# Phase-locking dynamics of heterogeneous oscillator arrays

Stefano Lepri  Consiglio Nazionale delle Ricerche, Istituto dei Sistemi
Complessi, Via Madonna del Piano 10 I-50019 Sesto Fiorentino, Italy  Istituto
Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1 I-50019,
Sesto Fiorentino, Italy  Arkady Pikovsky  Department of Physics and Astronomy,
University of Potsdam Karl-Liebknecht-Str 24/25, Bldg 28 D-14476, Potsdam,
Germany  Department of Control Theory, Lobachevsky University of Nizhny
Novgorod, Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia

###### Abstract

We consider an array of nearest-neighbor coupled nonlinear autonomous
oscillators with quenched random frequencies and purely conservative coupling.
We show that global phase-locked states emerge in finite lattices and study
numerically their destruction. Upon change of model parameters, such states
are found to become unstable with the generation of localized periodic and
chaotic oscillations. For weak nonlinear frequency dispersion, metastability
occur akin to the case of almost-conservative systems. We also compare the
results with the phase-approximation in which the amplitude dynamics is
adiabatically eliminated.

###### keywords:

Ginzburg-Landau lattice, Disorder , Localized chaos , Reactive coupling

##  1 Introduction

Nonlinear dynamics of systems driven out of equilibrium by external (possibly
non-conservative) forces is a fascinating research subject. The aim of this
vast, interdisciplinary research is to illustrate generic and universal
features by simple paradigmatic models. Among them, systems of classical
coupled oscillators are of particular interest as they represent a large
variety of different physical problems like atomic vibrations in crystals and
molecules or field modes in optics or acoustics.

Among a large number of nonlinear effects, synchronization of coupled
autonomous oscillators, first empirically observed by Huygens, is one of the
most remarkable. The basic theory has been advanced considerably [1] but still
attracts considerable attention, as demonstrated by the huge number of papers
published in the last decade.

A notable source of complexity in nonlinear systems is heterogeneity of
individual elements. For ensembles of linear, heterogeneous oscillators,
static disorder is known to lead to Anderson localization of matter, light,
and sound waves. Generically, the normal models (eigenstates) display a
localized pattern that usually hinders energy propagation [2].

As a matter of fact, in presence of disorder, nonlinearity, and interactions a
whole new set of phenomena may arise. The interplay between disorder and
nonlinearity, their localizing and delocalizing effects on lattice waves is
also an intriguing and challenging current research theme. For conservative
systems, it has been recognized that the addition of nonlinearity causes
interaction among the eigenmodes, which results in a slow wave diffusion [3,
4, 5]. For instance, two different regimes of destruction of Anderson
localization (asymptotic weak chaos, and intermediate strong chaos), separated
by a crossover condition on densities have been reported [6]. External driving
and dissipation can lead to even richer scenarios. For instance, disordered
nonlinear array with forcing and dissipation admit Anderson attractors [7] –
stationary multipeak patterns composed by a set of interacting, excited
Anderson modes. Such attractors emerge by joint effect of the pumping-induced
mode excitation, nonlinearity-induced mode interactions, and stabilization by
dissipation.

Besides intrinsic hetereogeneity of oscillators, another source of disorder
may arise from topology of connections, as it occurs for coupled nonlinear
oscillator on networks (see e.g. ref.[8] and references therein). From the
experimental point of view, we mention active optical systems, where joint
effect of disorder (both in frequencies and cavity topology) and nonlinearity
is relevant for random lasers [9] and lasing networks [10, 11].

If coupling of periodic self-sustained oscillators is weak, the dynamics can
be described in terms of the phase approximation, where only a variation of
oscillator phases are considered. For two coupled oscillators this leads to an
Adler-type equation [1]. The corresponding phase models are extensively used
to study oscillator lattices and globally coupled ensembles [12]. Much less
studied is the case in which the amplitude dynamics enters into play.

