11institutetext: Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B. T. Road, Kolkata-700108, India

# Dynamics of swarmalators: A pedagogical review

Gourab Kumar Sar and Dibakar Ghosh 111Email<EMAIL_ADDRESS>
###### Abstract

Swarmalators have emerged as a new paradigm for dynamical collective behavior
of multi-agent systems due to the interplay of synchronization and swarming
that they inherently incorporate. Their dynamics have been explored with
different coupling topologies, interaction functions, external forcing, noise,
competitive interactions, and from other important viewpoints. Here we take a
systematic approach and review the collective dynamics of swarmalators
analytically and/or numerically. Long-term states of position aggregation and
phase synchronization are revealed in this perspective with some future
problems.

###### pacs:

05.45.Xt

###### pacs:

05.90.+m

###### pacs:

89.75.Fb

Synchronization; coupled oscillators Nonlinear dynamical systems Structures
and organization in complex systems

##  1 Introduction

There is a common saying that “birds of a feather flock together” which is
often metaphorically used to highlight the similar attributes of people who
are of same sort. Not only humans, it is found in nature that most of the
living organisms have the tendency to stay in a colony and mimic the
activities of their neighbors. Flock of birds, school of fish, herd of sheep
are some very well-known examples of this very fact [1, 2, 3]. This phenomenon
is commonly known as swarming in the study of multi-agent systems [4, 5]. The
examples of swarming are pervasive in coordinated movement of group of
animals. To the best of our knowledge, there is no consistent definition of
swarming, but swarming systems typically inherit at least one of the two key
features: (i) aggregation and (ii) alignment. Similar to swarming, there is
another common occurrence which is synchronization, the examples of which are
ubiquitously found in nature and technology [6, 7, 8]. Chorusing frogs [9],
firing neurons [10], cardiac dynamics [11], the dynamics of power grids [12]
are some instances where synchronization spontaneously occurs. Synchronization
usually means self-organization behavior of entities of their internal states
in time.

Swarming and synchronization are two very similar, but different phenomena.
Swarming is self-organization behavior in space, whereas synchronization can
be considered as self-organization behavior in time. In other words, in
swarming, the external states of the entities are given core attention,
whereas in synchronization, it is the internal states of the units that takes
over. Over the past few decades, studies on these two fields have evolved
parallelly, though mostly they have remained disconnected. Although in the
studies of mobile agents or moving oscillators, the effect of agents’ motion
on the phase dynamics has been contemplated, the reverse scenario has not been
considered [13, 14, 15, 16, 17]. Oscillators which can swarm, named
swarmalators, were first introduced by O’Keeffe et al. [18] with the property
that the internal and external dynamics are affected by each other and can be
identified as a special class of active systems [19, 20, 21, 22]. This kind of
interplay between sync and swarming is prevalent in nature and has been
encountered in different fields, from biology [23, 24] and chemistry [25, 26]
to physics [27, 28] and robotics [29, 30] etc. Besides this, experimental
realizations of the swarmalators systems have been performed in the last
decade. Caenorhabditis elegans are found to synchronize their swimming gait
when they are within close proximity [31]. The feedback between the fluid flow
generated by catalytic reaction and flexible sheets immersed in a fluid filled
chamber enables synchronized motion of the sheets in space and time [32]. A
recent experimental study with the nematode Turbatrix aceti reveals the
formation of synchronized collective state whose strength and location are
controllable [33]. However, the pioneering work in this direction was carried
out by Tanaka et al. [34, 35] when studying “chemotactic oscillators” whose
movements in space are mediated by a surrounding chemical.

In this perspective article, we discuss the long-term asymptotic behaviors of
the swarmalators under the bidirectional effect of spatial and phase dynamics.
We review the theoretical aspects and numerically analyze the long-term states
for a lucid knowledge of such systems. By introducing a swarmalator model, we
analyze the basic properties, like collision avoidance and minimal inter-
particle distance. In the case of global interactions among the constituents,
we explore the dynamical states of the model with proper choices of
interaction functions and also under the introduction of external forcing. The
study is also accomplished with local interactions in the system and in
presence of thermal noise. We further introduce attractive-repulsive phase
coupling which brings competitive phase interaction and investigate such
effect on the model. Finally, for better theoretical understanding of the
emerging states, we study the dynamics of swarmalators when they are placed on
a $1$D ring.

##  2 Model of swarmalators

The bidirectional interplay between the spatial and phase dynamics of the
swarmalators is described by the following pair of equations,

$\dot{\textbf{x}}_{i}=\textbf{v}_{i}+\frac{1}{N}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\bigg{[}\text{I}_{\text{att}}(\textbf{x}_{j}-\textbf{x}_{i})\text{F}_{\text{att}}(\theta_{j}-\theta_{i})\\\
-\text{I}_{\text{rep}}(\textbf{x}_{j}-\textbf{x}_{i})\text{F}_{\text{rep}}(\theta_{j}-\theta_{i})\bigg{]},$
(1) $\dot{\theta}_{i}=\omega_{i}+\frac{K}{N}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\text{H}(\theta_{j}-\theta_{i})\text{G}(\textbf{x}_{j}-\textbf{x}_{i}),$
(2)

for $i=1,2,\ldots N$, where $N$ is the number of swarmalators and
$\textbf{x}_{i}\in\mathbb{R}^{d}$ ($d=2$ or $3$),
$\theta_{i}\in\mathbb{S}^{1}$ are the position vector and phase of the $i$th
swarmalator, respectively. $\textbf{v}_{i}$ is the self propulsion velocity
and $\omega_{i}$ is the natural intrinsic frequency of the $i$th swarmalator.
The alignment component of the swarming dynamics in Eq. (1) is reflected
through the velocity $\textbf{v}_{i}=(v_{0}\cos\eta_{i},v_{0}\sin\eta_{i})$,
where $\eta_{i}$ is the orientation of the $i$th swarmalator. The orientation
of particles (which is another state variable along with x and $\theta$) is
not taken into account by the current models of swarmalators and will not be
studied in this review. There are spatial attraction and repulsion among the
swarmalators, which are regulated by the functions
$\text{I}_{att}(\textbf{x})$ and $\text{I}_{rep}(\textbf{x})$, respectively.
Swarmalators in nearby phases can attract or repel themselves spatially. The
influence of phase similarity of spatial attraction and repulsion are governed
by the functions $\text{F}_{att}$ and $\text{F}_{rep}$, respectively. The
phase interaction is determined by the function H, where the phase coupling
strength is given by $K$. The spatial positions of the swarmalators affect
their phase couplings, which is taken into account in the model by the
function G. Note that, if there is no effect of phase similarity on spatial
attraction or repulsion, i.e., when $\text{F}_{att}(\theta)=1$ and
$\text{F}_{rep}(\theta)=1$, then Eq. (1) represents pure swarming dynamics. On
the other hand, when the effect of spatial position on the phase dynamics is
not considered in Eq. (2) (i.e., when $\text{G}(x)=1$), it represents the
dynamics of coupled phase oscillators which synchronize beyond a critical
phase coupling strength.

