\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,\\
\emph{Electronic Journal of Differential Equations},
Conference 22 (2015),  pp. 99--109.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document} \setcounter{page}{99}
\title[\hfilneg EJDE-2015/Conf/22 \hfil Oscillatory solutions induced by time delay]
{Comments on multiple oscillatory solutions in systems with time-delay feedback}

\author[M. Stich \hfil EJDE-2015/conf/22 \hfilneg]
{Michael Stich}

\dedicatory{Dedicated Alfonso Casal on his 70th birthday}

\address{Michael Stich \newline
Non-linearity and Complexity Research Group,
School of Engineering and Applied Science, 
Aston University, Aston Triangle, Birmingham B4 7ET, UK}
\email{m.stich@aston.ac.uk}

\thanks{Published November 20, 2015.}
\subjclass[2010]{35K57, 35B10, 35Q92}
\keywords{Pattern formation; reaction-diffusion system; control}

\begin{abstract}
  A complex Ginzburg-Landau equation subjected to local and global
  time-delay feedback terms is considered. In particular, multiple
  oscillatory solutions and their properties are studied. We present
  novel results regarding the disappearance of limit cycle solutions,
  derive analytical criteria for frequency degeneration, amplitude
  degeneration, and frequency extrema. Furthermore, we discuss the
  influence of the phase shift parameter and show analytically that
  the stabilization of the steady state and the decay of all
  oscillations (amplitude death) cannot happen for global feedback
  only. Finally, we explain the onset of traveling wave
  patterns close to the regime of amplitude death.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

It is well-known that the spatiotemporal dynamics of
reaction-diffusion systems in the vicinity of a Hopf bifurcation are
described by the complex Ginzburg-Landau
equation~\cite{CrossRMP93,Kuramoto84}.  In the relevant case of a
supercritical Hopf bifurcation, the primary solution of this system is
a stable limit cycle appearing at the Hopf bifurcation. However, in
some parameter range (if the Benjamin-Feir-Newell criterion
$1+\alpha\beta <0$ is fulfilled), the oscillations in the
spatially-extended system may be unstable, a phenomenon that is
induced by the diffusive coupling and that is therefore genuine to a
system with spatial degrees of freedom.

During many years, considerable efforts have been made to understand
this type of chaotic behavior and to apply methods to suppress this
kind of turbulence and replace it by simpler, more regular dynamics.
Consequently, control of chaotic states in nonlinear systems is a wide
field of research that we cannot review here~\cite{Scholl07}.  In the
context of the reaction-diffusion systems, the introduction of forcing
terms or global feedback terms have been shown to be efficient ways to
control turbulence \cite{MikhailovPR06}.

Global feedback methods, where a spatially independent quantity (in
many cases a spatial average of a space-dependent quantity) is coupled
back to the system dynamics, have attracted much attention since in
many cases the equations are easier to analyse and to be implemented
experimentally~\cite{Scholl07}.  Nevertheless, local methods have
gained interest in recent years since they allow to access other
solutions of the systems and may also be implemented, such as in the
light-sensitive BZ reaction or in neurophysiological experiments with
spatial arrays of electrodes.

Feedback methods with an explicit time delay widen the range of
possibilities of control that can be applied to the system and provide
the researcher with an additional adjustable parameter. On the level
of the mathematical description, the model equations become delay
differential equations.

Motivated by previous works 
\cite{BetaPD04,CasalMMMAS06,CasalDCDISA06,StichPD10,StichPRE13,StichPRE07},
the complex Ginzburg-Landau equation subjected to a time-delay
feedback with local and global terms is studied here.  Extensive
simulations showed the range of patterns that can be stabilized as
function of the local and global feedback terms~\cite{StichPRE07}. If
the feedback is global, local, or a combination of both, uniform
oscillations, the basic temporally-periodic solution of a system close
to a Hopf bifurcation, can be stabilized.  In~\cite{StichPD10}, we
studied the homogeneous periodic and the homogeneous stationary
(amplitude death) solutions from an analytical point of view,
performed linear stability analysis and derived the limiting curves of
the stability regions.  In this contribution, we dwell into previously
overlooked features of this model, that however have interesting and
important consequences for future applications.

