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\title{Limit Cycles and Complex Geometry}
\author{Youry R. Romanovsky\\
\normalsize
{\it Program Systems Institute of the Russian Academy of Sciences}\\
\normalsize
{\it Pereslavl--Zalessky 152140 Russia}\\
\normalsize
{{\sl E--mail\/}: \verb!romanov@math.botik.yaroslavl.su!}}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We study the relation between dynamical systems and linear
connections in domains of the complex plane {\bf C}. Any dynamical
system generates a unique connection without torsion such that the
operation of parallel displacement is compatible with the complex
structure and preserves the direction field of the system. The holonomy
group consists of homothetic transformations. The operation of parallel
displacement along each limit cycle does not depend on the choice of
any particular complex structure and is given by the eigenvalue of the
Poincar\'{e} map. We give an example of the dynamical system whose
connection is flat and limit cycles are structurally stable.
\end{abstract}

\section{Introduction}

Let $\gamma$ be a limit cycle of the dynamical system
$$
{dx\over dt}=P(x,y),\qquad{dy\over dt}=Q(x,y).
$$
Then the eigenvalue $\lambda$ of the Poincar\'{e} map,
{\it i.e.} the first return map, is given by
$$
\lambda=\exp\int_{\gamma}(P_x+Q_y)dt.
$$
Our main goal is to give a differential geometric interpretation of
this formula. In other words, we introduce the operator of parallel
displacement
$$
H_{\gamma}\colon T_{z_0}(M)\to T_{z_1}(M)
$$
along each path
$$
\gamma\colon [0,1]\to M,\qquad\gamma(0)=z_0,\quad\gamma(1)=z_1,
$$
in the domain
$$
M={\bf R}^2\setminus \{\mbox{\rm stationary points of the dynamical system}\},
$$
such that
\begin{itemize}
\item $H_{\gamma}$ is compatible with the complex structure $z=x+iy$,
      {\it i.e.}
      $H_{\gamma}$ is given by a matrix of the form
      $$
      \left(\begin{array}{cr}
            a & -b \\
            b &  a
            \end{array}
      \right)
      $$
      with respect to the tangent frame
      $\partial /\partial x,\;\partial /\partial y$;
\item $H_{\gamma}$ is a linear connection without torsion on the
      tangent bundle $T(M)$;
\item $H_{\gamma}$ preserves the field of directions $Qdx-Pdy=0$.
\end{itemize}
In Section 3, we prove that for each differential equation
$$
{dy\over dx}={Q(x,y)\over P(x,y)}
$$
there exists a unique such connection $H_{\gamma}$. The holonomy group
of $H_{\gamma}$ consists of homothetic transformations. In Section 4,
we show that for each limit cycle $\gamma$ the holonomy transformation 
$H_{\gamma}$ is given by the multiplicator $\lambda$, {\it i.e.}
$$
H_{\gamma}=\lambda I.
$$
Thus, we conclude that the operation of parallel displacement along
limit cycles does not depend on the choice of any particular complex
structure.

Finally, we study differential equations with a flat connection. Such
equations are generated by meromorphic differential 1-forms. The poles
correspond to the stationary points. The residues contain some information
about limit cycles. A spectacular example is
$$
(1+i\nu){dz\over z}.
$$
This meromorphic differential generates the dynamical system
$$
{dx\over dt}=x\cos\phi-y\sin\phi,\qquad
{dy\over dt}=x\sin\phi+y\cos\phi,
$$
where
$$
\phi=\nu\log\sqrt{x^2+y^2}.
$$
This system has a unique stationary point of index $1$ and has an infinite
set of limit cycles
$$
x^2+y^2=\exp{\pi(2k+1)\over\nu},\qquad k=0,\pm 1,\pm 2,\ldots,
$$
with the multiplicator
$$
\lambda=e^{2\pi\nu\sigma},\qquad\sigma=(-1)^{k+1}.
$$

We must emphasize that the idea to study differential equations within
the framework of the theory of connections dates back to Elie Cartan [1].
On the other hand, we would like to mention a sporadic example of
P.~Chirokoff [2] and P.~Hendl\'{e}~[3], which illustrates the relation
between differential equations, complex structures and linear connections.
We discuss this example in Section 2.

