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\begin{document}

\title{Numerical and Analytical Study of the 3$\omega$ Parametrically
Forced Complex Ginzburg-Landau Equation}
\author{Rafael Gallego$^1$, Daniel Walgraef$^2$, Maxi San Miguel$^1$ and 
Ra\'ul Toral$^1$}
\address{1.- Instituto Mediterr\'aneo de Estudios Avanzados, IMEDEA (CSIC-UIB), 
Campus UIB, 07071-Palma de Mallorca, Spain\\
2.- Centre for Non-Linear Phenomena and Complex Systems, Universit\'e Libre de
Bruxelles, Campus Plaine, Blv. du Triomphe B.P 231, B-1050 Brussels, Belgium}
\maketitle

%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
The effect of a temporal modulation at three times the critical frequency on a
Hopf bifurcation is studied in the framework of amplitude equations. In this
case, these equations are complex Ginzburg-Landau equations with an extra
quadratic term, resulting from the strong coupling between the external field
and unstable modes. On increasing the intensity of the forcing, one passes from
an oscillatory regime to an excitable one with three equivalent frequency
locked states. In the first regime, topological defects are one-armed phase
spiral, while in the second one they correspond to three-armed excitable
amplitude spirals. Analytical results show that the transition between these
two regimes occurs at a critical value of the forcing intensity. The transition
between phase and amplitude spirals is confirmed by numerical analysis.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%

\pacs{PACS: 05.45.-a,82.40.Bj,05.70.Ln }
\vskip 0.4cm

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%

In many cases, the nucleation of spatio-temporal patterns is associated with
continuous symmetry breakings, and these patterns are thus very sensitive even
to small perturbations or external fields. Perturbations may be induced by 
imperfections of the system itself (e.g.  impurities), of the geometrical
set-up (e.g. the boundary conditions),  of the control parameters, etc.
External fields, on the other hand, may  induce spatial or temporal modulations
of the control or bifurcation parameters. In fact, spatially or temporally
modulated systems are very common in nature, and the effect of external fields
on these systems has been studied for a long time. As a way of example, the
forcing of a large variety of nonlinear oscillators, from the pendulum to Van
der Pol or Duffing oscillators, has led to detailed studies of the different
temporal behaviors that were obtained. It has been shown that  resonant
couplings between the forcing and the oscillatory modes may lead to  several
types of complex dynamical behaviors, including quasi-periodicity,  frequency
lockings, devil's staircases, chaos and intermittency 
\cite{nonlinearosc,haobailin}.

In equilibrium systems, the importance of spatial modulations has been known
for  a long time. For example, in the case of spatial modulations occurring in 
equilibrium crystals, such as spin or charge density waves, the constraint 
imposed by the periodic structure of the host lattice leads to the now commonly
known commensurate-incommensurate phase transitions. The transition from the 
commensurate phase, where the wavelength of the modulated structure is a 
multiple of the lattice constant, to the incommensurate one occurs via the 
nucleation of domain walls separating domains which are commensurate with the 
host lattice \cite{bak80,bulaev78}.

In nonequilibrium systems, the systematic study of the influence of external 
fields on pattern forming instabilities is more recent. It has been first 
devoted to instabilities leading to spatial patterns. For example, the
Lowe-Gollub experiment \cite{lowe85} showed that, in the case of the 
electrohydrodynamic instability of liquid crystals, a spatial modulation of the
bifurcation parameter may induce discommensurations, incommensurate wavelengths
and domain walls. The similarities with analogous equilibrium phenomena rely on
the fact that, close to this instability, the asymptotic  dynamics is potential
\cite{lubensky,coullet86}. 

In the case of Hopf bifurcations, however, original effects occur as a 
consequence of the nonpotential character of the dynamics. In particular,
for wave bifurcations, unstable standing waves or two-dimensional wave
patterns may be stabilized by pure spatial or temporal modulations of suitable
wavelengths or frequencies
\cite{riecke88,walgraef88,rehberg88,coulletwal89}. The case of pure temporal
modulations of Hopf bifurcations in extended systems has been extensively
studied by Coullet and Emilsson \cite{kjartan} through amplitude equations
of the scalar  Ginzburg-Landau type. They considered periodic temporal
modulations of frequency $\omega_e = (n/m)(\omega_0 - \nu )$, where $(n/m)$
is an irreducible integer fraction, $\omega_0$ is the critical frequency of
the Hopf bifurcation, and $\nu $ is a small frequency shift. Such forcings
break the continuous time translation down to discrete time translations,
and the corresponding amplitude equations become, for the so-called
strongly resonant cases ($n=1,2,3,4$):

\begin{equation}\label{gleq}
\partial_t A= (\mu+i\nu) A + (1+i\alpha ) \nabla^2 A - (1+i\beta )A\vert A\vert^2 
+ \gamma_n\bar A^{n-1}.
\end{equation}

