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\begin{document}

\title{
Adler Equation with Delay}


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

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


\date{050303}

\begin{abstract}
I study the generalized Adler equation derived in the coupled laser paper for delay coupled phase synchronization with symmetric detuning.
\end{abstract}

\maketitle

I consider the equation
%
\begin{eqnarray}\label{eq:chi}
\dot{\phi} = \Delta  -K \left[\sin(\phi )+\sin(\phi^\tau  )\right],
\end{eqnarray}
%
which generalizes
the delay-free Adler equation. First, I shall consider in detail the special solutions corresponding to effectively no coupling or no delay. Then I use this knowledge in order to construct a numerical solution schema for the general case. 

\section{Special Frequency Solutions}
%=====================================

Solutions with
frequencies being integer multiples of $1/2\tau$ define
representative points in the system dynamics. Here, because of
$\phi=\phi^\tau +m\pi$, the coupling for $\phi$ (not $\varphi _k$)
is either zero ($m$ odd) or delay-free ($m$ even). With increasing
$\Delta$, the system meanders between those limits (Fig.\ 5d of PRL). 
Branches with negative slope are unstable,
providing the staircase feature. It is exactly this kind of
self-organization in the phase relation between the oscillators
that governs the synchronization. 

\subsection{zero-coupling frequencies}
%======================================

The solutions in this case are simply
%
\begin{eqnarray}
\phi_m(t)=\Delta_m \cdot t, \hspace{3em} \Delta_m=m \cdot\frac{\pi }{\tau }, \hspace{3em} (m \text{ odd}).
\end{eqnarray}
%

\subsection{zero-delay frequencies}
%======================================
Here we arrive at the orginial Adler equation in the form
%
\begin{eqnarray}
\dot{\phi} = \Delta  -2K \sin(\phi ).
\end{eqnarray}
%
The solution can be obtained by variable separation. It is in general
%
\begin{eqnarray}
\phi(t)=2~\arctan\left(\frac{\omega\tan(\omega t/2 )+ 2K }
                             {\Delta}
                  \right)
         +2\pi ~ \text{int}\left(\frac{\omega t-\pi }{2\pi  }\right), 
         \hspace{3em} \omega= \sqrt{\Delta^2-4K^2}.         
\end{eqnarray}
%
Here the principal value of arctan is to be taken and the int-term ensures continuous increase of the phase. 

The zero-delay case appears at 
%
\begin{eqnarray}
\omega=\omega_m:= m\frac{\pi }{\tau } \hspace{3em}\text{i.e. }  \Delta=\Delta_m:=\sqrt{\omega _m^2+4K^2}.
\end{eqnarray}
%
%
\begin{eqnarray}
\phi_m(t)=2~\arctan\left(\frac{m\tan(m\pi t/2\tau )+ 2K\tau/\pi }
                             {\sqrt{m^2+(2K\tau /\pi )^2}}
                  \right)
         +2~ \text{int}\left(\frac{mt-\tau }{2\tau }\right).         
\end{eqnarray}
%

\section{Numerics: stretching algorithm}
%========================================



\mbc{0.55}{
The idea is to calculate solutions having a given frequency $f_0$ by adjusting $\Delta$. An embedding is used increasing $K$ from 0 to a maximal value in small steps. In each step, $\Delta$ is adjusted until the given frequency $f_0$ is reached. 

The fig%
shows resulting lines of constant $f_0$ in the Detuning-coupling plane. 
The lines start at $K=0$ from $\Delta = 2\pi f_0$. They keep horicontal for odd $\Delta \tau /\pi = 2\tau f_0$ exhibit a maximum bend upwards for even ones, as expected. 

Ending of lines before $4\pi $ is due to unstability of either the solution itself or of the algorithm used. From this point of view, this approach is not suitable. 

} \hfill \mbc{0.4}{\includegraphics*[width=\textwidth]{ad}}

{\footnotesize Calculated with t.050302.r in directory \verb#D:\verb\ede\dfb\coupledlaser\oscis\adlerdelay#.} 


\section{Numerics: pathfollowing from special points}

\end{document}