\section{The Chirokoff equation}

In 1917, P.~Chirokoff [2] raised the following question.
{\it Trouver l'int\'{e}grale de l'\'{e}quation diff\'{e}rentielle suivante:}
$$
{dy\over dx}=\tan(xy).
$$
In 1918, P.~Hendl\'{e} [3] proved that the equation can be integrated
by quadrature, with solution
$$
\int_{0}^{y} e^{{}^{\scriptstyle {1\over2}(t^{2}-x^{2})}} \cos(xt)\,dt=\const.
$$
The integration procedure of Hendl\'{e} looks like a somewhat enigmatical
trick. An interesting feature of that trick is the use of complex numbers.
Keeping in mind this feature only, we will solve the Shirokoff equation
following the integration strategy of Sophus~Lie.

An arbitrary differential equation can be wtitten as
$$
{dy\over dx}=\tan(\alpha),\qquad\alpha=\alpha(x,y),
$$
or as a Pfaff equation defined by the 1-form
$$
\Omega^{\alpha}=\sin\alpha\,dx-\cos\alpha\,dy.
$$
According to Lie's integration strategy, we ought to find
a symmetry, {\it i.e.} a vector field
$$
V=\xi(x,y){\partial\over\partial x}+\eta(x,y){\partial\over\partial y}
$$
such that
$$
d\>{\Omega^{\alpha}\over\Omega^{\alpha}(V)}=0.
$$
Then the solution of the equation is given by
$$
\int{\Omega^{\alpha}\over\Omega^{\alpha}(V)}=\const.
$$
Now, the problem is to find the symmetry $V$. For this aim, we will
identify $V$ with the complex valued function
$$
w=\xi+i\eta
$$
and assume that $w$ holomorphically depends on the complex variable
$$
z=x+iy,
$$
that is
$$
{\partial w\over\partial\overline{z}}=0.
$$
A direct computation shows that the function $w(z)$ is a symmetry,
if and only if $\eta_x=\alpha_x\xi+\alpha_y\eta$, which is equivalent
to the condition
$$
\Im\left({dw\over dz}-\Gamma^{\alpha}w\right)=0,
$$
where the coefficient $\Gamma^{\alpha}$ is given by
$$
\Gamma^{\alpha}=2i{\partial\alpha\over\partial z}.
$$
In particular, if $\Gamma^{\alpha}$ depends holomorphically on $z$, then
we can construct the field of ``parallel'' vectors $w(z)$ such that
$$
w(z_1)=\lambda(z_0,z_1)\,w(z_0),\qquad
\lambda(z_0,z_1)=\exp\int_{z_0}^{z_1}\Gamma^{\alpha}dz.
$$
For the Chirokoff equation, we have
$$
\Gamma^{xy}=z,\qquad w(z)=w(0)\exp{z^2\over 2}.
$$
Setting $w(0)=i$, we get the Hendl\'{e} solution.

\section{Real Connections}

Consider the differential 1-forms
$$
\vartheta={dz\over w},\qquad\omega={dw\over w}-\Gamma dz,
$$
where $w=\xi+i\eta$ is the complex coordinate of the vector tangent to
$M\subseteq {\bf R}^2$ at the point with the complex coordinate $z=x+iy$,
and $\Gamma$ is a complex valued function of $(x,y)\in M$. Since $M$ is of
complex dimension one, the {\it complex tangent vector}
$$
w\,{\partial\over\partial z}
$$
can be identified with the {\it complex frame} of $M$. Therefore,
$\vartheta$ and $\omega$ can be viewed as 1-forms on the bundle of all
complex frames over $M$, which we denote by $F_c(M)$. This bundle is
a G-structure. Its structure group is the multiplicative group
${\bf C}\setminus\{0\}$. The 1-form $\vartheta$ is nothing else but the
{\it canonical differential form} on the G-structure $F_c(M)$. Roughly
speaking, the value $\vartheta_u(v)$ is the coordinate of the complex
tangent vector $v$ with respect to the complex frame $u$. The 1-form
$\omega$ defines a linear connection on the G-structure $F_c(M)$. The
{\it parallel displacement} along each path $\gamma$ is given by
$$
w\,{\partial\over\partial z}\;\;\mbox{\rm at}\;\; z_0\quad
\stackrel{H_{\gamma}}\longrightarrow\quad
\lambda(\gamma)\,w\,{\partial\over\partial z}\;\;\mbox{\rm at}\;\;z_1,
$$
where
$$
\lambda(\gamma)=a(\gamma)+i\,b(\gamma)=\exp\int_{\gamma}\Gamma dz.
$$
It is not hard to see that the 1-forms $\vartheta$ and $\omega$ satisfy
the structure equation of Elie Cartan
$$
d\vartheta+\omega\wedge\vartheta=0,
$$
which means that the connection $\omega$ has vanishing {\it torsion}.
The {\it curvature} of $\omega$ is given by the exterior derivative
$$
d\omega=K\,\vartheta\wedge\overline{\vartheta},
$$
where
$$
K=w\,\overline{w}\,{\partial\Gamma\over\partial\overline{z}}.
$$
We say that the connection $\omega$ is {\it flat},
if the curvature $K$ vanishes.