This dynamical system is of the relaxational type for $\alpha=\beta=0$.
In the case $\alpha=\beta$ it is still possible to find a Lyapunov
potential, whereas the case $\alpha\ne \beta$ is genuinely 
non-potential\cite{non-pot}.
If the forcing intensity $\gamma_n$ is sufficiently strong, this dynamics
admits asymptotically stable uniform steady states, corresponding to
frequency locked solutions. There are $n$ different frequency locked
solutions, which only differ by a phase shift of $2\pi /n$. In this 
regime, the dynamics resembles some sort of excitability. The locked solutions
may undergo various types of instabilities \cite{kjartan}. One of them is
of phase type and occurs when $1+\alpha\beta$ is sufficiently negative. In
this case, competition between phase instability and forcing leads to the
formation of stripes or hexagonal patterns, with their associated
topological defects. If the forcing is decreased, these structures break
down through spatio-temporal intermittency \cite{kjartan}. On the other
hand, in the phase stable regime, frequency locked solutions may undergo a
variety of bifurcations when forcing is decreased, leading to oscillation,
quasi-periodicity or chaos \cite{gambaudo}.

The equivalence between the different frequency locked states makes possible
the formation of stable inhomogeneous structures. These structures are
composed of domains of the locked states separated by abrupt interfaces.
Nonpotential effects may induce interface motion and, in particular, the
formation of $n$-armed spirals, each arm corresponding to a different frequency
locked solution.

These phenomena were studied in great detail by Coullet and Emilsson, for
$n=1$ and $n=2$, in one- and two-dimensional systems \cite{kjartan}. For
$n=3$, they only briefly mention the existence of three armed rotating
spirals in two-dimensional systems. It is the aim of this paper to study
the latter case in more details, and especially the transition from phase
spirals to amplitude spirals. The interest of this study is twofold. First,
it confirms the robustness of the Ginzburg-Landau dynamics, which is recovered
at low forcing, with all its complexity and its particular sensitivity to
kinetic coefficients. Second, it presents original dynamical behavior in the
excitable regime. This behavior presents interesting analogies with
Rayleigh-B\'enard convection in a rotating cell, described by three-mode
dynamical models \cite{BH,Tu,Gallego}. 

The paper is organized as follows. The dynamical model and its uniform
asymptotic solutions are presented in section \ref{sec:uniform}. Section
\ref{sec:phaseapprox} is devoted to the description of the dynamics in terms of
phase equations. The properties and possible development of front and spiral
solutions are discussed in section \ref{sec:fronts_spirals}. Numerical results,
for one- and two-dimensional systems, are presented in section
\ref{sec:numerical} and conclusions are drawn in section
\ref{sec:conclusions}.  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uniform solutions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:uniform}

Consider an extended system undergoing a Hopf bifurcation, and subjected to a
periodic temporal modulation of frequency $\omega_e = 3\omega_0$.
Sufficiently close to the bifurcation, its dynamics may be reduced to the
following complex Ginzburg-Landau equation \cite{daniel}:

\begin{equation}\label{gleq3}
\partial_t A= \mu A + (1+i\alpha ) \nabla^2 A - (1+i\beta )
  A\vert A\vert^2 + \gamma\bar A^2,
\end{equation}
where $\gamma$ is proportional to the external field intensity. The other
parameters are standard \cite{kjartan,daniel}.