##  3 Theoretical properties

The presence of multiple interaction functions in the model makes it difficult
to deal analytically. However, analytical progress can be made if we consider
a particular instance of the model with the following choices

$\text{I}_{att}(\textbf{x})=\frac{\textbf{x}}{|\textbf{x}|^{\alpha}},\,\text{I}_{rep}(\textbf{x})=\frac{\textbf{x}}{|\textbf{x}|^{\beta}},\,\text{G}(\textbf{x})=\frac{1}{|\textbf{x}|^{\gamma}},$
(3)

where $\text{I}_{att}$, $\text{I}_{rep}$, and G are chosen as power laws with
positive exponents $\alpha$, $\beta$, and $\gamma$, respectively, where
$|\cdot|$ represents the Euclidean norm. $\alpha$ and $\beta$ play a crucial
role in determining the aggregation structure of the system. We want to ensure
short-range repulsion and long-range attraction among the swarmalators so that
collision among them is avoided and the solution remains bounded [4]. For that
we assume the exponents $\alpha$ and $\beta$ satisfy the following condition

$1\leq\alpha<\beta.$ (4)

We further assume $\text{F}_{\text{att}}$ and $\text{F}_{\text{rep}}$ to be
even and bounded functions of their arguments. For notational simplicity we
set,
$\textbf{X}:=(\textbf{x}_{1},\textbf{x}_{2},\ldots,\textbf{x}_{N}),\;\textbf{V}:=(\textbf{v}_{1},\textbf{v}_{2},\ldots,\textbf{v}_{N}),\;\Theta:=(\theta_{1},\theta_{2},\ldots,\theta_{N}),\;W:=(\omega_{1},\omega_{2},\ldots,\omega_{N}),\;\mathcal{N}:=\\{1,2,\ldots,N\\}.$
We define the following functionals

$\displaystyle\mathcal{C}_{1}(\textbf{X},\textbf{V},t):=\sum_{i\in\mathcal{N}}\textbf{x}_{i}-t\sum_{i\in\mathcal{N}}\textbf{v}_{i},$
(5)
$\displaystyle\mathcal{C}_{2}(\Theta,W,t):=\sum_{i\in\mathcal{N}}\theta_{i}-t\sum_{i\in\mathcal{N}}\omega_{i}.$
(6)

Now, taking the sum over Eqs. (1) and (2) it is elementary to see that
$\dfrac{d}{dt}\mathcal{C}_{1,2}=0$. So, these two quantities $\mathcal{C}_{1}$
and $\mathcal{C}_{2}$ are conserved along the swarmalators dynamics. However,
the presence of singular functions like
$\frac{1}{|\textbf{x}_{j}-\textbf{x}_{i}|^{\mu}}$ in the model raises the
question whether it is well-posed or not. It is imperative to rule out the
inter-particle collision, which violates the well-posedness of the model.
Inter-particle collision can be avoided if we assume the initial data
$(\textbf{X}(0),\Theta(0))$ to be non-collisional i.e., $\min_{1\leq i,j\leq
N}|\mathbf{x}_{i}(0)-\mathbf{x}_{j}(0)|>0$ for all $i,j\in\mathcal{N}$ and
$i\neq j$, along with $\alpha,\beta$ satisfying Eq. (4). These conditions
further guarantee the existence of a minimal inter-particle distance among the
swarmalators which eliminates the possibility of unbounded solution in case of
swarmalators being asymptotically close. The detailed discussions of these
results are found in [36].

##  4 Swarmalator dynamics under global interaction

We consider a two-dimensional model of swarmalators [18] with specific choices
of coupling functions in Eqs. (1) and (2). The spatial functions
$\text{I}_{att}$, $\text{I}_{rep}$, and G are chosen (Eq. (3)) with
$\alpha=1,\beta=2$, and $\gamma=1$ which satisfy Eq. (4). The influence of
phase similarity on spatial attraction is considered by taking the function
$\text{F}_{att}(\theta)=1+J\cos(\theta)$ while the influence of phase dynamics
on the spatial repulsion is not considered (i.e., $\text{F}_{rep}(\theta)=1$
here). For $J>0$, like attracts like, i.e., swarmalators in nearby phases
attract each other. When $J$ is negative, swarmalators are preferentially
attached to the ones in opposite phases. $-1<J<1$ is considered so that
$\text{F}_{att}$ is always strictly positive. Phase interaction among the
swarmalators is taken in the spirit of Kuramoto model [7] by taking
$\text{H}(\theta)=\sin(\theta)$. Here identical swarmalators are chosen with
same velocity $\textbf{v}_{i}=\textbf{v}$ and frequency $\omega_{i}=\omega$
and by choice of reference frame both of them are set to zero. So, the system
now becomes

$\dot{\textbf{x}}_{i}=\frac{1}{N}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\bigg{[}\frac{\textbf{x}_{j}-\textbf{x}_{i}}{|\textbf{x}_{j}-\textbf{x}_{i}|}(1+J\cos(\theta_{j}-\theta_{i}))\\\
-\frac{\textbf{x}_{j}-\textbf{x}_{i}}{|\textbf{x}_{j}-\textbf{x}_{i}|^{2}}\bigg{]},$
(7)

$\dot{\theta}_{i}=\frac{K}{N}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\frac{\sin(\theta_{j}-\theta_{i})}{|\textbf{x}_{j}-\textbf{x}_{i}|}.$
(8)

The spatial and phase couplings among the swarmalators are all-to-all, which
means each swarmalator is influenced by the dynamics of other swarmalators
present in the system. The two controllable parameters $J$ and $K$ determine
the system’s long-term dynamical state. Five different asymptotic states are
found depending on the values of $J$ and $K$ which is contemplated in Fig. 1.
These states are static sync, static async, static phase wave, splintered
phase wave, and active phase wave. In the static states, the movement of the
swarmalators is ceased and the phases become stationary. There is only one
emerging state for $K>0$ which is the static sync state (see Fig. 1). The
positive value of phase coupling strength minimizes the phase difference among
the swarmalators and the phases are completely synchronized in this static
state. Three different long-term states are observed for negative values of
$K$ which are static async, splintered phase wave, and active phase wave (see
Fig. 1). In the latter two states, swarmalators exhibit movement in the
spatial position and their phases also keep evolving and we call them active
states. In the absence of phase coupling (when $K=0$), swarmalators’ phases
are locked to their initial values and for $J>0$ they arrange themselves in
the $2$D plane inside an annulus and this state is named static phase wave.
Initial positions of the swarmalators are chosen from $[-1,1]\times[-1,1]$ and
the initial phases are drawn from $[0,2\pi]$, both uniformly and randomly.