Suppose $\Gamma=\Gamma^{\alpha}$ is the function associated to the
differential equation $\Omega^{\alpha}=0$ (see Section 2). Then we have
$$
\Gamma^{\alpha}dz=i\,d\alpha-\ast\,d\alpha,
$$
where $\ast$ is the Hodge operator:
$$
\ast\left(p\,dx+q\,dy\right)=-q\,dx+p\,dy.
$$
Thus, if $\gamma$ is a closed path, then the imaginary part of the integral
$$
\oint_{\gamma}\Gamma^{\alpha}dz=
i\,\oint_{\gamma}d\alpha+\oint_{\gamma}(-\ast d\alpha)
$$
is an integer multiple of $2\pi$, and the holonomy transformation
$H_{\gamma}$ is given by the real number
$$
\lambda(\gamma)=\exp\oint_{\gamma}(-\ast d\alpha).
$$
This is why the differential 1-form
$$
\omega^{\alpha}={dw\over w}-\Gamma^{\alpha} dz
$$
will be called the {\it real connection} associated with the differential
equation $\Omega^{\alpha}=0$.

\bigskip

{\bf Proposition.} {\it There is an intrinsic correspondence between
differential equations and real connections.}

\medskip

Actually, the change of variables
$$
z\mapsto\varphi(z),\qquad w\mapsto\varphi'(z)w
$$
takes back the forms
$$
\Omega^{\alpha}\qquad{\rm and}\qquad\omega^{\alpha}
$$
to
$$
|\varphi'|\Omega^{\alpha-\beta}\qquad{\rm and}\qquad\omega^{\alpha-\beta}
$$
respectively, where
$$
\beta=\arg\varphi'.
$$
This proves that the
correspondence between differential equations and real connections
is independent of any particular coordinate system. Moreover, this
correspondence is analogous to the well known relation between
Riemannian metrics and Levi-Civita connections. Let us follow up
this analogy.
\begin{enumerate}
\item Any Riemannian metric is parallel with respect to the
associated Levi-Civita connection. This is a unique linear connection,
which has vanishig torsion and preserves the Riemannian metric.

{\it Any differential equation is  parallel with respect to
the associated real connection. This is a unique linear connection
on $F_c(M)$, which has vanishig torsion and preserves the differential
equation.}

{\it Proof.} An arbitrary linear connection on $F_c(M)$ is given by
$$
\omega={dw\over w}-\Gamma dz-\Pi d\overline{z}.
$$
The structure equation
$$
d\vartheta+\omega\wedge\vartheta=
{\rm T}\,d\vartheta\wedge d\overline{\vartheta},\qquad
{\rm T}=w\,\overline{w}\,\Pi,
$$
shows that $\omega$ has vanishing torsion, if and only if $\Pi=0$.
In this case, $\omega$ preserves the equation $\Omega^{\alpha}=0$,
if and only if $\Gamma=\Gamma^{\alpha}$.
\item The Levi-Civita connection is flat if and only if the
Riemannian metric is locally Euclidean.

{\it The real connection is flat if and only if the
differential equation is locally holomorphically equivalent to
the differential equation of straight lines.}

{\it Proof.} The curvature $K$ of $\omega^{\alpha}$ vanishes, if and
only if
$$
{\partial^2\alpha\over\partial z\,\partial\overline{z}}=0.
$$
This means that the angle $\alpha$ is a harmonic function and thereby
is the argument of a local holomorphic map $z\mapsto\varphi(z)$, which
takes back the equation $\Omega^{\alpha}=0$ to $\Omega^{0}=0$.
\item The holonomy group of the Levi-Civita connection consists
of rotations.