We look now for uniform solutions. By dropping the spatial derivative
terms, the corresponding uniform equations are, in phase and amplitude 
variables ($A=R_0(t)e^{i\Phi_0(t)}$):

\begin{eqnarray}
\dot{R}_0 &=&\mu R_0 - R_0^3 + \gamma R_0^2 \cos 3\Phi_0, \nonumber\\
\label{null}
\dot{\Phi}_0 &=&- \beta R_0^2 - \gamma R_0 \sin 3\Phi_0.
\end{eqnarray}
If we look at the stationary solutions (fixed points), eqs. (\ref{null}) give

\begin{equation}
(1+\beta^2)R_0^4 - (2\mu + \gamma^2 ) R_0^2 + \mu^2 = 0,
\end{equation}
from where the amplitudes of the uniform solutions are given by:

\begin{equation}
\label{rmm}
R_{\pm}^2= {1\over 2(1+\beta^2)} [2\mu  +\gamma^2  \pm
\sqrt  {\gamma^4 + 4\mu\gamma^2 - 4\mu^2\beta^2}]. 
\end{equation}
Such solutions exist provided that $\gamma^2 > \gamma_c^2$, with

\begin{equation}
\gamma^2_c= 2\mu (\sqrt{1+\beta^2} - 1 ).
\end{equation}
Once the amplitude is determined by (\ref{rmm}) the phase can be obtained
from the stationary version of (\ref{null}):

\begin{eqnarray}
\cos 3\Phi_0 & = & \frac{R_0^2-\mu}{\gamma R_0} \\
\sin 3\Phi_0 & = & \frac{-\beta R_0}{\gamma}.
\end{eqnarray}
Each value of $R_0=R_+,R_-$ gives rise to three solutions for the phase
$\Phi_0$ which only differ by a phase shift of $2\pi/3$.
Hence, 
for $\gamma^2 >\gamma_c^2$ the system has six uniform solutions:
$(\Phi_1^u,\Phi_2^u,\Phi_3^u)$ corresponding to $R_-$ and 
$(\Phi_1^e,\Phi_2^e,\Phi_3^e)$ corresponding to $R_+$.
A linear stability analysis shows
that the $\Phi_i^u$ are
always linearly unstable whereas the $\Phi_i^e$ 
are stable for $|\beta|<\sqrt 3$. The three $\Phi_i^e$ solutions
are called the frequency locked solutions. These become 
oscillatory unstable ($k=0$, $\omega\neq 0$) for
$\vert\beta\vert>\sqrt{3}$ in the range of forcing amplitudes
$\gamma_c\lesssim\gamma_1<\gamma<\gamma_2$ where
\begin{eqnarray}
\gamma_1 & = & = \sqrt{\gamma_c^2+\frac{\mu}{2(3\beta^2-1)}
\left[4\sqrt{1+\beta^2}(1-3\beta^2)+
      7\sqrt{3}\beta^3-\beta^2+3\sqrt{3}\beta-5
\right]}
   \nonumber \\
\gamma_2 & = & \sqrt{\mu(1+\beta^2)/2}. 
\end{eqnarray}

In the case where the frequency locked solutions are stable,
we can show that the system behaves as an excitable one:
let us construct the nullclines of the dynamical 
system (\ref{null}) and represent them in figure
\ref{fig:nullA}, for $\gamma = \beta = 4\mu = 1$,   or in figure
\ref{fig:nullB}, for $\gamma = \beta = 0.01$, $\mu = 0.25$.
In both figures, it is easy to see that the $R_-$ states (labeled $u$)
are unstable,
while the $R_+$ states (labeled $e$)
are stable for small perturbations. However, for 
perturbations larger than a well defined threshold, the latter are unstable and
the system makes an excursion in the phase space, before reaching another,
equivalent, steady state. It is a form of excitability. The excitability 
threshold can be explicitly computed in the limiting case $\beta,\gamma 
\ll \mu$. In this limit, and taking into account that $\gamma_c \simeq \beta
\sqrt \mu$, the adiabatic elimination of the amplitude
in (\ref{null}) leads to the phase equation:

\begin{equation}
\dot\Phi_0 = - \sqrt{\mu}(\gamma_c+\gamma \sin 3\Phi_0).
\end{equation}
from where the excitable stable steady states are given by
$\sin 3\Phi_i^e = - \gamma_c/\gamma$
and $\gamma \cos 3\Phi_i^e >0$, and the three unstable steady states
$\sin 3\Phi_i^u = -\gamma_c/\gamma$ and $\gamma \cos 3\Phi_i^u <0$, $i=1,2,3$. 
In this case, the excitability threshold is thus given by:

\begin{equation}
\Delta\Phi=\lvert\Phi_i^e -\Phi_i^u\rvert\simeq 
\frac{\pi}{3}-\frac{2}{3}\frac{\gamma_c}{\gamma}.
\end{equation}