Figure 1: $K$-$J$ parameter region for emerging collective states. Red dots
are the points where $U$ bifurcates from zero to non-zero, found numerically.
They are joined with dashed line to separate the regions between splintered
phase wave and active phase wave states. Blue dots are numerically calculated
points where $S$ bifurcates from non-zero to zero. Blue solid line is the
analytical $K=-1.2J$ line where the static async state loses its stability.

###  4.1 Order parameters

Different steady states emerge between the swarmalators in their spatial and
phase dynamics. To investigate the phase coherence, a complex order parameter
is defined as
$Re^{l{\Phi}}=\frac{1}{N}\sum_{\begin{subarray}{c}j=1\end{subarray}}^{N}e^{l\theta_{j}}\;(l=\sqrt{-1})$,
where $\Phi$ is the average phase of the swarmalators. $R\in[0,1]$ measures
the phase coherence among all the units. In the static sync state, where all
the swarmalators’ phases are same, the value of $R$ is $1$. In all the other
four states, $R$ is strictly less than $1$ as the phases are not synchronized.
The correlation between the phases ($\theta_{i}$) and the spatial angles
($\phi_{i}=\tan^{-1}({y_{i}}/{x_{i}})$) is measured with the help of another
order parameter, which is defined as

$S_{\pm}e^{l\Psi_{\pm}}=\frac{1}{N}\sum_{\begin{subarray}{c}j=1\end{subarray}}^{N}e^{l(\phi_{j}\pm\theta_{j})}.$
(9)

By definition, the magnitude of $S_{\pm}$ varies within $0$ and $1$. These
values depend on the choices of initial conditions and for this we define
$S=\max(S_{+},S_{-})$. Now, a nonzero value of $S$ indicates the phases and
spatial angles are correlated. In the static phase wave state, the phases are
perfectly correlated with the spatial angles, $\theta_{i}=\pm\phi_{i}+C$
($\pm$ and $C$ depend on the initial conditions), and $S$ is exactly $1$ here.
The correlation decreases when the value of $K$ is reduced from $0$ to
negative and it finally reaches the minimum value $0$ in the static async
state where swarmalators’ phase are distributed uniformly between $0$ to
$2\pi$. It is seen that the static async loses its stability beyond a critical
phase coupling strength $K_{c}=-1.2J$ [18] which is highlighted in Fig. 1. In
both the active states (splintered phase wave and active phase wave), the
value of $S$ is nonzero, but strictly less than 1. In the splintered phase
wave state, swarmalators splinter into clusters and inside each cluster
swarmalators move and the phases oscillate around their mean. Swarmalators do
not travel from one cluster to another and form disjoint clusters. The number
of such clusters and number of swarmalators inside each cluster depend on the
initial conditions. Cluster formation vanishes and swarmalators execute
regular circular motion both in phase and spatial angle in the active phase
wave state. Conveniently, an order parameter $U$ is defined which measures the
fraction of swarmalators that execute at least one full circle both in space
and phase, after transient period. We define $U=N_{rot}/N$, where $N_{rot}$ is
the total number of swarmalators executing at least one full circle in space
and phase. In Fig. 1, $K$-$J$ parameter region is demonstrated with the region
of occurrence of the five emerging states. Snapshots of the swarmalators’
position at $t=1000$ time unit are also included where they are colored
according to their phases.

###  4.2 Effect of phase similarity both on spatial attraction and repulsion

Now, we consider $\text{F}_{att}(\theta)=1+J_{1}\cos(\theta)$ and
$\text{F}_{rep}(\theta)=1-J_{2}\cos(\theta)$ to investigate the behavior of
swarmalators when phase similarity affects both spatial attraction and
repulsion, respectively [37]. A different set of values for $\alpha$, $\beta$,
and $\gamma$ is chosen ($\alpha=0$, $\beta=2$, and $\gamma=2$). Note that,
since Eq. (4) does not hold for these choices of $\alpha,\beta$, and $\gamma$,
collision avoidance can not be assured (although numerical simulations show
that the particles do not collide as long as $J_{2}\leq 1$), but these choices
simplify the mathematical analysis of the emerging dynamical states provided
the model is well-posed. Depending on the system parameters $J_{1}$, $J_{2}$,
$K$, and $N$, swarmalators display different behavioral dynamics. For certain
parameter values, we find a stationary state where swarmalators arrange
themselves on a ring centered around the origin. The phases are perfectly
correlated with the spatial angles (i.e., $\theta_{i}=\phi_{i}+C$, for some
constant $C$ depending on initial conditions). Accordingly, this state is
named ring phase wave. The position and phase of the $k$th swarmalator in this
state can be expressed as

$\displaystyle\textbf{x}_{k}=R\cos(2\pi k/N)\hat{i}+R\sin(2\pi k/N)\hat{j},$
(10) $\displaystyle\theta_{k}=2\pi k/N+C,$ (11)

where $R$ is the radius of the ring, $\hat{i}$ and $\hat{j}$ are unit vectors
along $x$ and $y$ directions, and $C$ is a constant determined by the initial
conditions. Following the nature of the ring phase wave, it is convenient to
move to complex notation where the vector
$\textbf{x}_{k}\equiv(x_{k},y_{k})\in\mathbb{R}^{2}$ is identified as a point
$z_{k}=x_{k}+iy_{k}$ in the complex plane. The existence and stability
analysis involves complex identities which are easier to handle with our
aforementioned choice $\alpha=0,\beta=2,\gamma=2$. We use Eqs. (10) and (11)
and substitute them into the governing equations to find the radius of the
ring state, which is $R=\sqrt{(N-1+J_{2})/(N(2-J_{1}))}$. So, ring state will
exist only in the parameter region
$\\{J_{1}<2,\,J_{2}>1-N\\}\cup\\{J_{1}>2,\,J_{2}<1-N\\}$. In Fig. 2, we
display the organization pattern and radius of this state. The stability
analysis of this state is done elaborately in ref. [37]. When $K=0$, this
state loses its stability beyond a critical value of $N$,
$N_{max}=8/((2-J_{1})(1-J_{2}))$. Above this value, static phase wave state
emerges. For $K<0$, it bifurcates into the splintered phase wave state. Ring
phase wave can also be found for the previously chosen values of
$\alpha,\beta,\gamma$ ($\alpha=1,\beta=2,\gamma=1$), where the expression of
the radius $R$ takes slightly complicated form [40].