{\it The holonomy group of the real connection consists
of homothetic transformations.}

\item Christoffel's symbols of the Levi-Civita connection can
be expressed in terms of the Riemannian metric.

{\it Christoffel's symbols of the real connection can
be expressed in terms of the differential equation as follows}
$$
\left( \begin{array}{cc}
\Gamma^1_{11}dx+\Gamma^1_{12}dy & \quad\Gamma^1_{21}dx+\Gamma^1_{22}dy \\
                                &                                 \\
\Gamma^2_{11}dx+\Gamma^2_{12}dy & \quad\Gamma^2_{21}dx+\Gamma^2_{22}dy
       \end{array}
\right) =
\left( \begin{array}{cc}
\ast d\alpha & \quad d\alpha \\
             &         \\
-d\alpha     & \quad\ast d\alpha
       \end{array}
\right).
$$
\end{enumerate}

\section{Limit Cycles}

Each dynamical system on the plane {\bf R}$^2$
$$
{dx\over dt}=P(x,y),\qquad{dy\over dt}=Q(x,y),
$$
defines on the open set
$$
M={\bf R}^2\setminus \{(x,y)\mid P(x,y)=Q(x,y)=0\}
$$
the field of directions
$$
\Omega^{\alpha}\equiv{Q\,dx-P\,dy\over\sqrt{P^2+Q^2}}=0,
$$
where the angle $\alpha$ is defined locally by the cocycle
$$
d\alpha={P\,dQ-Q\,dP\over P^2+Q^2}.
$$
Suppose $M$ is supplied with the {\it complex structure} $z=x+iy$.
Then, as we have seen in Section 3, the differential equation
$\Omega^{\alpha}=0$ generates on $F_c(M)$ a real connection
$\omega^{\alpha}$. The holonomy transformations of $\omega^{\alpha}$
define on the loop space a real valued functional
$$
\gamma\;\mapsto\;\lambda(\gamma)=\exp\oint_{\gamma}\Gamma^{\alpha}dz
=\exp\oint_{\gamma}(-\ast d\alpha).
$$

\bigskip

{\bf Theorem.} {\it If $\gamma$ is a limit cycle of the dynamical system,
then $\lambda(\gamma)$ is the eigenvalue of the Poincar\'{e} map.}

\medskip

{\it Proof.} Applying the Hodge star to the cocycle $d\alpha$, we find
$$
-\ast d\alpha={(PQ_y-QP_y)dx+(QP_x-PQ_x)dy\over P^2+Q^2}.
$$
Using the time $t$ as a parameter on the limit cycle $\gamma$, we compute
the restriction
$$
(-\ast d\alpha)|_{\gamma}=(P_x+Q_y)dt+d\log{1\over\sqrt{P^2+Q^2}}.
$$
Therefore, we have
$$
\lambda(\gamma)=\exp\int_{\gamma}(P_x+Q_y)dt.
$$
The proof is complete.

\bigskip

{\bf Corollary.} {\it If $\gamma$ is a limit cycle of the dynamical system,
then the number $\lambda(\gamma)$ does not depend on the choice of any
particular complex structure.}

\bigskip

{\bf Remark.} This Corollary is analagous to the Gauss--Bonnet theorem,
which states that the integral of the curvature over a compact surface
does not depend on the choice of any particular Riemannian metric, because
this integral is the Eulerian characteristic of the surface
multiplied by $2\pi$.