For $\gamma^2 <\gamma_c^2$ there are no fixed points of (\ref{null})
and asymptotic solutions
correspond to temporal oscillations of the limit cycle type. For $\gamma=0$ the
limit cycle is a circle that becomes deformed for
$0<\vert\gamma\vert<\gamma_c$ (see fig. \ref{fig:uniform}). On increasing
$\gamma$, the period of the oscillations increases and diverges for $\gamma^2
\to \gamma_c^2$. The stability of these oscillatory solutions can be better
analyzed in the framework of phase equations. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Phase approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:phaseapprox}


In this section we present several phase equations each one valid in a
different region of parameters. As mentioned in the previous section,
phase equations can be used to analyze the stability of the uniform
patterns.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The oscillatory regime}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the oscillatory regime ($\gamma^2<\gamma_c^2$), the phase dynamics can
be obtained by perturbing the uniform solution $(R_0(t),\Phi_0(t))$, and writing
$R = R_0(t) + \rho (\vec r, t)$, $\Phi = \Phi_0(t +\phi(\vec r,t))$. Following
Hagan \cite{hagan}, the adiabatic elimination of the amplitude perturbations
in the regime $\beta,\gamma\ll\mu$ leads to the following phase dynamics:

\begin{equation}
\partial_t\phi = (1+ \alpha \bar\beta )\nabla^2 \phi + 
\kappa (\vec\nabla\phi)^2 + ...
\end{equation}
where $T$ is the period of the oscillations, and

\begin{equation}
\bar\beta = {{\int_0^T dt \, {2\beta R_0 + \gamma \sin 3\Phi_0 \over 2R_0 -  
\gamma
\cos \Phi_0}\, \dot\Phi_0^2}\over {\int_0^T dt\, \dot\Phi_0^2}}, \quad
\kappa = {{\int_0^T dt\,({2\beta R_0 + \gamma \sin 3\Phi_0 \over 2R_0 -  
\gamma
\cos \Phi_0}-\alpha )\, \dot\Phi_0^3}\over {\int_0^T dt\, \dot\Phi_0^2}} 
\end{equation}

\noindent For $\gamma \to 0$, one recovers the usual Burgers equation

\begin{equation}
\partial_t\bar\phi = (1+ \alpha \beta )\nabla^2 \bar\phi +(\alpha - \beta 
)(\vec\nabla\bar\phi )^2 + ...
\end{equation}
with $\bar\phi = \beta\mu\phi$.

Hence, in the regime where $1+\alpha\bar\beta >0$, stable (phase) spiral waves
may be expected, with wavenumber proportional to $\kappa$, and thus depending
on the characteristics of the oscillations \cite{hagan,Yamada}. In this regime,
the qualitative behavior and interaction between these topological defects
should thus be almost insensitive to the forcing \cite{rica,kramer,pismen}.
Furthermore, in the regime where $1+\alpha\bar\beta < 0$, defect mediated
turbulence should also be expected \cite{coulletgilega}. In the oscillatory
regime, the system presents thus qualitatively the same complexity and the same
spatio-temporal behaviors than self-oscillating systems. Only quantitative
aspects are affected by the forcing. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The excitable regime}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the excitable regime, $\gamma^2>\gamma_c^2$ and the phase dynamics can be
obtained in the limit $\beta,\gamma \ll \mu$, $\beta\ll 1$, by eliminating
adiabatically the amplitude of the field. Taking into account that, in this
regime,  $R^2\simeq\mu$ and $\gamma_c\simeq |\beta|\sqrt\mu$, we are left with
the following phase equation:

\begin{equation}\label{phasedyn}
\partial _t\Phi = -\sqrt{\mu} (\gamma_c + \gamma  \sin 3\Phi )+
(1+ \alpha \beta )\nabla^2 \Phi -(\alpha - \beta )(\nabla\Phi  )^2
 + \frac{\alpha^2(1+\beta^2)}{2\mu}\nabla^4\Phi 
\end{equation}
Besides the homogeneous solutions discussed in the previous section, this 
equation admits front solutions connecting stable states asymptotically at
$x=\pm \infty$. In the case $\alpha=\beta= 0$ the phase equation is
relaxational and the fronts connect two states with the same value of the
potential and are, therefore, stationary. In the case $\alpha=\beta\ne 0$, the
phase equation is still relaxational but now the steady states have different
value of the potential and the front moves. Moreover, when $\alpha\ne \beta$
there is a purely non-potential induced front motion. Equation (\ref{phasedyn})
will be used in the next section as the starting point to compute the velocity
of the front solution.