Figure 2: (a) Scatter plot of ring phase wave state. Here the phase of each
swarmalator is represented by a blue ray, and is given by the angle the ray
makes with the positive $x$ axis. Ring state is shown for $J_{1}=0$,
$J_{2}=1$, $K=-0.003$, and $N=100$. (b) Radius of the ring state is plotted
against $J_{1}$. Red dots are numerically calculated values of radius for
$J_{2}=1$ and $N=100$. Black curve is the theoretical one. We refer the reader
to ref. [37] for further details.

###  4.3 Swarmalators under external forcing

Think about a group of fireflies in a forest flashing simultaneously. If a
flashing LED is placed in the center of the group, the flashing rhythms of the
individuals get affected and synchronize with the LED flash. This kind of
external perturbation can introduce new phenomena to the swarmalator system.
To study the dynamics of swarmalators under external force, Lizarraga et al.
[38] introduced periodic forcing which directly affects the phases. This
external stimulus is placed at the center of the initial positions of the
swarmalators. The phase dynamics under the effect of periodic forcing is now
given by

$\dot{\theta}_{i}=\frac{K}{N}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\frac{\sin(\theta_{j}-\theta_{i})}{|\textbf{x}_{j}-\textbf{x}_{i}|}+F\frac{\cos(\Omega
t-\theta_{i})}{|\textbf{x}_{0}-\textbf{x}_{i}|},$ (12)

where $F$, $\Omega$, and $\textbf{x}_{0}$ represent the amplitude, frequency,
and spatial position of the stimulus, respectively. The spatial dynamics here
is same as Eq. (7).

When the amplitude of the force $F$ increases, swarmalators which are near the
center of the force start to synchronize with the external frequency $\Omega$.
All the five states of non-forced model (static sync, static async, static
phase wave, splintered phase wave, and active phase wave) go through phase
transition from partial to full synchronization with increasing $F$. Complete
synchronization of swarmalators’ phases takes place after a critical value of
$F$ which is independent of $\Omega$. In the static sync state, the
swarmalators near the position of the stimulus get synchronized with the
stimulus forming small cluster around it. As the amplitude of the force
increases, the cluster also expands until encompassing the whole system (see
first column of Fig. 3). Same phenomena is observed when external forcing is
introduced to the static async state only with the exception that
swarmalators’ phases are now desynchronized in the absence of external force.
The dynamics for static phase wave and splintered phase wave state are more
complex. When $F=0.5$, swarmalators near the source of the stimulus begin to
rotate around it clockwise, whereas those at the boundary show counter-
clockwise motion. As the intensity of the force increases to $F=1$,
swarmalators divide into two clusters which rotate slowly around the source
(see third and fourth columns of Fig. 3). When $F$ is increased further, the
clusters reassemble to synchronize their phases and build small cluster around
the source. The size of the cluster grows with increasing $F$ until it covers
the whole system. The size of the synchronized state is much smaller here
compared to static sync and static async states since the value of $J$ for
static phase wave and splintered phase wave are much higher. The transition of
the active phase wave state under the stimulus is similar to the splintered
phase wave state, but without splitting in groups. These phase transitions are
depicted in Fig. 3.

Figure 3: Transition of states for each of the five non-forced states as a
function of force amplitude $F$ for $\Omega=3\pi/2$. $J=0.1$, $K=1$ for static
synchrony, $J=0.1$, $K=-1$ for static asynchrony, $J=1$, $K=0$ for static
phase wave, $J=1$, $K=-0.1$ for splintered phase wave, and $J=1$, $K=-0.75$
for active phase wave state are chosen. For further details, see ref. [38].

##  5 Dynamics under local interaction

In the previous section, we have studied the swarmalators where the spatial
and phase interactions among them were global. But, in reality most of the
real-world multi-agent systems manifest local neighbor interaction among the
agents. This motivates us to study the swarmalators where the spatial or phase
interaction among the units are restricted to local neighbors. Taking a step
in this direction, Lee et al. [39] studied the steady state patterns of
swarmalators under finite cutoff interaction in the spatial dynamics. Now the
swarmalators spatially interact with each other only within a finite range
$r$. The spatial dynamics in this scenario is expressed as,

$\dot{\textbf{x}}_{i}=\frac{1}{N_{i}(r)}\sum_{j\in\Lambda_{i}(r)}\bigg{[}(\textbf{x}_{j}-\textbf{x}_{i})(1+J\cos(\theta_{j}-\theta_{i}))\\\
-\frac{\textbf{x}_{j}-\textbf{x}_{i}}{|\textbf{x}_{j}-\textbf{x}_{i}|^{2}}\bigg{]},$
(13)

where $\Lambda_{i}(r)$ is the set of indices of swarmalators lying within a
distance $r$ from the $i$th swarmalator (except itself) and $N_{i}(r)$ is the
cardinality of this set. The phase dynamics is kept same as Eq. (8). The
linear attraction kernel with $\alpha=0$ is considered here to simplify the
analysis, although the results remain invariant for $\alpha=1$. When an
infinite cutoff is considered, i.e., $r\rightarrow\infty$, all the steady
states (static sync, static async, and static phase wave) for global
interaction reappear. The system reveals fascinating results when a finite
value of $r$ is considered. For $r>D_{\infty}$, we observe multiple identical
copies of the steady states, where $D_{\infty}$ is the diameter of the
respective steady states in the $r\rightarrow\infty$ limit. The number of such
copies depends crucially on the choice of initial conditions. The reason of
such occurrences of multiple copies can be justified by the following reason:
Once a group of swarmalators assemble in a circular region of radius $r$ where
they are separated from others by a minimum distance $r$, they evolve as a
separate group independently. Each group follows the identical dynamics for
$r\rightarrow\infty$ limit and accordingly, the diameter of each of them is
$D_{\infty}$. Consequently, the minimal separation between the groups is
$D_{\infty}$. For $r<D_{\infty}$, anomalous version of static sync and static
async states are reported where the density of swarmalators are not uniform.
Within a certain interval of $r$ ($1<r<1.8$), bar-like patterns emerge from
the static phase wave state. If we consider local attractive phase coupling
among the swarmalators, the emergence of multiple non-identical static
clusters is observed [40].

