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\begin{document}

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\title{
Delay coupled phase oscillators
}


\author{
H-J. W\"{u}nsche$^{1}$
}

\affiliation{
$^{1}$ Institut f\"{u}r Physik, Humboldt-Universit\"{a}t zu Berlin, Newtonstr.15,  12489 Berlin, Germany\\
}

%\pacs{42.65.Sf, 05.45Xt, 42.55Px, 02.30Ks}

\date{041226}

\begin{abstract}
\end{abstract}

\maketitle

\section{introduction}
%---------------------

To find out how general our findings for the delay-coupled lasers are, I have tried to simplivy the model as much as possible. This is the result. 

\section{model}
%--------------

I consider two coupled phase oscillators
%
\begin{eqnarray} \label{eq:model}
\partial_t {\varphi }_k = \omega _k + K \sin\left(\varphi _l^\tau -\varphi _k\right)
\end{eqnarray}
%
with $k=1,2, l=3-k$. $\varphi (t)$ ist the actual phase of oscillator $k$ and $\varphi _k^\tau (t)=\varphi _l(t-\tau )$ the phase of oscillator $l$ before a delay time $\tau $. This is just the model already assumed by Schuster and Wagner \cite{schuster89} who considered the synchronised states. 

\section{numerical solution}
%---------------------------

I use $2\tau $ as time unit, i.e. $\tau =1/2$. For clarity, I consider symmetrically detuned oscillators $\omega _2=-\omega _1= \Delta$. The detuning $\Delta/\pi$ is varied within $\pm 6$.  With $K=2\pi $, the locking range without delay is exactly $\pm 2$ (see below). This is well reproduced by the numerical solutions. With delay, the frequencies show the clearly the staircases already known from the lasers.

{\bf Conclusion:} the staircases are the general and typical scenario for nearly locked delay-coupled oscillators.

%
\begin{figure}[b] \center
\includegraphics*[width=8cm]{K2pisym.eps}
\caption{\label{fig:K2pisym} 
Detuning scenario of the symmetrically detuned phase oscillators. Thick black: frequencies of both oscillators for increasing detuning. Thick grey: same for decreasing detuning. Thin lines: no coupling (grey) and no delay (red).  
}
\end{figure}
%


\section{analysis}
%-----------------

We introduce phase difference $\Phi=(\varphi _2-\varphi _1)/2$ and mean $\Psi=(\varphi _2+\varphi _1)/2$ as new variables. With some algebra,  Eqs. (\ref{eq:model}) transform to 
%
\begin{eqnarray} \label{eq:adlerdelay}
\partial _t {\Phi} &=& \Delta - K \cos(\Psi-\Psi^\tau ) \sin(\Phi+\Phi^\tau ) 
\\ \label{eq:adlerdelay2}
\partial _t {\Psi} &=& \Omega - K \sin(\Psi-\Psi^\tau ) \cos(\Phi+\Phi^\tau ).
\end{eqnarray}
%
These are the central equations of delayed phase synchronisation. In our example, the mean frequency $\Omega$ is zero. 

\subsection{delay free limit}
%------------------------------

With $\tau =0$, these equations reduce to
%
\begin{eqnarray} \label{eq:adler}
\partial _t {\Phi} = \Delta - K \sin(2\Phi) ~~~ \text{ and } ~~~
\partial _t {\Psi} = \Omega.
\end{eqnarray}
%
This is the well known Adler equation for the phase difference $2\phi$ between the two oscillators. 

\subsubsection{locking}
%----------------------

If $|\Delta| \le K$, the right hand side approaches to zero at $2\Phi = 2\Phi_\text{locked}= $acrsin$(\Delta/K)$, which is the constant phase difference of the locked oscillators. Since $\Psi=\Omega t$, it holds $\varphi _{1,2}=\mp \Phi_\text{locked} +  \Delta \cdot t$.

\subsubsection{unlocking}
%---------------------

Still keeping $\tau =0$, the unlocking of the two oscillators for $|\Delta| > K$ is also well understood. I suppress here the known analytic solutions of the equations in this regime. 

Generic scenario: The two ''frequencies'' split beyond the locking range. The splitting $\delta \omega_0 $ increases in a hyperbolic manner and approaches $2 \Delta$ (see Fig. \ref{fig:K2pisym}). But: these frequencies are average frequencies, i.e. phase increments over a long time divided by this time. Just beyond the locking range, the phase difference $\Phi$ raises extremely nonuniformly. Long plateaus of nearly constant $\Phi$, i.e. of a nearly locked state, are interrupted by short phase slips by $2\pi $. Approximately, the plateau length $T_\text{plateau}$ goes inversely with the frequency splitting,  $T_\text{plateau}\approx2\pi /\varphi \omega_0 $.
\\[2ex] Abbildung machen!!

\subsection{nonzero delay }
%--------------------------

Hypothesis: the staircases are something like synchronisation between the phase-slips and the round trip frequency. 

\subsubsection{locking}
%----------------------

This has been treated by Schuster and Wagner, I should check whether my formulae (\ref{eq:adlerdelay}) yield the same. My fieling: they are simpler. 

\subsubsection{unlocking}
%---------------------

Fig. \ref{fig:K2pisym} shows: 
\begin{itemize}
\item
the ''plateaus'' meander between the zero delay and zero coupling lines.
\item
zero coupling is touched exactly at odd multiples of the round-trip frequency  $\nu_\text{rt}$.
\item
zero delay is touched when the splitting equals even multiples of  $\nu_\text{rt}$.
\item
hysteresis is seemingly due to double-folds at smaller detunings.
\end{itemize}

I can explain the touching points analytically: 

I restrict to symmetric detuning ($\Omega =0$). In this case, $\Psi= $const. is always a solution of Eq. (\ref{eq:adlerdelay2}). Hence, the other equation simplifies to $\partial _t \Phi = \Delta - K \sin(\Phi+\Phi^\tau )$. It follows
%
\begin{eqnarray*}
\partial _t (\Phi+\Phi^\tau ) = 2\Delta - K \left[\sin(\Phi+\Phi^\tau )+\sin(\Phi^\tau +\Phi^{2\tau} )\right].
\end{eqnarray*}
%
If now $\Phi^{2\tau }=\Phi+ m\pi  $ with integer $m$, then 
\begin{itemize}
\item
the $[\ldots]$ vanishes for odd $m$, i.e. the oscillator frequencies equal the no coupling case
\item
$(\Phi+\Phi^\tau )$ obeys the Adler equation (\ref{eq:adler}) for even $m$, i.e., the delay-free frequency. 
\end{itemize}



\section{conclusion}
%-------------------

\begin{thebibliography}{9}
%
\bibitem{schuster89}
H. Schuster and P. Wagner, Progr. Theor. Phys. 81, 939
(1989).
%
\end{thebibliography}
\end{document}