\section{The model system}

Reaction-diffusion systems can display various types of oscillatory
dynamics.  However, close to a supercritical Hopf bifurcation, all
such systems are described by the complex Ginzburg-Landau equation
(CGLE)~\cite{AransonRMP02},
\begin{equation}
  \frac{\partial A}{\partial t}=(1-{\rm i}\omega )A
 -(1+{\rm i}\alpha )|A|^{2}A+(1+{\rm i}
  \beta )\Delta A,
\end{equation}
where $A$ is the complex oscillation amplitude, $\omega$ the linear
frequency parameter, $\alpha$ the nonlinear frequency parameter,
$\beta$ the linear dispersion coefficient, and $\Delta$ the Laplacian
operator. For $1+\alpha\beta<0$ (the Benjamin-Feir-Newell criterion),
the homogeneous periodic solution $A_u={\mathrm e}^{-{\mathrm
    i}(\omega+\alpha)t}$ is unstable and a regime of spatiotemporal
chaos is observed.

The CGLE for a one-dimensional medium with a combination of local and
global time-delayed feedback has been introduced in
 \cite{StichPRE07} and reads
\begin{equation}  \label{eq:model}
  \begin{gathered}
    \frac{\partial A}{\partial t}=(1-{\rm i}\omega )A-(1+{\rm i}\alpha )|A|^{2}A+(1+{\rm i}
    \beta ) \frac{\partial^2 A}{\partial x^2}
    +F,\\
    F = \mu {\mathrm e}^{{\mathrm i}\xi}
    \left[m_l(A(x,t-\tau)-A(x,t))
    +m_g(\bar{A}(t-\tau)-\bar{A}(t))\right],
  \end{gathered}
\end{equation}
where
\begin{equation}
\bar{A}(t)=\frac{1}{L}\int_0^L A(x,t) \, {\rm d}x %\nonumber
\end{equation}
denotes the spatial average of $A$ over a one-dimensional medium of
length $L$. We assume Neumann boundary conditions
\begin{equation}
 \frac{\partial A}{\partial x} \Big|_{0,L}=0,
\end{equation}
%
which is a reasonable choice for reaction-diffusion systems.  The
parameter $\mu$ describes the feedback strength and $\xi$
characterizes a phase shift between the feedback and the current
dynamics of the system.  The parameters $m_g$ and $m_l$ denote the
global and local contributions to the feedback, respectively.  If
$m_l=0$, the case of global time-delay feedback is
retrieved~\cite{BetaPD04}.

As result of extensive simulations, many different spatiotemporal
patterns occurring in this system have been
identified~\cite{StichPRE07}. Besides the basic, homogeneous periodic
solution, we mention standing waves, traveling waves, amplitude death,
complex localized patterns, and spatiotemporal chaos.  The
stabilization of an homogeneous periodic solution and amplitude death
have been considered in~\cite{StichPD10}.  First, we introduce the
homogeneous periodic solution.

\section{Homogeneous periodic solution}

The feedback-induced, homogeneous periodic solution is given by
\begin{equation}
\label{homogen}
A_0(t)=\rho {\mathrm e}^{-{\mathrm  i}\Omega t}.
\end{equation}
It is a solution of \eqref{eq:model} with the
amplitude and frequency given by
\begin{subequations}  \label{eq:uniform}
\begin{gather}    \label{rho}
    \rho = \sqrt{1 + \mu (m_g+m_l)\chi_1},\\
    \label{Omega}
    \Omega = \omega + \alpha + \mu (m_g+m_l)(\alpha\chi_1 -\chi_2)\,.
\end{gather}
\end{subequations}
Here, $\chi_{1,2}$ denote effective modulation terms that can be
positive or negative.  They arise from the feedback and hence depend
on $\xi$ and $\tau$,
\begin{subequations}  \label{eq:chi}%
  \begin{gather}
    \chi_1 = \cos(\xi+\Omega\tau)-\cos\xi,\\
    \chi_2 = \sin(\xi+\Omega\tau)-\sin\xi.
  \end{gather}
\end{subequations}
Equations \eqref{rho},\ref{Omega} do not explicitly depend on $\rho$, only
on $\Omega$.  However, no explicit analytic solution for
\eqref{Omega} can be given because $\chi_{1,2}$ also depend on
$\Omega$. Nevertheless, the solution can be computed using
root-finding algorithms, as done in~\cite{StichPD10}.

