+
\sum_k a_k y_k(\tau,s)\,, \eqno (3.5)
$$
where the sum includes the 1--periodic solutions $y_k(\tau,s)$ of (2.5),
$x_s(\tau,s)=x_s(\tau+1,s)$.
From (2.8), (2.9) and (3.5) we obtain
$$
h_s(s)=(H_x(x(\tau,s)),x_s(\tau,s))=-T_s(s)T^{-1}(s)(H_x(x(\tau,s)),
y_2(\tau,s))\ne 0\,. \eqno (3.6)
$$
Thus, $h_s(s)\ne 0$ for $T_s(s)\ne 0$. The lemma is proved.
\hfill\fbox{\ }\smallskip
Let $x_k^p (t,s)$ and $ x_r^q (t,s)$ be Lyapunov families; if they belong
to the same snake, they merge under continuation in $s$.
The following lemma (having a dominant role in the proof of
the above Theorem) indicates cases when different families certainly belong
to
different snakes.
Let $\mu_l^m=1$ and $\mu_l^m=2$ for a frequency
$\omega_l^m$ of first and second kind, respectively. Recall that
$d^m$ denotes the number of negative eigenvalues of the matrix
$J^{-1}H_{xx}(x^m)$.
\begin{lemma} If the values $\mu_k^p +d^p$ and $\mu _ r^q+d^q$
are both odd or even, the families $x_k^p (t,s)$ and $ x_r^q(t,s)$
belong to different snakes. \end{lemma}
\paragraph{Proof} Suppose $x_k^p(t,s)$ and $x_r^q(t,s)$ belong to the same
snake $x(t,s)$. We also assume that $x(t,s)=x^p_k(t,s)$ for small $s$,
then $x(t,s_*-s)= x^q_r(t,s)$, $h(0)=H(x^p)$ and $h(s_*)=H(x^q)$
for some $s_*$. Let $s_k,\, k=1,2,\dots$ be successive critical points
of $h(s)$ on $(0,s_*)$, ($h_s (s_k)=0$).
As is shown above, $U(s_k)= 0$; generically, $h_{ss}(s_k)\ne 0$
and $U_s(s_k)\ne 0$.
Suppose first that $\mu_k^p=\mu_r^q$ (i.e., the frequencies $\omega_k^p$ and
$\omega_r^q$ are of the same kind). By Lemma 1, the signs of
$h_{ks}^p(s)=h_s(s)$ and $h^q_{rs}(s)=-h_s(s_*-s)$ for small $s$ coincide;
therefore, the total number of the points $s_k$ is odd. It follows that
the signs of $U(0)$ and $U(s_*)$ are different.
Clearly, $U(s)=(\rho_3(s)-1)\dots(\rho_{2n}(s)-1)$. The complex
multipliers $\rho_k(s)$ are conjugate, so
$U(s)>0 $ or $U(s)<0$ when, respectively,
the number of multipliers $\rho_k(s)\in (0,1)$ is even or odd. Observing
that
$A(\tau,0)=H_{xx}(x^p)$ and $A(\tau,s_*)=H_{xx}(x^q)$, we find that
the respective numbers of the multipliers equal $d^p$ and $d^q$. Hence,
one of these values and, therefore,
one of the sums $\mu_k^p +d^p$ and
$\mu _ r^q+d^q$ is odd and another is even.
Suppose now that $\mu_k^p\ne \mu_r^q$. Here the signs of
$h_{ks}^p(s)$ and $h^q_{rs}(s)$ are different, so the number
of $s_k\in (0,s_*)$ is even. Therefore,
the signs of $U(0)$ and $U(s_*)$ coincide, so, both of the values
$d^p$ and $d^q$ are odd or even and, thus, as in the previous case, one of the
sums $\mu_k^p +d^p$ and
$\mu _ r^q+d^q$ is odd and another is even. Thus, only under this condition
different families may belong to the same snake.