##  6 Competitive phase interaction

So far we have discussed the results for swarmalators’ long-term states where
the phase coupling among the units was either positive (attractive) or
negative (repulsive). But the coupling characteristic among the units of many
systems is often more complex. It can be found in the neuronal networked
systems [41] and in the calling behavior of Japanese tree frogs [9]. This has
motivated the research community to explore the dynamics of swarmalators in
presence of mixed coupling in the system. In ref. [42], the phase coupling
strength $K_{ij}$ between the $i$th and the $j$th swarmalators is chosen
randomly from a two-peak distribution

$h(K_{ij})=p\delta(K_{ij}-K_{a})+(1-p)\delta(K_{ij}-K_{r}),$ (14)

where $K_{a}$ and $K_{r}$ are attractive and repulsive coupling strengths,
respectively and $p$ is the probability of attractive coupling. The
swarmalator dynamics in this case is governed by Eqs. (7) and (8), except the
fact that now the phase coupling strengths are chosen randomly and with the
symmetric property $K_{ij}=K_{ji}$. The ratio $Q=-{K_{r}}/{K_{a}}$ ($>0$) can
be chosen as a control parameter of the system. Taking annealed approximation
of the quenched coupling, it can be shown that the system undergoes phase
transition from desynchronized state to the sync state at a critical value
$p_{c}=Q/(1+Q)$. In the incoherent regime $0<p<p_{c}$, the existence of
nonstationary states like splintered phase wave and active phase wave for
suitable values of $J$ is found even with positive coupling among swarmalators
with $p>0$. To find the possibility of the nonstationary states for $p>0$, we
find the mean velocity $\bar{v}$ defined by
$\bar{v}=N^{-1}\sum_{i=1}^{N}\sqrt{\dot{x_{i}}^{2}+\dot{y_{i}}^{2}}$, where
$\text{x}_{i}=(x_{i},y_{i})$. The order parameters $S$, $R$, $U$, and
$\bar{v}$ are used to distinguish the states and observe the transition.
Introducing mixed randomness in the system, several combinations of deformed
patterns are encountered which are analogous to the “chimera state” observed
in oscillators with identical frequency [43, 44]. These deformed patterns are
understood with each annealed average of the mixed couplings.

###  6.1 Time-varying phase interaction

The studies on swarmalators discussed so far are mainly based on static
network formalism, i.e., the coupling scheme among the units does not change
over time. It would be interesting to see what happens when we consider time-
varying interactions [45] in the system. Sar et al. [40] introduced time-
varying competitive phase interactions in the swarmalator model and studied
their long-term states. Every oscillator moves in the $2$D plane with a
uniform circular interaction range, given by the vision radius $r$. A
particular swarmalator is attractively phase coupled with another swarmalator
only if it lies inside the interaction range of the former. Otherwise, phase
coupling between them is repulsive. Since the swarmalators move in space, the
coupling among them changes at every instant of time depending on their
spatial positions. This time-varying competitive phase dynamics is given by
the equation,

$\dot{\theta}_{i}=\frac{K_{a}}{N_{i}(r)}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\text{A}_{ij}\frac{\sin(\theta_{j}-\theta_{i})}{|\textbf{x}_{j}-\textbf{x}_{i}|}+\\\
\frac{K_{r}}{N-1-N_{i}(r)}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\text{B}_{ij}\frac{\sin(\theta_{j}-\theta_{i})}{|\textbf{x}_{j}-\textbf{x}_{i}|},$
(15)

where $K_{a}$ and $K_{r}$ are strengths for attractive and repulsive
couplings, respectively and A, B are the corresponding adjacency matrices.
$N_{i}(r)$ is same as the one used in Eq. (13). The spatial dynamics is same
as Eq. (7). In the extreme limits of the vision radius $r$, all the long-term
states of the model defined by Eqs. (7) and (8) are reproduced. By running
simulations it is found that, for appropriate values of system parameters
$K_{a}$, $K_{r}$, $J$, and $r$ the system settles into a stationary state
where swarmalators break into two clusters. Within each cluster the phases are
totally synchronized, but there is a phase difference of $\pi$ between the
clusters. This state is termed as the static $\pi$ state (see Fig. 4(a)).
Analytically it is found that the center of positions of the clusters are
always at a distance $d_{\pi}=1/(1-J)$ away from each other which is showed in
Fig. 4(b). For small values of $J$, swarmalators gather themselves in space
with the ones in similar phases but do not form clusters. Their phases are
neither fully synchronized nor fully desynchronized. The phase difference is
small for spatially nearby ones and it increases when the distance increases.
In this state, swarmalators show movement in space and this active state is
named as mixed phase wave. In Fig. 4(c) and (d) snapshots of this state for
different parameter values are shown.

Figure 4: (a) Snapshot of static $\pi$ state for
$(J,r,K_{a},K_{r})=(0.5,0.8,0.5,-0.5)$. (b) Distance between the center of
positions of the clusters in static $\pi$ state. Red dots are numerically
calculated values. Black curve is theoretical prediction given by
$d_{\pi}=1/(1-J)$. Snapshots of the mixed phase wave state for (c)
$(J,r,K_{a},K_{r})=(0.1,1.38,0.5,-0.5)$ and
(d)$(J,r,K_{a},K_{r})=(0.1,0.36,0.1,-1.0)$ at $t=1000$ time unit. Simulations
are done with $N=100$ swarmalators.

##  7 Dynamics of swarmalators on a ring

Although swarmalators show fascinating long-term behavior arising from the
bidirectional interplay between spatial and phase dynamics, most of the active
states lack analytical support. To address this issue, we can try to reduce
the spatial dimension of the system and treat them as positioned on a $1$D
ring. This decreases the complexity of the system and allows us to study the
states analytically. Now, the model becomes a pair of Kuramoto model which is
governed by the equations [46],

$\displaystyle\dot{x}_{i}=\frac{J}{N}\sum_{j}^{N}\sin(x_{j}-x_{i})\cos(\theta_{j}-\theta_{i}),$
(16)
$\displaystyle\dot{\theta}_{i}=\frac{K}{N}\sum_{j}^{N}\sin(\theta_{j}-\theta_{i})\cos(x_{j}-x_{i}),$
(17)

where $(x_{i},\theta_{i})\in(\mathbb{S}^{1},\mathbb{S}^{1})$ denotes the
position and phase of the $i$th swarmalator. The parameters $J$ and $K$
control the phase dependent spatial coupling and position dependent phase
coupling, respectively. This reduced model given by Eqs. (16) and (17)
exhibits five different dynamical behavior of the swarmalators on a ring. They
are named analogously to their $2$D model counterparts as static sync, static
$\pi$, static phase wave, static async, and active async following their
behavior on the ring. The stability of these observed states are analyzed
elaborately in ref. [46]. Similar studies on the ring have been carried out by
considering non-identical oscillators [47] and distributed couplings [48].

##  8 Conclusions

In this perspective, we have made a concise review to explore the dynamical
behavior of swarmalator systems. Theoretical analyses have revealed that under
certain assumptions such systems possess some conserved quantities and finite-
time collision avoidance can be assured. The long-term states of these systems
have been analyzed numerically and theoretically under global interaction,
addition of external forcing, local interaction, and competitiveness in the
phase coupling with both static and time-varying coupling topologies. These
studies have enfolded some captivating dynamics. More studies like oscillatory
behavior with repulsive short-range interaction [49], active phase wave with
attractive phase coupling [50], dynamics under stochastic coupling and memory
[51], and mean-field limit [52] have enriched this field which we mention here
and were unable to discuss in our main text.