In the absence of feedback ($\mu=0$), the standard solution for the
CGLE is recovered: $\rho_0=1$, $\Omega_0=\omega+\alpha$. Applying the
parameters used in~\cite{StichPRE07}, and in particular $\omega=2\pi
-\alpha$, the period of the oscillations is chosen to be
$T_0=2\pi/\Omega_0=1$. As already mentioned, this solution is unstable
for $1+\alpha\beta<0$, also assumed here. We assume $m_l+m_g=1$, but a
generalization to other combinations of $m_l$ and $m_g$ is
straightforward~\cite{CasalDCDISA06}.

\section{Existence of multiple oscillatory solutions}

The multiplicity of uniform oscillatory solutions is a direct
consequence of the above-mentioned Equation \eqref{Omega} and which is an
established fact for various versions of the CGLE at least since
\cite{BetaPD04}. Surprisingly, the shape of the curves that
describe these equations has not been studied in more detail, which
will be done here.

Following \cite{StichPD10}, we show how multiplicity of solutions is
introduced as we vary $\tau$ and $\mu$ as control parameter. If the
feedback strength is small and we vary $\tau$, the oscillation
frequency is not constant, but remains relatively close to the value
of the frequency of oscillations for $\mu=0$. However, if the value of
$\mu$ is larger than a critical threshold, there is an interval of
$\tau$ for which the \eqref{Omega} has three solutions, the center
one being unstable always. This is shown in Figure \ref{fig111}(a). If
an even larger value of $\mu$ is chosen, the interval of $\tau$ in
which multiplicity of solutions occurs becomes larger.

It is important to note that we consider small to medium values of
$\mu$, up to $\mu\approx 2$ (as in \cite{StichPD10}). The
rationale behind this is that we are interested in the constructive
interplay between the terms on the right-hand side of \eqref{eq:model},
not in the case where the feedback terms will strongly dominate the native
dynamics.

The multiplicity of solutions is a direct consequence of the equation
for the frequency,  \eqref{Omega}. However, the equation for the
amplitude, \eqref{rho}, has also to be taken into account. In
particular, the value of the amplitude has to take a positive value if
it should correspond to an actual oscillation. In
Figure \ref{fig111}(b), we show the amplitudes and periods of the
solutions for a fixed $\mu$. The solid curves represent the periods,
and the dashed curves the amplitudes. Color codes for corresponding
parts of the solution. Let us consider first large values of
$\tau$. The blue (thick) curve represents the oscillation with a
relatively high frequency and large amplitude. Around $\tau\approx
1.06$, we enter the regime of birhythmicity, as also red (upper) and
green (middle) branches appear, green (red) referring to the unstable
(stable) branch. Our focus here is on the amplitude and we see that
the limit cycle corresponding to the unstable oscillations disappears
close to $\tau\approx 0.87$. Furthermore, around $\tau\approx 0.84$,
the stable high-frequency solution disappears as well. This means that
although we are in an apparent regime of birhythmicity due to the
multiplicity of frequency solution, actually below $\tau\approx 0.84$
only the low-frequency solution is left as unique solution. The parts
of the frequency curves that actually do not correspond to any
existing limit cycle solution are marked as black.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig1}  %figt01
\end{center}
\caption{(a) The period of oscillations as determined by
  \eqref{Omega} as a function of $\tau$ for $\mu=0.3$ (dashed),
  $\mu=0.66$ (solid), $\mu=1.0$ (dotted-dashed). (b) Periods (solid)
  and amplitudes (dashed) as a function of $\tau$ are shown for
  $\mu=1.5$. For more information see text. The other parameters are
  $\alpha=-1.4$, $\omega=2\pi-\alpha$, $\xi=\pi/2$.}
 \label{fig111}
\end{figure}

Below $\tau\approx 0.79$, the frequency solution is again
single-valued until around $\tau\approx 0.23$ even that solution
disappears as the amplitude of the oscillations approaches zero. Only
for $\tau$ below $0.09$, we observe again a limit cycle. The finite
interval of $\tau$ for which no limit cycle is found and the
stationary state is stabilized is called amplitude death. For this
model, it has been studied in  \cite{StichPD10} and we show some
novel results below.