A related important issue is the possibility of observing states with a non-
trivial collective dynamics. In the context of coupled phase oscillators, the
case of short-ranged coupling (e.g. to nearest-neighbor on a lattice) has been
less considered than the globally coupled one. A general question regards the
distinction between synchronization and frequency entrainment and the mapping
to the diffusion equation [13]. The linear theory predicts frequency
entrainment in any dimension, for any non zero coupling but the role of
nonlinear terms is still to be understood. It is known [14] that up to certain
values of disorder in natural frequencies of phase oscillators, one can find
synchronous states. Also, it has been proved that the probability of frequency
entrainment is generically exponentially small in the number of oscillators
[15].

In the present work we report a numerical study of a one-dimensional lattice
of self-sustained heterogeneous oscillators described by their phase and
amplitudes with purely reactive (i.e.conservative) coupling. More precisely,
we consider a one-dimensional Ginzburg-Landau lattice [16] where each site has
a different (random) natural frequency. In the literature, the case of purely
diffusive coupling and global coupling it is mostly considered, see e.g. [17,
18, 19]. An example of global reactive coupling is in Ref. [20]. The choice of
a purely conservative and local coupling is motivated by the experiments on
lasing networks, where the individual elements (non-identical active and
passive fibers) are coupled by non-dissipative optical couplers [21] but
applies to more general nonlinear multimode photonic networks [22]. A similar
coupling has been recently realized for nanoelectromechanical oscillators
[23]. We will illustrate that non-trivial state exist, including frequency-
entrained domains and localized chaotic states. The paper is organized as
follows. In Section 2 we define the model and in Section 3 we describe some of
its phenomenology. Section 4 deals with average properties with respect to the
disordered realizations. A brief discussion on the phase approximation and its
continuum limit is given in Section 5.

##  2 Model

In the present work we considered a specific paradigmatic model namely the
discrete Ginzburg-Landau (or locally coupled Stuart-Landau) array of
oscillators with with random intrinsic frequencies

$\dot{\psi}_{n}=\left[i\omega_{n}+\gamma+(i\alpha-\beta)|\psi_{n}|^{2}\right]\psi_{n}+(c_{2}+ic_{3})(\psi_{n-1}+\psi_{n+1})$
(1)

where $\psi_{n}$ is the complex oscillation amplitude. Frequencies
$\omega_{n}$ are chosen as independent random numbers uniformly distributed in
$[-w/2,w/2]$, they produce quenched disorder. The index $n=1,\ldots,N$ labels
the lattice sites and periodic boundary conditions are assumed. The parameter
$\alpha$ describes nonlinear frequency shift while $\beta>0$ is the nonlinear
dissipation which is necessary to counterbalance the growth term $\gamma>0$.

We focus on the purely reactive coupling case namely we set $c_{2}=0$ and
$c_{3}\equiv c\neq 0$. The model has two interesting limits. For vanishing
$\alpha,\beta,\gamma$ it reduced to the well-known one-dimensional Anderson
model (linear disordered lattice). For $\gamma=\beta=0$ and $\omega_{n}$ non-
random, it is the so-called Discrete Nonlinear Schrödinger equation [24],
which has been widely investigated in many different nonequilibrium conditions
[25, 26, 27]. The DNSE with random frequencies has been considered in [3, 4].

For later reference, we also write the equations for amplitude and phases of
the complex field ${\psi}_{n}=a_{n}e^{i\theta_{n}}$. Introducing the phase
differences $\varphi_{n}=\theta_{n+1}-\theta_{n}$ and the relative detunings
$\delta_{n}=\omega_{n+1}-\omega_{n}$ between neighbouring oscillators, we
obtain the exact equations

$\displaystyle\dot{a}_{n}=(\gamma-\beta
a_{n}^{2})a_{n}+c\left[a_{n+1}\sin\varphi_{n}-a_{n-1}\sin\varphi_{n-1}\right]$
$\displaystyle\dot{\varphi}_{n}=\delta_{n}+\alpha(a_{n+1}^{2}-a_{n}^{2})+$ (2)
$\displaystyle+c\left[\frac{a_{n+2}}{a_{n+1}}\cos\varphi_{n+1}+\left(\frac{a_{n}}{a_{n+1}}-\frac{a_{n+1}}{a_{n}}\right)\cos\varphi_{n}-\frac{a_{n-1}}{a_{n}}\cos\varphi_{n-1}\right]$

Note that the $\delta_{n}$ are random (with uniform distribution) but for
periodic boundary they must satisfy the constraint $\sum_{n}\delta_{n}=0$
(only the first $N-1$ differences are indeed independent). Correspondingly,
the number of equations for the phase differences $\varphi_{n}$ is $N-1$.
Those equation will be the basis of some approximate treatment below.