The lemma is proved. \hfill\fbox{\ }\smallskip
Note that from Lemma 3 it follows that
families $x_k^p (t,s)$ and $ x_r^p(t,s)$ emanating from the same
equilibrium position $x^p$
and corresponding to frequencies $\omega_k^p$ and $\omega_r^p$ of
the same kind cannot merge together.
The above results enable us to prove readily the Theorem.
\paragraph{Proof of Theorem 1} By definition, $n_1^p(T)$ and $n_2^p(T)$
are, respectively, the numbers of the frequencies of first and second kind
satisfying the inequality $\omega_k^p>2\pi/T$. Taking into account Lemma~3,
we find that at least $m(T)$ of the corresponding families
$x^p_k(t,s)$ with the initial periods $T_k^p(0)T $ or is continuable to an arbitrary
large value of the modulus $M^p_k(s)$. Clearly, in the first case $T_k^p(s)=T $
for some $s$. Let us prove that the same is true for the second case.
Let $X(t)=\| x(t)\|$, $X_- = X(t_-) = \min_t X(t) $ and
$X_+ = X(t_+) = \max_t X(t) $
where $x(t)=x(t+T)$ is a solution of (1.1), $0\le t_-r\,, \forall t$.
By $(H)$, the value $\alpha$ may serve
as a Lipschitz constant for the function $J^{-1}H_x(x)$ with
$\| x \|>r $; so
from a theorem by Yorke \cite{Yo} it follows that
$$
T_k^p(s)>2\pi/\alpha\,. \eqno (3.7)
$$
As mentioned above, at some points $s_r$ the minimal period of the solution
$x^p_k(t,s_r)$ may be equal $T_k^p(s_r)/q$ where $q $ is an integer \cite{Ma}.
Let us show that $T_k^p(s_r)$ is the minimal period of $x^p_k(t,s_r)$ for
some $s\ne s_r$.
Really,
for $T=T_k^p(s_r)/q$, variational equation
(2.4) has a multiplier $\rho_{m}^{\prime}=\exp(2\pi i/q) \quad (m>2)$ \cite{Ma};
so, for $T=T_k^p(s_r)$, the corresponding multiplier
$\rho_{m}=(\rho_{m}^{\prime})^q=1$. Therefore, $h_s(s_r)=0$ and, by Lemma 2,
$T^p_{ks}(s_r)=0$. Generically, $s_r$ is a local extremum of the function
$T_k^p(s)$,
so, for $T\in (2\pi/\omega_k^p,2\pi/\alpha)$, there exists
$s\ne s_r$ such that $ T_k^p(s)=T_k^p(s_r)$
is the minimal period of $x^p_k(t,s)$.
The theorem is proved.
\section{Conclusion}
Theorem 1 gives a lower bound for the number of periodic
solutions with a prescribed minimal period $T$.
Note that if a system is asymptotically subquadratic (i.e.,
$H(x)\| x\|^{-2}\to 0 $ as $\| x\| \to \infty$), then the value
$\alpha$ in $(H)$ may be taken as small as one likes, so Theorem 1
enables one to establish the existence of periodic solutions
with an arbitrary large minimal period.
Suppose that a system has a
unique equilibrium position $x=0$ and $H_{xx}(0)>0$ or $H_{xx}(0)<0$.
As shown above, the frequencies $\omega_k,\, k=1,\dots,n $ are of first or
second kind. The corresponding families cannot coalesce as $s$ increases.
Therefore,
for any $T<2\pi/\alpha$, there exist at least $n_1(T)$ periodic solutions
with the minimal period $T$ where $n_1(T)$ is the number of frequencies
$\omega_k >2\pi/T$. In particular, for an asymptotically subquadratic system,
there exist at least $n$ periodic solutions with any minimal period
$T>2\pi/\omega_1$ where $\omega_1$ is the smallest frequency of the
linearized
system.
Note that these results
cannot be improved without additional information about $H(x)$.