Let us now consider again the frequency equation for a larger range of
$\tau$. In Figure \ref{fig222}(a) we show the results for three values
of $\mu$: While for the smallest feedback strength there is no
multiplicity around $\tau\approx 1$, there is actually a range around
$\tau\approx 2$ where this is found. The central observation here is
that the interval sizes where multiplicity is found become larger as
$\tau$ increases and that these intervals are located around $\tau
\approx n T_0$. As we go to larger $\tau$, the branches become more
and more inclined such that also for a fixed $\tau$, multiplicity of
more than 3 solutions is possible. For example, for $\mu=1.5$ and
$\tau=3.0$ there are 5 frequency solutions (whether all of them
correspond to valid limit cycle solutions has not been checked
here). Therefore, both, increasing $\tau$ or $\mu$ leads to an
increase of the number of frequency solutions.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig2} % figt02
\end{center}
\caption{(a) Frequency as a function of $\tau$ for $\mu=0.5$ (red thin
  curve), $\mu=1.5$ (black solid curve), $\mu=2.5$ (blue dashed
  curve). (b) Frequency as a function of $\tau$ for $\mu=1.5$ (black
  solid curve) and the curves for which $\rho=0$ (red dashed). The
  black curve enclosed by the first and second red curves (from left)
  denotes the first area where no oscillation is possible. The next
  area is defined by where the black curve is enclosed by the third
  and fourth red curves, etc.  The other parameters are as in
  Figure \ref{fig111}.}
 \label{fig222}
\end{figure}

\subsection*{Frequency degeneration}

Another observation to make in this figure is that the curves for
different $\mu$ intersect in the same points. Let us calculate their
values through analysis of \eqref{Omega}. First, we clarify for
which $\tau$ the period of the oscillations coincides with the one of
uniform oscillations in the absence of feedback.
\smallskip

\noindent\textbf{Case 1:} $\tau=0$. Then
\begin{equation}
\Omega(\tau=0)=\omega+\alpha=\Omega_0.
\end{equation}
\smallskip

\noindent\textbf{Case 2:}  $\tau=n\cdot 2\pi/\Omega$ with $n=1,2,\dots$. Then
\begin{subequations}
\begin{gather}
\chi_1 = \cos\Big(\xi+\Omega\frac{2\pi n}{\Omega}\Big)-\cos\xi
=\cos(\xi+2\pi n)-\cos\xi=0,\\
\chi_2 = \sin\Big(\xi+\Omega\frac{2\pi n}{\Omega}\Big)-\sin\xi
=\sin(\xi+2\pi n)-\sin\xi=0,
\end{gather}
\end{subequations}
and therefore
\begin{equation}
\Omega=\omega+\alpha+\mu(m_l+m_g)(\alpha\chi_1-\chi_2)=\omega+\alpha=\Omega_0.
\end{equation}
Therefore, the value of $\tau$ is actually:
\begin{equation}
\tau=\frac{2\pi n}{\Omega_0}=\frac{2\pi n}{\omega+\alpha}.
\end{equation}
Case 2 includes Case 1 if we allow for $n=0$. We call this value of $\tau$:
\begin{equation}
\tau_0^{\ast}=n T_0,
\end{equation}
for $n=0,1,2,\dots$.
\smallskip

\noindent\textbf{Case 3:}
 Another -- and more interesting -- possibility to obtain
$\Omega\neq \Omega(\mu)$ (and therefore $\Omega=\Omega_0$) is to ask
for
\begin{equation}
\alpha\chi_1-\chi_2=0,
\end{equation}
where we require $\alpha\ne 0$. First we write
%
\begin{subequations}
\begin{gather}
\chi_1 = \cos(\xi+\Omega\tau)-\cos\xi
=-2\sin\Big(\frac{2\xi+\Omega\tau}{2}\Big)
\sin\Big(\frac{\Omega\tau}{2}\Big),\\
\chi_2 = \sin(\xi+\Omega\tau)-\sin\xi
=2\sin\Big(\frac{\Omega\tau}{2}\Big)
 \cos\Big(\frac{2\xi+\Omega\tau}{2}\Big),
\end{gather}
\end{subequations}
and therefore
\begin{equation}
\tan \Big( \xi+\frac{\Omega\tau}{2}\Big)=-\frac{1}{\alpha}.
\end{equation}
The cases where this reduction cannot be made correspond to the cases
discussed above. Since we also know that $\Omega=\Omega_0$, we can
reformulate and give an explicit expression for $\tau$, denoted as:
\begin{equation}
\tau_1^{\ast}=\frac{1}{\omega+\alpha}
 \Big( 2\arctan\big(-\frac{1}{\alpha}\big)-2\xi+ 2\pi n\Big),
\label{taueqT0}
\end{equation}
with $n=0,\pm 1,\pm 2,\dots$ (but restricted to results with
$\tau_1^{\ast}>0$).  For the parameters used in Figure \ref{fig111}, the
first intersection point lies at $\tau\approx 0.69$ and corresponds to
the solution with $n=1$.