The model has five control parameters but, upon rescaling, one may fix two to
unity. In the following, we choose to leave the gain parameters $\gamma$ free,
and we set $c=1$, $\beta=1$ so that $w$ and $\gamma$ are in units of $c$.

##  3 Simulations

We have first performed a series numerical simulations for different gain
parameters $\gamma$ and $N=200$, and fixed $w,\alpha$. Several lattice sizes
have been considered with similar outcomes. The monitored quantities are the
time averages (over the simulation time $T$) of the amplitudes
$\langle|\psi_{n}|^{2}\rangle$ and the mean frequencies

$\langle\dot{\theta_{n}}\rangle=\frac{1}{T}[\theta_{n}(T)-\theta_{n}(0)]\;,$

as well as fluctuations of these quantities
$\langle\dot{\theta}_{n}^{2}\rangle-\langle\dot{\theta}_{n}\rangle^{2}$ and
$\langle|\psi_{n}|^{4}\rangle-\langle|\psi_{n}|^{2}\rangle^{2}$, respectively.

Figure 1: Space-time plots of $|\psi_{n}|^{2}$ for different $\gamma$ and
$w=1.5$ $\alpha=2$, chain length $N=200$, after discarding a transient of
$10^{4}$ time units. At large $\gamma=1$ a stationary state is observed. At
$\gamma=0.74$, periodic modulation close to $n=170$ occurs. At $\gamma=0.46$,
several domains with perdiodic and chaotic modulations are observed. This
state can be described as a coexistence of phase-locked clusters separated by
localized chaotic or periodic oscillations. Finally, at $\gamma=0.3$, the
regime is rather turbulent.

Figure 2: Spatial configurations for $\gamma=0.74$ (upper panel)
and$\gamma=0.46$ (lower panel) of the simulations presented in figure 1. In
each panel the average amplitudes and their fluctuations, the average
frequencies and their fluctuations. For $\gamma=0.74$ there is only one
irregular region; for $\gamma=0.46$ there are four such domains.

Figure 3: Time evolution of amplitudes and phase velocities at the site
$n=166$ of the chain for the simulation reported above. Upon decreasing
$\gamma$ there is a bifurcation from periodic to chaotic. Parameters are the
same as in fig.1.  Figure 4: Localized chaos for $\alpha=2$ and $w=1.5$,
$\gamma=0.46$, chain length $N=200$, calculated after discarding a transient
of $10^{4}$ time units. Left: Space-time plot of the Lyapunov vector
displaying a localized structure around site $n=100$. Right: time traces of
the oscillator amplitudes on two sites belonging to the chaotic region.
Figure 5: Space -time plots of $|\psi_{n}|^{2}$ showing multistability for
four different initial conditions, at $\alpha=0.03$ and $w=1.5$, $\gamma=2$,
chain length $N=200$. A transient of $10^{4}$ time units has been discarded.

The main phenomenology is exemplified by Figures 1, 2 and 3 for a particular
realization of the disorder. The first observation is that, for relatively
large $\gamma$, the system approach a fully synchronized regime whereby all
the oscillators have the same common frequency but different amplitudes
$a_{n}$ and constant phase shifts $\varphi_{n}$. Such a state depends on the
specific disorder realization and should correspond to a stable stationary
solution of eqs. (2) (a uniformly rotating solution of eqs. (1)).