There are plenty to explore about the swarmalators given their complex
behavior and interesting self-organizing patterns. Swarmalators with endowed
orientation could reveal new collective states where the swarming properties
include both aggregation and orientation. The role of the exponents
$\alpha,\beta$, and $\gamma$ in determining the asymptotic states is not fully
studied yet which demands more attention. A new but impactful direction of
research would be to consider non pairwise interaction [53] among the units
where higher order interactions takes over. The stability properties of active
states of the $2$D swarmalator model remain one of the aspects which demand
sincere attention. Looking at the difficulty of analyzing the $2$D model, one
can ask: Can we model a solvable system for the swarmalators? The $1$D model
discussed in the main text is a pivotal step towards this goal which sustains
the properties of the $2$D model. The interdependence between the spatial and
phase dynamics of the swarmalator systems leads to diverse collective states
which emerges as an active field of research.

###### Acknowledgements.

## References

  * [1] Bialek W. et al., Proc. Natl. Acad. Sci., 109 (2012) 4786.
  * [2] Hemelrijk C. K. et al., Interface focus, 2 (2012) 726.
  * [3] Sumpter D. J. T., Collective animal behavior (Princeton University Press) 2010.
  * [4] Fetecau R. C. et al., Nonlinearity, 24 (2011) 2681.
  * [5] Reynolds C. W., in Proceedings of the 14th annual conference on Computer graphics and interactive techniques, 1987, pp. 25-34.
  * [6] Winfree A. T., J. Theor. Biol., 16 (1967) 15.
  * [7] Kuramoto Y., in International symposium on mathematical problems in theoretical physics (Springer), 1975, pp. 420-422.
  * [8] Pikovsky A. et al., A universal concept in nonlinear sciences 2001.
  * [9] Aihara I. et al., Artificial Life and Robotics, 12 (2008) 29.
  * [10] Montbrió E. et al., Phys. Rev. X, 5 (2015) 021028.
  * [11] Gois S. R. et al., Chaos, Solitons & Fractals, 41 (2009) 2553.
  * [12] Motter A. E. et al., Nat. Phys., 9 (2013) 191.
  * [13] Vicsek T. et al., Phys. Rev. Lett., 75 (1995) 1226.
  * [14] Frasca M. et al., Phys. Rev. Lett., 100 (2008) 044102.
  * [15] Chowdhury S. N. et al., New J. Phys., 21 (2019) 073048.
  * [16] Majhi S. et al., Phys. Rev. E, 99 (2019) 012308.
  * [17] Majhi S. et al., Chaos, 27 (2017) 053115.
  * [18] O’Keeffe K. P. et al., Nat. Commun., 8 (2017) 1.
  * [19] Ramaswamy S., Annu. Rev. Condens. Matter Phys., 1 (2010) 323.
  * [20] Marchetti M.C. et al., Rev. Mod. Phys., 85 (2013) 1143.
  * [21] Needleman D. et al., Nat. Rev. Mater., 2 (2017) 1.
  * [22] Shaebani M.R. et al., Nat. Rev. Phys., 2 (2020) 181.
  * [23] Yang Y. et al., Phys. Rev. E, 78 (2008) 061903.
  * [24] Riedel I. H. et al., Science, 309 (2005) 300.
  * [25] Zhang B. et al., Nat. Commun., 11 (2020) 1.
  * [26] Bricard A. et al., Nat. Commun., 6 (2015) 1.
  * [27] Hrabec A. et al., Phys. Rev. Lett., 120 (2018) 227204.
  * [28] Haltz E. et al., Phys. Rev. B, 103 (2021) 014444.
  * [29] Barciś A. et al., in 2019 International Symposium on Multi-Robot and Multi-Agent Systems (MRS), 2019, pp. 98-104.
  * [30] Monaco J. D. et al., Biological cybernetics, 114 (2020) 269.
  * [31] Yuan J. et al., Proc. Natl. Acad. Sci., 111 (2014) 6865.
  * [32] Manna R. K. et al., Proc. Natl. Acad. Sci., 118 (2021) e2022987118.
  * [33] Peshkov A. et al., Soft Matter, 18 (2022) 1174.
  * [34] Tanaka D., Phys. Rev. Lett., 99 (2007) 134103.
  * [35] Iwasa M. et al., Phys. Rev. E, 83 (2011) 036210.
  * [36] Ha S. Y. et al., Math. Models Methods Appl. Sci., 29 (2019) 2225.
  * [37] O’Keeffe K. P. et al., Phys. Rev. E, 98 (2018) 022203.
  * [38] Lizarraga J. U. F. et al., Chaos, 30 (2020) 053112.
  * [39] Lee H. K. et al., Chaos. 31 (2021) 033134.
  * [40] Sar G. K. et al., New J. Phys., 24 (2022) 043004.
  * [41] Hopfield J. J., Proc. Natl. Acad. Sci., 79 (1982) 2554.
  * [42] Hong H. et al., Phys. Rev. E, 104 (2021) 044214.
  * [43] Kuramoto Y. et al., arXiv preprint cond-mat/0210694, (2002).
  * [44] Abrams D. M. et al., Phys. Rev. Lett., 93 (2004) 174102.
  * [45] Ghosh D. et al., Physics Reports, 949 (2022) 1.
  * [46] O’Keeffe K. P. et al., Phys. Rev. E, 105 (2022) 014211.
  * [47] Yoon S. et al., arXiv preprint arXiv:2203.10191, (2022).
  * [48] O’Keeffe K. P. et al., Phys. Rev. E, 105 (2022) 064208.
  * [49] Jiménez-Morales F., Phys. Rev. E, 101 (2020) 062202.
  * [50] Hong H., Chaos, 28 (2018) 103112.
  * [51] Schilcher U. et al., in 2021 IEEE International Conference on Autonomic Computing and Self-Organizing Systems (ACSOS), 2021, pp. 90-99.
  * [52] Ha S. Y., Kinetic & Related Models, 14 (2021) 429.
  * [53] Majhi S. et al., J. R. Soc. Interface, 19 (2022) 20220043.