At this point we can search more generally for those values of
$\tau_1 \ne \tau_2$, for which $\Omega_1=\Omega_2=\Omega$. A calculation
analogous to the one above yields
\begin{equation}
\tan \Big(\xi+\frac{\Omega(\tau_1+\tau_2)}{2}\Big)=-\frac{1}{\alpha}.
\end{equation}
We leave the result in this form since in the general case we do not
have an analytical expression for $\Omega$. But for $\tau_2=0$ (and
therefore $\Omega=\Omega_0$) we recover Case 3.

\subsection*{Extrema of the frequency curve}

We can obtain the extrema of the frequency curve $\Omega(\tau)$
through implicit differentiation. After some transformations, the
condition $d\Omega/d\tau=0$ yields for non-zero $\alpha$ and non-zero
$\mu$ the expression
\begin{equation}
\Omega\tau=\arctan\big(-\frac{1}{\alpha}\big)-\xi \pm n \pi ,
\label{extrem}
\end{equation}
where for our set of parameters ($\xi=\pi/2$, $\alpha=-1.4$) the first
maximum corresponds to $n=1$ and the first minimum to $n=2$. Maxima
and minima are obtained by~\eqref{extrem} alternatingly. As example,
in Figure \ref{fig222}(a) we show the hyperbola for the first
minimum. Note that again this curve does not depend on $\mu$.

\subsection*{Condition for vanishing amplitude}

As we have already seen in Figure \ref{fig111}(b) that the amplitude of
the oscillations can vanish, we evaluate \eqref{rho} in order to
find an analytic criterion. The condition $\rho=0$ yields two
expressions:
\begin{subequations} \label{rhozero}
\begin{gather}
\Omega\tau = \arccos\Big(\cos\xi -\frac{1}{\mu(m_l+m_g)}\Big)-\xi \pm 2\pi n,\\
\Omega\tau = -\arccos\Big(\cos\xi -\frac{1}{\mu(m_l+m_g)}\Big)+\xi \pm 2\pi n,
\end{gather}
\end{subequations}
with $n=0,1,2,\dots$ and restricted to $\tau>0$ and $\Omega \ne
0$. These functions are shown in Figure \ref{fig222}(b) as dashed
curves. Where these curves intersect with the curve for the period
(solid curve), we find the limits where $\rho=0$. For example, the
interval where amplitude death is observed is found in between the
first and second dashed curve (counted from left). For this interval,
the lower bound is given by the first expression for $n=0$, and the
upper bound by the second expression for $n=1$. The second interval
(lying in the birhythmicity area) is limited by the curves given for
$n=2$ and $n=3$, etc..

\subsection*{Amplitude degeneration}

Above we have seen that besides $\tau_0^{\ast}=nT_0$ there is a
$\tau_1^{\ast}$ where the oscillation frequency is independent of
$\mu$ and identical to $2\pi/T_0$. We can check whether there is also
a $\tau$ for which the amplitude is independent of $\mu$ and identical
to $\rho_0=1$. A short inspection of Eq.~\eqref{rho} shows that this
is only the case for $\tau=\tau_0^{\ast}$, not for
$\tau=\tau_1^{\ast}$. This means that although we can create different
feedback-induced limit cycles with same frequency, we cannot create
feedback-induced limit cycles with identical amplitudes.