In order to study pattern forming instabilities, we can use (\ref{phasedyn}) in
the limit of small $\Phi$. Expanding the $\sin 3\Phi$ up to linear order in
$\Phi$, we are led to a damped Kuramoto-Sivashinsky phase
equation~\cite{kjartan}. It follows that frequency locked solutions are stable
for $1+\alpha\beta >0$. If $1+\alpha\beta <0$, a pattern forming instability of
the locked states would occur for~\cite{kjartan}\footnote{This corrects the
misprint of reference \cite{kjartan} in $\epsilon$ and $k_0$ after eq. (28).}

\begin{equation}
\mu > 36\gamma^2\frac{\alpha^4(1+\beta^2)^3}{(1+\alpha\beta )^4}.
\end{equation}
Since, in this regime, $\gamma^2>\gamma_c^2$, a necessary condition for this
instability is thus

\begin{equation}
(1+\alpha\beta )^4 > 72(\sqrt{1+\beta^2} - 1 ) \alpha^4(1+\beta^2)^3,
\end{equation}
and this condition cannot be realized in the $\{1+\alpha\beta<0\}$ domain.
Therefore, the frequency locked solutions are stable so that pattern forming
instabilities are ruled out within the phase approximation. It is also possible
to prove that the locked solutions are always stable in the limit of large
forcings~\cite{kjartan}. 

Note that in reference \cite{Elphick} the forced CGLE is studied. Special
emphasis is put in a front instability which may cause the decomposition of
phase fronts with a difference of $\pi$ between the phase of the uniform phase
states into several fronts with a smaller shift phase. The transition occurs
when the strength of the forcing is reduced below some critical value.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fronts and Spirals}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:fronts_spirals}

For $\vert\gamma\vert > \gamma_c$, the forced Ginzburg-Landau equation
possesses three equivalent excitable steady states. The excitability mechanism
described in the previous section provides a natural way of building fronts
between these steady states. Despite the equivalence of the fixed points, such
fronts are expected to move, as a result of the nonpotential character of the
dynamics. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{One-dimensional systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Consider a front solution of eq. (\ref{phasedyn}), e.g. $\Phi_{12}(x - vt)$, 
joining the states 1 (at $x \to -\infty $) and 2 (at $x \to + \infty$), such 
that $\Phi_2^e > \Phi_1^e$. Its velocity may be computed along the standard
procedures, and is such that (at leading order in perturbation)

\begin{equation}\label{vel}
 v =  { 2\pi\gamma_c\sqrt{\mu}
+3(\alpha - \beta )\int^{+\infty}_{-\infty} (\partial_x \Phi_{12})^3 \over
3\int^{+\infty}_{-\infty} (\partial_x \Phi_{12})^2 }.
\end{equation}
Hence, for $\alpha >\beta$, the fronts $\Phi_{12}$, $\Phi_{23}$ and $\Phi_{31}$
move to the right, while the fronts $\Phi_{21}$, $\Phi_{13}$ and $\Phi_{32}$
move to the left. Hence, any domain of one steady state, embedded in a domain
of another one, either expands or shrinks, leaving the system in one steady
state (domains of 2 embedded into 1, 3 into 2 and 1 into 3 shrink  while
domains of 1 into 2, 2 into 3 and 3 into 1 expand). However, a succession (from
left to right) of domains with states in the order 1, 2, 3, 1, etc. moves as a
whole to the right. When it is in the order 1, 3, 2, 1, etc., it moves as
a whole to the left (see figure \ref{fig:phase1d}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two-dimensional systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In two-dimensional systems, straight linear fronts have the same behavior as
in one-dimensional systems. Furthermore, sets of two inclined fronts
separating domains with  different steady states also move away or
annihilate, leaving the system in one steady state only (see figure
\ref{fig:2dfronts}).