# Real-time atmospheric monitoring for the Cherenkov Telescope Array using a
wide-field optical telescope

, Petr Janeček, Michael Prouza and Jiří Blažek for the CTA Consortium
Institute of Physics of the Czech Academy of Sciences, Prague
E-mail Full consortium author list at http://cta-observatory.org

###### Abstract:

The Cherenkov Telescope Array (CTA) is the next generation of ground-based
very high energy gamma-ray instruments and is planned to be built on two sites
(one in each hemisphere) in the coming years, with full array operation
foreseen to begin 2020. The goal of performing high precision gamma-ray energy
measurements while maximizing the use of observation time demands detailed and
fast information about atmospheric conditions. Besides LIDARs designed to
monitor clouds and aerosol content of the atmosphere in the pointing direction
of the CTA telescopes, we propose to use the ”FRAM” (F(/Ph)otometric Robotic
Atmospheric Monitor) device, which is a small robotic astronomical telescope
with a large field of view and a sensitive CCD camera that together ensure
precise atmospheric characterization over the complete field-of-view of the
CTA. FRAM will use stellar photometry to measure atmospheric extinction across
the field of view of the CTA without interfering with the observation (unlike
laser-based methods). This allows FRAM to operate with high temporal
resolution and provide both real-time data for on-the-fly scheduling decisions
and an offline database for calibration and selection of scientific data. The
fast robotic mount of the telescope supports quick observation of multiple
fields when the array is split and even a check of the conditions in the
directions of the upcoming observations is possible. The FRAM concept is built
upon experience gained with a similar device operated at the Pierre Auger
Observatory. A working prototype of FRAM proposed for CTA is being built in
Prague for extensive testing before deployment on site; first results and
experiences with this prototype are presented.

##  1 Introduction

The Cherenkov Telescope Array (CTA) is expected to enter full operation in
2020 as the next generation of ground-based very high energy gamma-ray
instruments [1]. Its unprecedented scientific potential comes not only from
the sheer size, with tens of telescopes of different sizes across two sites
(one in each hemisphere), but also from the strict requirements established on
the precision of measurements and efficiency of use of the observation time.
To meet these requirements, careful calibration of the instrument is key,
including precise knowledge of momentary atmospheric conditions, in particular
the presence of clouds and the aerosol content of the atmosphere. From the
devices proposed to be installed at the CTA sites [2], the LIDARs can measure
clouds and aerosols with high accuracy in a given direction [3] and the All-
Sky Cameras can monitor the whole sky at all times without interfering with
the CTA observations [4] – however neither of those devices can provide
detailed information about the conditions in the current CTA field of view
with high spatial and temporal resolution. Thus we propose a complementary
device – a small robotic optical telescope, the ”FRAM” (F(/Ph)otometric
Robotic Atmospheric Monitor), which will measure the transparency of the
atmosphere in the direction of the CTA field of view using stellar photometry.
In isolation, this method is not without shortcomings, because only the
integral extinction from ground to the upper edge of the atmosphere can be
provided (without a detailed altitude profile), but if operated together with
profiling atmospheric devices (like to LIDARs), it will help provide a
complete picture of the atmospheric conditions across the CTA field of view,
in real time during observations.

The CTA FRAM concept draws upon experience gained over a decade with a
pioneering device [5] that has been operated at the Pierre Auger Observatory
[6] – more details on the current implementation of the Auger FRAM can be
found in [7] and an explanation of its use for rapid atmospheric monitoring at
Auger in [8]. The differences between the application of stellar photometry at
Auger and CTA as well as the changes in the hardware setup that follow from
them are explained in detail in [9]. In this paper, we briefly review the
current status of the design of the FRAM for CTA and then we discuss the
performance of the partial prototype, which has been assembled at the
Institute of Physics in Prague and has received first light at the beginning
of June 2015, and advances in data analysis.

##  2 Hardware and software

We currently foresee having three identical FRAM devices at the CTA sites (one
in the northern site, two in the larger southern site to cover for spatial
variations in atmospheric conditions over the extended area). Based on the
extensive testing carried out over years at the Auger FRAM, we have chosen
components that have proven reliable and efficient. Apart from the housing and
roof, which are designed and built according to our specifications by local
specialist companies, and some custom structural elements, all the other
components are readily available commercial products – a fact that has allowed
us to have a working partial prototype (Fig. 1) within a couple of months from
receiving funding for it. The main components of the prototype (and thus of
the proposed future CTA FRAMs) are:

Figure 1: The partial prototype of the FRAM for CTA (without housing, pillar
and dedicated computer) receives its first light in the backyard of the
Institute of Physics in Prague.

  * •

Paramount MYT robotic astronomical mount. The combination of camera and lens
used is lightweight and could be easily used with a less robust equatorial
mount; however the Paramount mounts have the clear advantage of time-proven
reliability and ability to autonomously regain orientation after a power cut
or loss of communication thanks to their absolute homing system.

  * •

Moravian Instruments G4-16000 large-format CCD camera equipped with standard
BVRI photometric filters. The KAF-16803 sensor used in this camera spans
36$\times$36-mm, and thus allows the FRAM to reach large field of view while
keeping a relatively big physical aperture of the lens. Collecting enough
starlight in a short time window is crucial for the envisaged high temporal
resolution of the measurements.

  * •

Zeiss 135 f/2.0 photographic lens. This particular combination of sensor size
and lens focal length provides a field of view of $15\times 15$ degrees and
thus covers the whole expected CTA field of view (expected to be up to 10
degrees) with a sufficient margin for the parallax with respect to the more
distant telescopes. The Zeiss lens is reportedly unrivaled in sharpness at
full aperture at this focal length, particularly considering the large sensor
area to cover – the first light results have even exceeded our expectations in
this respect.

  * •

An external focuser which mechanically couples to the focus ring of the lens
and thus allows remote re-focusing; this component has not yet been delivered,
but non-remote operation of the prototype is easily done using manual focus.

Figure 2: 3D model of the house and roof

Eventually, each FRAM will be located in a small house with a roof that fully
unfolds in the night to allow unobstructed observation of the full sky. The
design of the housing (by Elya Solutions) is now finished (Fig. 2) and the
production of the first house and roof has started. We have chosen a footprint
that is rather large with respect to the FRAM device ($2.5\times 2.5$ meters)
to allow reasonable maintenance access to FRAM with the dome closed for
environmental protection. With such a large footprint already set, the design
of the roof was chosen to be flexible so that it can be used for other robotic
telescopes – the housing is suitable for any telescope that fits into a sphere
with a radius of 80 cm during operation (for example a Schmidt-Cassegrain with
30 cm diameter as in the Auger FRAM).

Both the walls and the moveable roof will be made of aluminium composite
panels (a sandwich structure of plastic between two aluminium sheets)
reinforced with aluminium profiles; all parts can be easily bolted together
and to the concrete foundations on site. The roof will be opened and closed
using a hydraulic system with electric valves controlled by the Schneider
Zelio controller. In case of a power outage, a pressure storage reservoir
(hydraulic accumulator) will provide the required pressure to close the dome,
while the valves will run from a battery. For additional safety, a manual
hydraulic pump and mechanical valve controls will be also provided. When
closed, the roof will be weather-proof and its operation is possible with
winds up to 60 km/h; it can also be opened even after heavy snowfall without
danger of snow falling on the telescope.