\subsection*{Birhythmicity area revisited}

With the obtained knowledge about the disappearance of limit cycle
solutions within the apparent birhythmicity region, e.g., as shown in
\cite{StichPD10}, we may have to correct the area where two
stable limit cycles are found. This is shown in
Figure \ref{fig333}(a). The thin black curves limits the area -- on
basis of the frequency equation~\eqref{Omega} -- where three limit
cycle solutions were believed to exist, two of them stable. The two
new curves were derived from \eqref{rhozero} and show the locus of
$\rho=0$ on the center branch (blue curve with circles) and on the
high-frequency branch (red curve with squares). Since the center
branch is unstable always, only to the left of the red branch the
stability scenario can be modified effectively. Note that both curves
end at $\mu=1$ since $\xi=\pi/2$ and $m_l+m_g=1$. Since the solution
of uniform oscillation depends on the sum $m_l+m_g$, the shown curves
do not depend on the specific choice of $m_l$. However, this changes
once we consider \emph{stability} of uniform oscillations. The
stability curves depend on $m_l$ and in Figure \ref{fig333}(a) we show
as green dashed curves the stability limits (as derived in
\cite{StichPD10}) of the high-frequency solution for $m_l=0.6$
(hence $m_g=0.4$). The high-frequency solution becomes actually
unstable {\emph{before}} crossing the red curve, so that the stability
scenario is effectively unchanged in this case. However, no extensive
analysis is performed here, so we cannot exclude parameter values
where a stable limit cycle disappears.


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig3} % figt03
\end{center}
\caption{(a) Birhythmicity area for $\alpha=-1.4$,
  $\omega=2\pi-\alpha$, $\xi=\pi/2$ and $m_l+m_g=1$. The thin black
  curves that denote the limits of the multiplicity area for the
  frequency equation~\eqref{Omega}, as shown in~\cite{StichPD10}. To
  the left of the blue curve (with circles), the center solution
  (always unstable) is absent, to the left of the red curve (with
  squares), the high-frequency solution. The green dashed curves show
  where the high-frequency solution becomes unstable (stable in the
  upper right area). (b) The frequency equation~\eqref{Omega} for
  $\mu=1$ and for $\xi=0$ to $\xi=7\pi/4$ in steps of $\pi/4$ in
  order red, blue, black, green, then dashed red, dashed blue, dashed
  black, dashed green.}
 \label{fig333}
\end{figure}


\subsection*{Influence of $\xi$}

In previous work on that model~\cite{StichPRE07,StichPD10,StichPRE13},
the parameter controlling the shift between the system dynamics and
the feedback $\xi$ has been taken the value $\pi/2$. Here, we want to
study the influence it has on the shape of the frequency curve. This
is displayed in Figure \ref{fig333}(b) for a fixed $\mu=1$ for 8 values
of $\xi$ from $0$ to $7\pi/4$. The first observation is that the
oscillation frequency can be tuned to be much larger than the native
frequency $\Omega_0$ over almost the whole range of $\tau$, as for
example for $\xi=\pi/4$ (blue curve). Inspection of \eqref{Omega}
shows that this also depends strongly on $\alpha$. The second
observation is that for a given $\xi$ the curves $\Omega(\xi)$ and
$\Omega(\xi+\pi)$ intersect where $\Omega=\Omega_0$. This reflects the
fact that $\tau_1^{\ast}$ is invariant to a shift of $\xi\to\xi+n\pi$
(cf. \eqref{taueqT0}). This also implies that for a fixed $\tau$ a
shift $\xi\to\xi+n\pi$ represents a switch from a low-frequency
solution to a high-frequency solution (or vice versa). This is a
slight over-simplification since the areas of existence of solutions
do not match in general under such as shift (not to mention
stability).