\bigskip

Now, suppose the real connection is flat. Then, for any closed path $\gamma$,
homotopic in $M$ to a given limit cycle, $\lambda(\gamma)$ is also equal to
the eigenvalue of the Poincar\'{e} map. Moreover, in the flat case the
functional $\lambda(\gamma)$ defines a homomorphism
$$
\lambda\colon\pi_1(M)\to{\bf R}^+
$$
of the fundamental group $\pi_1(M)$ to the group of positive real numbers
${\bf R}^+$. Consider the induced homomorphism
$$
h\colon\pi_1(M)\to{\bf C}
$$
such that
$$
h(\gamma)=\log\lambda(\gamma)+i\,2\pi n=\oint_{\gamma}\Gamma^{\alpha}dz,
$$
where $n$ is an interger equal to the index of the stationary point
surrounded by the generator $\gamma\in\pi_1(M)$, the orientation of
$\gamma$ is supposed to be positive. In the theory of Riemann surfaces,
the homomorphism $h$ is called the {\it period homomorphism}. The problem
is to restore the dynamical system with a prescribed period homomorphism.

\bigskip

{\bf Example.} Suppose we want the dynamical system to have one stationary
point and a srtucturally stable limit cycle. Then we have
$$
M={\bf C}^{\ast},\qquad\pi_1({\bf C}^{\ast})={\bf Z},\qquad
h({\bf 1})=\chi+i\,2\pi,\quad\chi\neq 0.
$$
In other words, the index of the stationary point should be equal to one,
the number
$$
\lambda(\pm{\bf 1})=e^{\pm\chi}
$$
is supposed to be the eigenvalue of the Poincar\'{e} map. Clearly, the
described homomorphism $h$ can be given by the meromorphic differential
$$
\Gamma^{\alpha}dz = \left(1-i{\chi\over 2\pi}\right){dz\over z},
$$
where
$$
\alpha=\arg z+\phi,\qquad\phi={\chi\over 2\pi}\log{1\over |z|}+\const.
$$
The corresponding dynamical system is
$$
{dz\over dt}=\rho e^{i\phi}z,\qquad\rho=\rho(x,y)>0.
$$
In this system the dynamics is oscillatory, that is every trajectory is
asymptotic to a structurally stable periodic orbit
$$
\phi={\pi\over 2}+\pi k,\qquad k=0,\pm1,\pm2,\ldots.
$$

\bigskip

Finally we note that in the same way one can construct more
complicated systems with oscillatory dynamics.
Let $M={\bf C}\setminus\{z_1,\ldots,z_s\}$. Then each period
homomorphism $h$ can be given by
$$
\Gamma^{\alpha}dz=\sum_{j=1}^{s}{h_j\over 2\pi i}\,{dz\over z-z_j}+df(z),
$$
where $f(z)$ is a polynomial, $h_j=\chi_j+i\,2\pi n_j$,
$$
\alpha=\sum_{j=1}^{s} n_j\,\arg(z-z_j)+\phi,
$$
$$
\phi=\sum_{j=1}^{s}{\chi_j\over 2\pi}\,\log{1\over |z-z_j|}+\Im f(z).
$$
The corresponding dynamical system is
$$
{dz\over dt}=\rho e^{i\phi}(z-z_1)^{n_1}\cdots(z-z_r)^{n_r}
(\overline{z}-\overline{z}_{r+1})^{-n_{r+1}}\cdots
(\overline{z}-\overline{z}_s)^{-n_s}.
$$
We assume here that
$$
n_1>0,\ldots,n_r>0\qquad {\rm and} \qquad n_{r+1}<0,\ldots,n_s<0.
$$
If $n_1=1$ and $\chi_1\neq 0$, then, in a small neighbourhood of $z_1$,
the dynamical system takes the form
$$
{d\over dt}(z-z_1)=\rho_1 e^{i(\phi_1+\varepsilon_1)}(z-z_1),
$$
where
$$
\phi_1={\chi_1\over 2\pi}\log{1\over |z-z_1|}+c_1,
$$
and $\varepsilon_1\rightarrow 0$ as $|z-z_1|\rightarrow 0$. This means that
the stationary point $z_1$ is surrounded by a structurally stable periodic
orbit with the multiplicator
$$
\lambda=e^{\pm\chi_1}.
$$

\section*{References}

\begin{enumerate}
\item E. Cartan, {\it Le\c{c}ons sur la Th\'{e}orie des Espaces \`{a}
      Connexion Projective}, Gauthier--Villars, Paris, 1937.
\item P. Chirokoff, {\it Interm\'{e}diaire Math.} {\bf XXIV} (1917) 104.
\item P. Hendl\'{e}, {\it Interm\'{e}diaire Math.} {\bf XXV} (1918) 45--46.
\end{enumerate}

\end{document}