While the computationally demanding image processing would need to be handled
by a server in the CTA data centre, a small local computer is necessary to
control the mount, camera, focuser, roof and auxiliary devices. Intel
DE3816TYKHE, a low-power, passively cooled industrial computer was chosen
because of its relatively strong-for-its-size I/O capabilities (6 USB ports,
gigabit Ethernet), necessary for controlling all the connected devices and for
quick transfer of the large images from CCD camera (32 MB) to the data centre.
It will run Linux and the RTS2 software [10] for robotic observatories. The
communication of FRAM with the central CTA array control system would be
implemented (similarly to many other CTA devices) using the OPC UA interface.

##  3 First light and image processing

Figure 3: All identified stars from an image taken within the milky way in
Cygnus. The color code shows the difference between measured and catalog
brightness for each star; the value is plotted before zero-point calibration
and thus has an arbitrary offset. Note the vignetting clearly visible in the
corners and the residual artifacts from the 6$\times$6 matrix of sub-images
used for star identification (stars cannot be identified very close to the
border of a sub-image).

During a very clear night at the beginning of June 2015 we have taken first
images of the sky with the FRAM prototype. Here we use them to demonstrate the
methods of image processing. All images shown are taken in the V filter with a
15-second exposure.

Because image distortion over the large field of view makes star
identification difficult, each image taken is first split into a matrix of
6$\times$6 sub-images. Across a single sub-image, the distortion is negligible
and thus the relation between the position of a star on the image in pixels
and its celestial coordinates can be taken as linear. For each point source
detected in the image, the apparent brightness is measured and the Tycho
catalog [11] is searched for a corresponding star (with preference to brighter
stars when more catalog entries are very close to each other). For a random
field within the milky way in Cygnus (Cyg), about 20 thousand stars were
identified; when observing perpendicularly to the galactic plane in Ursa Major
(UMa), we detected roughly 4000 stars.

The lens produces useful stellar images at full aperture even in the corners
of the CCD chip (which is slightly larger than 35 mm film), however there is
significant vignetting (up to 2 magnitudes) in the corners (Fig. 3). While the
vignetting can be removed using flat-fielding, it greatly reduces precision of
the measurement in the affected areas; however these are only a very small
part of the field of view.

Figure 4: Identified stars brighter than 9.5 mag from a flat-field corrected
image. The visible gradient is due to the zenith-angle dependence of the
extinction (the image spans altitudes between roughly 15 to 30 degrees above
the horizon).

Fig. 4 shows the same image after applying the flat-field correction and
selecting only stars brighter than 9.5 mag, for which both the catalog and
measured brightness are reasonably precise (1825 such stars in the Cyg field
and 638 stars in the UMa field). The spread of the values of the difference
between catalog and measured brightness is visibly smaller (note the different
scale), but a gradient remains, due to the dependence of the atmospheric
extinction on the traversed airmass $X$ (and thus the zenith angle), which is
for such a large field already clear from one image. If we express catalog vs.
observed brightness as

$m_{obs}=m_{cat}+a+kX$

and we fit the zero-point $a$ and the extinction coefficient $k$ within the
image, we obtain a reasonable value of $k=0.075$ for the night of observation.

Figure 5: Identified stars brighter than 9.5 mag from an image corrected for
flat-field and the dependence of atmospheric extinction on airmass. The
extinction fit also removes the arbitrary offset due to the zero-point of the
camera.

Using this fit, we can correct for atmospheric extinction (and apply the
obtained zero-point to convert the magnitude scale), as shown in Fig. 5. Here
we see a residual gradient (likely due to inadequate flat-field caused by
parasitic light from nearby artificial sources) and some outliers (likely due
to mis-identified stars, catalog errors, and stars with large color indices),
but for most of the stars, the difference between catalog and observed
brightness lies within 0.1 mag from the mean value; we expect further
improvements after we perform detailed corrections for star colors and the
spectral response of the system, as well as more careful flat-fielding of the
camera.

Ironically, the very clear weather during the time available for testing
prevented us from detecting actual clouds with the prototype. As a substitute,
we have created an ”artificial thin cloud” by hanging a dark cloth across the
field of view for 5 seconds during a 15-second exposure. This method simulated
a cloud or other source of absorption which does not remove the stars
completely and thus cannot be detected by simply counting the number of
identified sources vs. the expected number from the catalog. Fig. 6 shows such
an ”artificial cloud” as a decrease of observed brightness of a large number
of stars.

Figure 6: An example of an ”artificial thin cloud” detected using stellar
photometry.

##  4 Conclusions

We have introduced the FRAM device proposed for atmospheric monitoring at the
future CTA sites using stellar photometry. We have shown using the partial
prototype of the FRAM that the technique can deliver detailed information on
atmospheric extinction across a large field of view without interfering with
the operation of other optical detection devices. This information can be used
by the CTA to evaluate the atmospheric conditions, make operative decisions
based on the current situation and to assess the quality of data and select
those that are not significantly affected by atmospheric irregularities.

## Acknowledgements

We gratefully acknowledge support from the agencies and organizations listed
under Funding Agencies at this website: http://www.cta-observatory.org/. The
authors are grateful for the support by the grants of the Ministry of
Education of the Czech Republic MSMT-CR LE13012 and LG14019.

## References

  * [1] Acharya, B.S. et al. (CTA Consortium), Astroparticle Physics 43, 3–18 (2013)
  * [2] Gaug, M. et al. (CTA Consortium), ”Calibration of the Cherenkov Telescope Array”, these proceedings (2015)
  * [3] M. Doro et al., Proceedings of the AtmoHEAD Conference, Saclay (France), arXiv:1402.0638 (2013)
  * [4] C. Fruck et al., JINST 10 P04012 (2015)
  * [5] Prouza, M. et al., Advances in Astronomy 2010, 849382 (2010)
  * [6] Abraham, J. et al. (The Pierre Auger Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 523, 50 (2006)
  * [7] Ebr, J. et al., Rev. Mex. AA (SC) 45, 114 (2014)
  * [8] Ebr, J. et al. (The Pierre Auger Collaboration), Rev. Mex. AA (SC) 45, 53 (2014)
  * [9] Janeček, P., Ebr, J., Prouza M., EPJ WoC 89, 03006 (2015)
  * [10] Kubánek, P., Jelínek, M., Nekola, M. et al., Gamma-Ray Bursts: 30 Years of Discovery, AIP Conference Proceedings 727, 753–756 (2004)
  * [11] Høg et al., Astronomy and Astrophysics 355, L27 (2000)