\section{Amplitude death}

Amplitude death represents a collective breakdown of oscillations and
the simultaneous stabilization of a stationary state. Therefore, it
can be studied by checking for the stability of the fixed point
solution $\rho=0$.  Let us recall some results
from~\cite{StichPD10}. By using a mode separation ansatz
\begin{equation}
  A(x,t)= H(t)+A_+(t)e^{{\mathrm i}\kappa x} + A_-(t)e^{-{\mathrm i}\kappa x} \,
  \label{ansatz_lsa}
\end{equation}
we obtain  to lowest order in $H$ and $A_{\pm}$
\begin{subequations}
  \begin{align}
    \label{eq:Had}
    &\dot{H} = (1-{\mathrm i} \omega)H
    + \mu (m_l+m_g) e^{{\mathrm i}\xi}(H(t-\tau)-H(t)),\\
    \label{eq:A+ad}
    &\dot{A}_+ = (1-{\mathrm i} \omega)A_+
    -(1+{\mathrm i} \beta)\kappa^2 A_+ + \mu m_l e^{{\mathrm i}\xi}(A_+(t-\tau)-A_+),\\
    \label{eq:A-ad}
    &\dot{A}_-^{\ast} = (1+{\mathrm i} \omega)A_-^{\ast}
    -(1-{\mathrm i} \beta)\kappa^2 A_-^{\ast}
    +\mu m_l e^{-{\mathrm i}\xi}(A_-^{\ast}(t-\tau)-A_-^{\ast}).
  \end{align}
  \label{eq:modesad}%
\end{subequations}
These three equations are decoupled and can be studied independently.
In order to investigate the linear stability of the state $H=0$ with
respect to uniform perturbations, we set
%
\begin{equation}
H=H_0\exp(\lambda t),
  \label{eq:ansatzad}%
\end{equation}
with $H_0$ an initial amplitude and $\lambda$ a complex
eigenvalue. Inserting \eqref{eq:ansatzad} into \eqref{eq:Had},
yields the following characteristic equation
\begin{equation}
\lambda = 1-{\mathrm i}\omega + \mu(m_l+m_g){\mathrm e}^{{\mathrm i}\xi}
({\mathrm e}^{-\lambda \tau}-1).
  \label{eq:charad}
\end{equation}
This equation is equivalent to the amplitude death condition for a
single Hopf oscillator and can be solved with the Lambert W function
or numerically.  Note that the homogeneous mode behaves the same for
all relative weights between local and global feedback as long as the
sum $m_l+m_g$ is constant (as assumed here).

We now turn to the inhomogeneous modes with wavenumber $\kappa$.
Since \eqref{eq:A+ad} and \eqref{eq:A-ad} are decoupled and
identical, it is sufficient to consider only one of them. Substituting
the ansatz
\begin{equation}
A_+=A^{0}_{+}\exp(\lambda t)
  \label{eq:ansatzad2}
\end{equation}
into \eqref{eq:A+ad}, we obtain
\begin{equation}
\lambda = 1-{\mathrm i}\omega -\kappa^2 - {\mathrm i} \beta \kappa^2
+ \mu m_l{\mathrm e}^{{\mathrm i}\xi} ({\mathrm e}^{-\lambda \tau}-1).
  \label{eq:charad2}
\end{equation}
The $-\kappa^2$ term indicates that the mode with $\kappa=0$ is the
most unstable one: If the fixed point is stable respect to
perturbations with $\kappa=0$, then it is also stable with respect to
all perturbations $\kappa>0$. Note that the inhomogeneous mode depends
on $m_l$ only.

While these results have been correctly stated in
\cite{StichPD10}, there is an interesting case that had been
overlooked there: the obvious choice $m_l=0$ (thus corresponding to
the case of purely global feedback, since $m_l+m_g=1$). In this case,
the eigenvalue of the inhomogeneous mode becomes
\begin{equation}
\lambda = 1-{\mathrm i}\omega -\kappa^2 - {\mathrm i} \beta \kappa^2,
  \label{eq:charad2zero}
\end{equation}
which means (a) that the (global) feedback does not have any impact on
the stability of the inhomogeneous mode, and (b) that the stationary
state is unstable for all perturbations with $\kappa<1$. Since global
stability requires stability with respect to the homogeneous mode and
the most critical inhomogeneous mode ($\kappa=0$), we see that
amplitude death is impossible to achieve for $m_l=0$ and that $m_l\ne
0$ is necessary in order to observe amplitude death, being $m_l=1$ the
most favorable case. As a matter of fact, this agrees completely with
simulations shown in Figure 2 of~\cite{StichPRE07} where the area of
amplitude death is largest for $m_l=1$, decreases for smaller $m_l$,
and is eventually absent for $m_l=0$.