New phenomena may arise when the three steady states coexist in the system. In
this case, three fronts, which separate the respective domains, coalesce in one
point (a vertex). The three fronts are expected to rotate around this
point. The result is a rotating spiral whose angular velocity increases with
the forcing amplitude. Spirals corresponding to sequences of states in the
order $1\to 2 \to 3$ or  $1\to 3 \to 2$ around the center have opposite senses
of rotation. Isolated vertices remain immobile, but nonisolated ones have a
dynamical evolution induced by mutual interactions, which may even lead to the
annihilation of counter-rotating spirals. This dynamical behavior is
illustrated by the results of the numerical analysis presented in the next
section.

The situation is similar to that observed in system with competing fields.
For example, in the context of fluid dynamics, a three mode model has been
proposed to study Rayleigh-B\'enard convection in a rotating
cell~\cite{BH,Tu}. The fields represent the amplitudes of three set of
convection rolls oriented $60^{\circ}$ at each other. In the two-dimensional
system, vertices may form when the three different types of roll domains meet
at one point (notice that this is not possible in one dimensional
systems~\cite{Gallego}). Then, the nonpotential dynamics induces the
rotation of the interfaces around the vertices preventing the system from
coarsening. At long time scales, the vertices diffuse throughout the system.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:numerical}

In this section, we present numerical results in two spatial dimensions which
illustrate the various dynamical behaviors described in the preceding sections.
We have simulated the forced CGLE in two spatial dimensions by using a
pseudospectral method with periodic boundary conditions. We discretized the
system in a square mesh of $256\times 256$ points. Cases within and beyond the
phase approximation and with $1+\alpha\beta> 0$ and $1+\alpha\beta<0$ were
considered. In all the cases the parameter $\mu$ was taken fixed and equal to
$1$. In table \ref{numericalcases} we give the parameters chosen for each one
of the cases studied.

We start the discussion in cases for which the phase approximation applies
($\beta,\gamma\ll \mu$). Below the Benjamin-Fair (BF) line ($1+\alpha\beta>0$)
we took \{$\alpha=2$, $\beta=-0.2$; $\gamma_c=0.2$\}. This case corresponds to
the \emph{frozen states} regime with respect to the no forcing situation. In
figure \ref{fig:snapsCaseI} we show the modulus and phase of the complex field
$A$ in the cases no forcing, oscillatory, $\gamma\simeq\gamma_c$ and excitable.
As expected~\cite{Chate}, spiral defects surrounded by shock lines occur in the
no forcing regime. When the strength of the forcing $\gamma$ is increased but
still being below the critical value $\gamma_c$ (oscillatory regime), the phase
dynamics does not change significantly. However, amplitude spirals appear in
the modulus of the field. The split of the phase into three locked states is
observed approximately at the predicted theoretical value of the forcing
$\gamma_c$. For forcings slightly greater than $\gamma_c$, we observe 
annihilation of vertices until an homogeneus state is achieved. For larger
forcings, the nonvariational dynamics is capable to stop vertex annihilation
and therefore the coarsening process.

Above the BF line ($1+\alpha\beta<0$), we took $\alpha=5.5$ and kept the rest
of parameters as before. The most notorious difference with the previous case
is the existence of asymptotic frozen states in the oscillatory regime and also
close to the critical forcing $\gamma_c$ (see figure \ref{fig:snapsCaseVI}).
Below $\gamma_c$, we observe frozen targets while close to the transition the
frozen patterns hold three locked phase states but without vertices. As
expected, large enough forcings give rise to a time-dependent dynamics with
three-armed spirals rotating around vertices.

Beyond the phase approximation, different phenomena may occur. In particular,
pattern forming instabilities may take place for small and moderate forcings
above the critical value $\gamma_c$. In the case above the BF line with
parameters \{$\alpha=1$, $\beta=-0.76$; $\gamma_c=0.72$\} (\emph{phase
turbulence} regime in the absence of external forcing), oscillating targets
that coexist with vertices are observed close to the transition (see figure
\ref{fig:snapsCaseV}). In the modulus, pulses form at the center of the targets
and move away, while the phase oscillates between $-\pi$ and $\pi$ with a
certain period.  