Upon decreasing $\gamma$ the global-phase locked state destabilizes gradually.
There is the spontaneous creation of a “defect” at some random location, where
the amplitudes at a few neighboring sites start to oscillate in time (see the
case $\gamma=0.74$ in figure 1, where the position of the defect is around
$n=170$). Further decrease of $\gamma$ leads to appearance of new defects and
to more complex dynamics at each defect (figure 2). As a result, a regime
appears where patches of phase-entrained oscillators (frequency plateaus), are
separated by defects that may oscillate periodically or chaotically (see the
labeled arrows in figure 1).

An heuristic explanation of the existence of the localized instability can be
given by the following argument. Let denote by $\psi_{n}=u_{n}\exp(-i\mu t)$
the global phase-entrained solution (existing at large values of $\gamma$),
with $\mu$ being its frequency. The linear stability analysis is performed by
letting $\psi_{n}=(u_{n}+\chi_{n})\exp(-i\mu t)$ with $\chi_{n}$ complex, and
linearizing the equation of motion to obtain

$\dot{\chi}_{n}=[i(\omega_{n}-\mu)+\gamma]\chi_{n}+ic[\chi_{n+1}+\chi_{n-1}]+(i\alpha-\beta)\left(2|u_{n}|^{2}\chi_{n}+u_{n}^{2}\chi_{n}^{*}\right).$
(3)

The above eigenvalue problem contains random terms (quenched disorder). It is
thus akin to a random matrix problem where we expect to have localized
eigenvectors. Also, it is not self-adjoint so we expect complex eigenvalues
leading to oscillatory instability and thus localized modulations of the
solution.

The localized nature of chaotic states is illustrated in the left panel of
Fig. 4. We show the space-time evolution of the Lyapunov vector associated
with the largest Lyapunov exponent, computed by evolving the linearized
equation (same as eq. (3), but on top of a chaotic solution $\psi_{n}(t)$).
The appearance of chaotic oscillations in usually associated to the fact that
amplitudes tend to become small (see figure 3 and the right panels in fig. 4).
So it is evident that the amplitude dynamics plays a crucial role in this
regime.

The same type of scenario is observed upon changing $\alpha$ at fixed $\gamma$
(data not reported). A novel feature however emerges in the limit of very
small $\alpha$ when the steady state depends on the initial conditions (see
figure 5). The observed final state entails phase-entrained cluster with
chaotic ones that resemble spatio-temporal intermittency. Multistable behavior
is confirmed by computing the Lyapunov exponent $\lambda$. As seen in figure
6, regular trajectories with $\lambda\approx 0$ coexist with chaotic ones for
the same value of the parameter $\alpha$ suggesting coexistence of regular and
chaotic attractors.

Figure 6: (a) The largest Lyapunov exponent versus $\gamma$ for the same
disorder realization and parameters as in previous figures. (b) The largest
Lyapunov exponent versus $\alpha$ for four different initial conditions
showing the multistable behavior $N=200,w=1.5,\gamma=2$. The error on the
exponent is roughly of the size of the symbols.

The above results have been obtained for an uniform and thus bounded
distribution of frequencies. Actually, the choice of the distribution is
relevant. For instance, a set of simulations performed with a Gaussian
distribution with the same width $w$ indicate that it is somehow easier to
find unlocked states there.

##  4 Ensemble properties

In the section above we have focused on a specific realization of the random
frequencies $\omega_{n}$. For a statistical characterization, we measured the
length of the phase-locked clusters $l_{i}$ defined by the condition that the
mean frequency difference is less that a given threshold. Let us denote by $M$
their number and by $N_{s}$ the size of the largest one. In fig. 7 we report
two series of simulations averaged over disorder realizations. The data show a
significant dependence on the lattice size: full phase-locking is achieved
provided that $\gamma$ or $\alpha$ are large enough but the value required
increases systematically with $N$. On the other hand, the number of clusters
$M$ seems to be extensive in the size. This also means that the average
density of localized (regular or chaotic) defects is roughly constant but
vanishes for large $\gamma$ or $\alpha$.