We need not to stop here and observe that all critical curves of
\eqref{eq:charad2} for $\kappa=0$ and different $m_l$ lie
{\emph{within}} the area where $H=0$ as given by
\eqref{eq:charad}.  Thus, stability with respect to
space-dependent perturbations is obtained only if the system is
already stable with respect to uniform perturbations. Therefore, we
can actually go beyond the analysis of  \cite{StichPD10} and can
revise the equations for the inhomogeneous modes taking into account
that in the case of criticality $H=0$ within the equations for
$A_{\pm}$.  We obtain to lowest order
\begin{subequations}
  \begin{align}
    \label{eq:A+ad2}
    \dot{A}_+ &= (1-{\mathrm i} \omega)A_+
    -(1+{\mathrm i} \alpha)(|A_+|^2+2|A_-^{\ast}|^2)A_+
    -(1+{\mathrm i} \beta)\kappa^2 A_+ \nonumber \\
&\quad  + \mu m_l e^{{\mathrm i}\xi}(A_+(t-\tau)-A_+),\\
    \label{eq:A-ad2}
    \dot{A}_-^{\ast} &= (1+{\mathrm i} \omega)A_-^{\ast}
    -(1-{\mathrm i} \alpha)(|A_-^{\ast}|^2+2|A_+|^2)A_-^{\ast}
    -(1-{\mathrm i} \beta)\kappa^2 A_-^{\ast} \nonumber  \\
&\quad +\mu m_l e^{-{\mathrm i}\xi}(A_-^{\ast}(t-\tau)-A_-^{\ast}).
  \end{align}
  \label{eq:modesad2}
\end{subequations}
This is also the lowest order at which the two modes are coupled.  Of
course, if we assume $|A_{\pm}| \ll 1$, we recover the decoupled
system~\eqref{eq:A+ad}, \ref{eq:A-ad} studied above.  More
interestingly, if we neglect the time-delay terms and the term
proportional to $\kappa$, the system reduces to the normal form of a
Hopf bifurcation with circular
symmetry~\cite{CrossRMP93,DanglmayrPROC98}. This equation is known to
have either standing or traveling waves as fundamental solutions,
depending on the prefactors of the cubic terms, generally to be of the
form $a|A_+|^2+b|A_-^{\ast}|^2$. If $b>a>0$ -- as in our case --
traveling waves are a stable solution of~\eqref{eq:modesad2}. Indeed,
traveling waves of small amplitude (around $<10^{-4}$) were observed
in a number of simulations performed close to the area of amplitude
death (\cite{StichPRE07} and unpublished results) which suggest that
they were created in this scenario.


\section{Conclusion}

Following previous work~\cite{StichPRE07,StichPD10,StichPRE13}, a
complex Ginzburg-Landau equation subjected to local and global
time-delay feedback terms has been considered. In particular, we have
revisited the multiple oscillatory solutions that exist in this
model. First result was the observation that the amplitude of a limit
cycle can vanish (and hence the limit cycle disappear) while the
frequency solution still suggests a valid solution. Secondly, we have
derived several analytical criteria for the disappearance of limit
cycle solutions, frequency degeneration, amplitude degeneration, and
frequency extrema. Only the frequency degeneration condition had been
mentioned in previous work~\cite{StichPD10}, albeit not in
detail. This gives us the possibility to select for limit cycles with
desired properties, e.g., for birhythmicity where the limit cycles
should have a maximum difference in frequency and minimum difference
in amplitude.  Then, we have discussed the influence of the phase
shift parameter $\xi$, which in previous work had always taken a fixed
value $\xi=\pi/2$. In principle, $\xi$ can be chosen freely between
$0$ and $2\pi$, but we point out some invariances and symmetries for
$\xi$.  Finally, we discuss the phenomenon of amplitude death, the
stabilization of the steady state and the decay of all
oscillations. Going beyond the analysis in~\cite{StichPD10}, we show
that amplitude death cannot happen if $m_l=0$, i.e., for global
feedback only. In the parameter space $(\tau,\mu)$ where the
stationary state is stable with respect to homogeneous perturbations,
but unstable with respect to inhomogeneous perturbations, we predict
the onset of traveling wave patterns.

\section*{Acknowledgments}
M. S. acknowledges very gratefully many discussions and continuous
support by Alfonso Casal.

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\end{document}