We also considered, out of the phase approximation, a case below the BF line.
We took the set of parameters \{$\alpha=0$, $\beta=-1.8$; $\gamma_c=1.45$\},
which correspond to a \emph{defect turbulence} regime at $\gamma=0$. As
$1+\alpha\beta>0$, well-developed spirals can be observed (see figure
\ref{fig:snapsCaseIV}). The modulus of the field is characterized for
$\gamma<\gamma_c$ by amplitude spirals that rotate around defects whereas the
phase field shows a similar behaviour to the no forcing situation. On the other
hand, since $\vert\beta\vert>\sqrt{3}$, and according to the discussion of
section \ref{sec:uniform}, there exists a range of forcing intensities for
which the locked solutions are oscillatory unstable at zero wavenumber. This is
seen in figure \ref{fig:zero}. This instability is observed after the
annihilation of two counter-rotating defects. An oscillating target develops
and its central part spreads over the system as it oscillates, and eventually
it dissapears.  

In all the cases studies, the phase is locked approximately at the value
theoretically predicted of the forcing amplitude. When the phase approximation
is valid, the locked phase states are seen to be stable but excitable spirals
may be absent near the transition for system parameters such that
$1+\alpha\beta<0$. Beyond the phase approximation, instabilities of the phase
homogeneous states may take place and give rise to complex patterns. It is
important to emphasize that for $\gamma\gg\gamma_c$ the physics is essentially
the same regardless of the no forcing regime, with interfaces rotating around
vertices. This rotation, which is due to the underlying nonpotential dynamics,
inhibits phase coarsening which would take place through vertex annihilation.
The vertices are essentially pinned and the resulting pattern is nearly time
periodic at relative short time scales.


%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%
\label{sec:conclusions}

We studied in this paper a special case of temporal forcing of nonlinear
oscillators beyond a Hopf bifurcation. Temporal modulations with frequencies
nearly equal to three times the critical one, may be strongly coupled with the
unstable modes associated with the Hopf instability. It modifies the
character of the bifurcation and the resulting spatio-temporal patterns.

For forcing amplitudes below a critical value, the system is in an oscillatory
regime, where spatio-temporal behavior strongly depends on the parameters of
the associated Ginzburg-Landau equation. In particular, topological defects
correspond to one-armed phase spirals. For forcing amplitudes above the
critical one, the system is in a phase locked regime with three equivalent
steady states. There are a number of analogies with the patterns observed in
rotating Rayleigh-B\'enard convection. In this case the domains correspond to
sets of paralell convection rolls with a certain orientation and the vertices
to points at which the roll amplitudes take the same value. As in the case of
the CGLE, the rotation of interfaces around vertices is due to nonpotencial
effects.

Like in the $n=1$ and $n=2$ cases of strongly resonant forcings, a form of
excitability may also be observed. However, contrary to the $n=1$ and $n=2$
forcings, no pattern forming instability of frequency locked states occurs, in
this case, in the phase only regime. Due to the nonpotential character of the
dynamics, fronts between equivalent steady states are always moving. The result
is that, when the three equivalent steady states coexist in the system, three
armed rotating spirals are generated around vertices where the fronts
separating each domain coalesce. Hence, we predict a transition
between one-armed phase spirals and three-armed excitable amplitude spirals,
which occurs when the forcing amplitude passes through a critical value. This
transition is confirmed by numerical analysis of the corresponding
Ginzburg-Landau equation.

\acknowledgements
We acknowledge financial support from DGES projects PB-94-1167 and
PB-97-0141-C02-01.