To appreciate the role of the disorder strength and fluctuations, we report in
Figure 8 the number $M$ of phase-locked clusters as a function of $w$. To have
an idea of the sample-to-sample statistics, we draw $M$ for each disorder
realization. There is a clear transition from full entrainment ($M=1$) upon
increasing $w$ that for smaller $\alpha$ appears to be more abrupt.

Figure 7: Upper panels: the disorder-averaged size of the larger phase-locked
cluster sizes; lower panels: the average number of clusters as a function
$\gamma$ and $\alpha$. Different symbols refer to different lattice sizes (see
legend). The threshold to locate the clusters is set to $10^{-3}$.

Figure 8: The dependence of phase-locked cluster number on the disorder
strength $w$ for a chain of length $N=200$, for two different values of
$\gamma=1$ and $5$ and different $\alpha$. Each point represents a different
disorder realization, the solid lines are the corresponding averages. The
threshold to locate the clusters is set to $10^{-3}$. Zero number of clusters
(at small disorder) means full synchrony, the number of clusters close to
$N=200$ (at large disorder) means almost complete asynchrony.

##  5 Phase approximation

An interesting question is to what extent the dynamics can be described in
term of the phase approximation, where the amplitude evolution is neglected.
This approach allows also to rationalize the phenomenology observed in the
above model.

For weak coupling and large enough gains, the system can indeed be
approximated in terms of an effective equation for the relative phases
$\varphi_{n}$ between neighboring oscillators. If the amplitude relaxes
quickly enough and $c$ is small, one can perform a straightforward adiabatic
elimination of the amplitudes in eqs. (2): letting first
$a_{n}=\sqrt{\gamma/\beta}(1+r_{n})$ we approximate eqs. (2) as

$\displaystyle\dot{r}_{n}\approx-2\gamma
r_{n}+c\left[\sin\varphi_{n}-\sin\varphi_{n-1}\right]$ (4)
$\displaystyle\dot{\varphi}_{n}\approx\delta_{n}+2\alpha{\frac{\gamma}{\beta}}(r_{n+1}-r_{n})+c\left[\cos\varphi_{n+1}-\cos\varphi_{n-1}\right]$

Eliminating the first equation adiabatically, we get
$r_{n}=c\left[\sin\varphi_{n}-\sin\varphi_{n-1}\right]/(2\gamma)$ and

$\dot{\varphi}_{n}=\delta_{n}+{\frac{\alpha
c}{\beta}}(\sin\varphi_{n+1}+\sin\varphi_{n-1}-2\sin\varphi_{n})+c\left[\cos\varphi_{n+1}-\cos\varphi_{n-1}\right]$
(5)

which is akin to the Kuramoto-Sagacuchi model [28, 29], In the case with no
disorder $\delta_{n}=0$, this equation has bee studied in detail in ref. [30].
As discussed there, the terms in $\sin\varphi$ (dissipative couplings) favor
synchronization while those in $\cos\varphi$ (dispersive or conservative
couplings) hinders it. This is best understood by considering the
“conservative limit” where the frequency dispersion is small $\alpha\to 0$.
The phase equation is further approximated as

$\dot{\varphi}_{n}\approx\delta_{n}+c\left[\cos\varphi_{n+1}-\cos\varphi_{n-1}\right]$
(6)

which has vanishing phase-space divergence, meaning that the dynamics is
similar to a (disordered) conservative system. The physical interpretation is
that in this limit, the amplitude and the phase dynamics decouple from the
beginning, and the phase dynamics is purely conservative because the coupling
is purely reactive in eq. (1). Since the phase space is compact (variables are
angles), a small dissipation will stabilize motion on tori and we may expect
some form of multistability as in the case of Hamiltonian systems with weak
dissipation [31]. This qualitatively fits with the phenomenology observations
made in the previous sections, see again figures 5 and 6.

An analysis of eq. (5) in presence of the random detunings $\delta_{n}$ has
been given recently in ref.[29]. It is there shown that the stationary
synchronized solution persists upon increasing the level of disorder in the
natural frequencies. and that the phase shifts. Also, the domain of existence
of the stable synchronous regime widens in comparison to the case of zero
phase shift. This is quite counterintuitive, because a phase shift terms
usually act against synchronization.