One can easily construct a Hamiltonian such that
$\| x_k(t,s)\|\to \infty$ and the periods $T_k(s)$ increase
monotonically to $2\pi/\alpha$ as $s\to\infty$. Clearly, here the number
of solutions with a minimal
period $T$ equals $n_1(T)$, i.e., coincides with the lower bound obtained.
\begin{thebibliography}{00}
\bibitem{Am} Ambrosetti A. and G.Mancini: {\em Solutions of minimal period
for a class of convex Hamiltonian systems}, Math. Ann.,
{\bf 255}(1981), 405-421.
\bibitem{Au} Aubin J.-P. and I.Ekeland., {\em Applied Nonlinear Analysis},
New York, Wiley, 1988.
\bibitem{Cl} Clarke F.H. and I.Ekeland, {\em Hamiltonian trajectories having
prescribed minimal period}. Comm. Pure Appl. Math., {\bf 33}(1980), 103-116.
\bibitem{Ek} Ekeland I. and H.Hofer, {\em Periodic solutions with prescribed
minimal period for convex autonomous Hamiltonian systems}, Invent.
Math., {\bf 81}(1985), 155-188.
\bibitem{Gi} Girardi M. and M.Matseu, {\em Periodic solutions of convex
Hamiltonian systems with a quadratic growth at the origin and superquadratic
at infinity}, Ann. Mat. Pura. Appl., {\bf 147}(1987), 21-72.
\bibitem{Ha} Hartman P., {\em Ordinary differential equations}, Wiley,
New York, 1964.
\bibitem{Kr} Krein M.G., {\em Principles of the theory of $\lambda$-zones of
stability of a canonical system of linear differential equations with
periodic coefficients} (in Russian), In Memory of A.A.Andronov,
Moscow, Izd AN SSSR, (1955), 413-498.
\bibitem{Ly} Lyapunov A., {\em Probleme generale de la stabilite du
mouvement}, Ann. Fac. Sci., Toulouse, {\bf 2}(1907), 203-474.
\bibitem{Ma} Mallet-Paret J. and J.Yorke, {\em Snakes: oriented families of
periodic orbits, their sources, sinks and continuation}, J. Diff. Eq.,
{\bf 43}(1982), 419-450.
\bibitem{Po} Poincar\'e H., {\em Les methodes nouvelles de la mecanique
celeste}, {\bf 1}, Paris, Gauthier-Villars, 1892.
\bibitem{Ra} Rabinowitz P.H., {\em Critical point theory and applications
to
differential equations: a survey. Topological Nonlinear
Analysis. Progress in Nonlinear Differential Equations and their
applications}, Birkhauser, Bosotn, {\bf15}(1995), 463-513.
\bibitem{Va} Van Groesen E.W.C., {\em On small period, large amplitude
normal modes of natural Hamiltonian systems}, Nonlinear Analysis TMA,
{\bf 10}(1986), No. 1, 41-53.
\bibitem{Ya} Yakubovitch V.A. and V.M.Starzhinsky, {\em Linear differential
equations with periodic coefficients}, Wiley, New York, 1980.
\bibitem{Yo} Yorke J.A., {\em Periods of periodic solutions and the Lipschitz
constant}, Proc. Amer. Math. Soc., {\bf 22}(1969), 509-512.
\bibitem{Z1} Zevin A.A., {\em Qualitative analysis of periodic oscillations
in classical autonomous Hamiltonian systems}, Int. J. Non-Linear Mech.,
{\bf 28}(1993), No. 3, 281-290.
\bibitem{Z2} Zevin A.A., {\em Nonlocal generalization of the Lyapunov theorem},
Nonlinear Analysis TMA, {\bf 28}(1997), No. 9, 1499-1507.
\bibitem{Z3} Zevin A.A., {\em On the theory of linear gyroscopic systems},
J. Appl. Maths. Mechs., {\bf 60}(1996), No. 2., 227-232.
\end{thebibliography}
\bigskip
{\sc A.A. Zevin}\\
Transmag Research Institute, Ukrainian Academy of Sciences\\
320005 Dniepropetrovsk, Piesarzhevsky 5, Ukraine\\
e-mail address: zevin@transmag.vidr.dp.ua
\end{document}