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\clearpage   

\begin{figure}[htp]
\centerline{\epsfig{figure=fig1.eps,width=10cm}}
\caption{Uniform soluitons of eq. (\ref{null}) for several values of the
forcing parameter $\gamma$. System parameters are $\mu=1$, $\beta=-2.0$ (so
$\gamma_c=1.57$). Note that for $\gamma<\gamma_c$ uniform solutions are of the
cycle limit type whereas for $\gamma>\gamma_c$ they become fixed points.}
\label{fig:uniform}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig2.eps,width=10cm}}
\caption{Nullclines and fixed points (in rectangular coordinates) of the forced
Ginzburg-Landau equation for $\gamma = \beta = 4\mu = 1$ in the $R,\phi $
plane.}
\label{fig:nullA}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig3.eps,width=8cm}}
\caption{Nullclines and fixed points (in polar coordinates) of the forced
Ginzburg-Landau equation for $\gamma = \beta = 0.01$, $ \mu = 0.25$ in the
$R,\phi $ plane. Note that for these parameter values $R_+\approx R_-$. }
\label{fig:nullB}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig4.eps,width=\textwidth}}
\caption{Plot of the phase field in the excitable regime in 1d. Note the
existence of three homogeneous phase states. The arrows indicate the direction
of motion of the several fronts. Parameter values are $\mu=1$, $\alpha=2$,
$\beta=-0.2$, $\gamma=0.5$. }
\label{fig:phase1d}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig5.eps,width=10cm}}
\caption{Examples of motion of pairs of inclined fronts separating domains
with equivalent steady states of the forced Ginzburg-Landau equation for
$\gamma >\gamma_c$.}
\label{fig:2dfronts}
\end{figure}
%
\begin{table}
\newlength{\LL} \settowidth{\LL}{Defect Turbulencex}
\begin{center}
\begin{tabular}{|m{1.2cm}<{\centering}>{\centering}m{1.2cm}>{\centering}m{1.2cm}
>{\centering}m{1.2cm}>{\centering}m{2.75cm}>{\centering}m{1.5cm}
>{\centering}m{\LL}m{1.5cm}<{\centering}|}
\hline $\mu$ & $\alpha$ & $\beta$ & $\gamma_c$ & Phase approx. valid? &
$1+\alpha\beta$ & Regime ($\gamma=0$) & Figure \\ \hline\hline
$1.0$ & $2.0$ & $-0.20$ & $0.20$ & Yes & $>0$ & \emph{Frozen states} &
\ref{fig:snapsCaseI} \\
$1.0$ & $5.5$ & $-0.20$ & $0.20$ & Yes & $<0$ & \emph{Frozen states} &
\ref{fig:snapsCaseVI} \\ \hline
$1.0$ & $2.0$ & $-0.76$ & $0.72$ & No  & $<0$ & \emph{Phase turbulence} &
\ref{fig:snapsCaseV} \\
$1.0$ & $0.0$ & $-1.80$ & $1.45$ & No  & $>0$ & \emph{Defect turbulence} &
\ref{fig:snapsCaseIV}, \ref{fig:zero} \\ \hline
\end{tabular}
\end{center}
\caption{Parameters of the various cases discussed in section
\ref{sec:numerical}.}
\label{numericalcases}
\end{table}
%
\begin{figure}
\centerline{\epsfig{figure=fig6.eps,width=\textwidth}}
\caption{Modulus and phase of the complex field $A$ in the cases $\gamma=0$
(no forcing), $\gamma<\gamma_c$ (oscillatory), $\gamma\simeq\gamma_c$ and
$\gamma>\gamma_c$ (excitable). Parameter values are $\mu=1$,
$\alpha=2$, $\beta=-0.2$ (so $\gamma_c\simeq 0.2$), and 
$\gamma=0.1\ (0.25)$ for the oscillatory (excitable) case.}
\label{fig:snapsCaseI}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig7.eps,width=\textwidth}}
\caption{Same as in figure \ref{fig:snapsCaseI}. Parameter values are
$\mu=1$, $\alpha=5.5$, $\beta=-0.2$ ($\gamma_c\simeq 0.2$), and 
$\gamma=0.1\ (0.25)$ for the oscillatory (excitable) case.} 
\label{fig:snapsCaseVI}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig8.eps,width=\textwidth}}
\caption{Same as in figure \ref{fig:snapsCaseI}. Parameter values are
$\mu=1$, $\alpha=2$, $\beta=-0.76$ ($\gamma_c\simeq 0.72$), and 
$\gamma=0.5\ (1.5)$ for the oscillatory (excitable) case.}
\label{fig:snapsCaseV}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig9.eps,width=\textwidth}}
\caption{Same as in figure \ref{fig:snapsCaseI}. Parameter values are
$\mu=1$, $\alpha=0$, $\beta=-1.8$ ($\gamma_c\simeq 1.45$), and 
$\gamma=1\ (1.6)$ for the oscillatory (excitable) case.}
\label{fig:snapsCaseIV}
\end{figure}
%
\begin{figure}
\centerline{\epsfig{figure=fig10.eps,width=0.75\textwidth}}
\caption{Snapshots of the modulus of the field in a regime of
parameters where an oscillatory instability at zero wave number occurs. The
square encloses an oscillating region. Time increases when going from (a)
to (d).}
\label{fig:zero}
\end{figure}


\end{document}