Note that equation (5) is, in this approximation, independent on $\gamma$. We
remark that in the case in which $\gamma\to\gamma_{n}$ is not uniform, we
expect the nearest-neighbor couplings to be site dependent, thus representing
a further source of inhomogeneity in the phase equations. Moreover, as in the
case of the well-know Adler equation, it gives a rough criterion: stationary
solutions exist only if $\delta_{n}$ is not too large. A necessary condition
for locking would be $w<4\frac{\alpha c}{\beta}$ as the frequencies are
distributed uniformly.

To conclude this Section we add some remarks on the continuum limit of the
derived phase equations. For small $\varphi_{n}$ the equation is approximated
to quadratic order as

$\dot{\varphi}_{n}\approx\delta_{n}+{\frac{\alpha
c}{\beta}}(\varphi_{n+1}+\varphi_{n-1}-2\varphi_{n})-\frac{c}{2}(\varphi^{2}_{n+1}-\varphi^{2}_{n-1})$
(7)

the first term reminds of a diffusion equation with random sources. The
nonlinear term is dispersive. Taking the continuum limit, letting
$D={\frac{\alpha c}{\beta}}$ and denoting by $\Delta(x)$ the random frequency
differences, we have

$\dot{\varphi}=\Delta(x)+D\partial_{x}^{2}\varphi-c\partial_{x}(\varphi^{2})$
(8)

which is incidentally a Burgers equation with quenched static disorder. As it
is well know, the Burgers equation can be mapped onto the Kardar-Parisi-Zhang
(KPZ) equation via the transformation $\varphi=\partial_{x}h$ yielding

$\dot{h}=\omega(x)+D\partial_{x}^{2}h-c(\partial_{x}h)^{2}+F$ (9)

where $F$ is an integration constant. The relation to coupled oscillator
lattices with KPZ dynamics has been pointed out before [1] and has been
invoked to explain phase dynamics in the noisy Kuramoto-Sakaguchi model with
local couplings [32]. It should be emphasized that here we are dealing with
the case of KPZ with quenched additive noise it usually referred to as
columnar disorder in the literature [33].

Finally, An alternative formulation is obtained taking the time derivative of
(8) and introduce the new field $v$

$\displaystyle\dot{\varphi}=v$ (10)
$\displaystyle\dot{v}=D\partial_{x}^{2}v-2c\,\partial_{x}(v\varphi)$

from which it is seeen that $v=0$, $\bar{\varphi}(x)$ is the stationary
frequency-entrained solution. This introduces some form of inertia in the
phase dynamics. This will not be discussed further but we remark that this
equation is different from the one given in eqs. (36-37) of ref. [13]. This
suggests that the model considered here may belong to a different class.

##  6 Conclusions

In summary, we have studied synchronization transitions in a disordered
Ginzburg-Landau lattice of coupled limit-cycle oscillators. We have
demonstrated that at large values of the activity parameter $\gamma$ a fully
synchronous state in the lattice establishes. For smaller values of $\gamma$,
transition to turbulence occurs through a sequence of appearing isolated
periodic or chaotic “defects”. At the intermediate stage the patches between
the defects remain synchronized, so that the dynamics can be described as a
combination of several synchronous clusters having different frequencies, plus
several localized regions with weak chaos. In the latter regions the dynamics
of the amplitudes appears to be crucial. Thus, in the phase approximation,
where the amplitudes are considered as slaved variables, the features of
localized chaotic defects are not properly reproduced. We have also
demonstrated how the number of different clusters in the lattice grows with
the growing strength of disorder.

## Declaration of Competing Interest

The Authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

SL acknowledge the research scholarship 91673361, DAAD funding programme
Research stays for university academics and scientists, 2017 for granting the
sojourn at Potsdam University where this work was initiated. AP was supported
by Russian Science Foundation (grant Nr. 17-12-01534) and by DFG (grant PI
220/22-1).

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