# A FEW REMARKS ON NON-BAIRE SETS IN CATEGORY BASES

Sanjib Basu Department of Mathematics,Bethune College,181 Bidhan Sarani
<EMAIL_ADDRESS>,  Atasi Deb Ray Department of Pure Mathematics,
University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West
Bengal, India<EMAIL_ADDRESS>and  Abhit Chandra Pramanik Department of
Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road,
Kolkata 700019, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first establish some equivalent formulations of non-Baire
sets in category bases. We then introduce the notion of an uniform non-Baire
family of sets and show that there is an uniform non-Baire family inducing a
decomposition of the whole space. This phenomenon is then interpreted in the
context of the famous Banah-Mazur game.

###### Key words and phrases:

Category base;non-Baire sets;($\star$)-property;point-meager base;Baire
base;completely non-Baire;test region;uniform non-Baire family;Banach-Mazur
game

###### 2020 Mathematics Subject Classification:

03E20,28A05,54E52,91A44

The third author thanks the CSIR, New Delhi – 110001, India, for financial
support

## PRELIMINARIES AND RESULTS

The idea of category base is a generalization of both measure and topology and
its main objective is to present measure and Baire category(topology) and also
some other aspects of point set classification within a common framework. It
was introduced by J.C.Morgan II in the mid seventies of the last century and
has developed since then through a series of papers $[1],[2],[3],[4],[5],$
etc. To start with, we recall some basic definitions and theorems which may be
found in the above references and also in the monograph [6].

######  Definition 1.

A category base is a pair (X,$\mathcal{C}$) where X is a non-empty set and
$\mathcal{C}$ is a family of subsets of X, called regions satisfying the
following set of axioms:

  1. (1)

Every point of X belongs to some region; i,e., X=$\cup$$\mathcal{C}$.

  2. (2)

Let A be a region and $\mathcal{D}$ be a non-empty family of disjont regions
having cardinality less than the cardinality of $\mathcal{C}$.
i) If A$\cap$($\cup$$\mathcal{D}$) contains a region, then there is a region
D$\in$$\mathcal{D}$ such that A$\cap$D contains a region.
ii) If A$\cap$($\cup$$\mathcal{D}$) contains no region, then there is a region
B$\subseteq$A that is disjoint from every region in $\mathcal{D}$.

Several examples of category bases can be found in [6].

######  Definition 2.

In a category base (X,$\mathcal{C}$), a set is called singular if every region
contains a subregion which is disjoint from the set. Any set which can be
expressed as countable union of singular sets is called meager. Otherwise, it
is called abundant.

If countable sets are meager, then the category base is point-meager base and
a category base whose regions are abundant sets is a Baire base.

######  Definition 3.

In a category base (X,$\mathcal{C}$), a set S is called Baire if in every
region, there is a subregion in which either S or its complement X$-$S is
meager.

######  Theorem 4.

The intersection of two regions either contains a region or is a singular
sets.

######  Theorem 5.

(The Fundamental Theorem) Every abundant set in a category base is abundant
everywhere in some region. This means that for any abundant set A, there
exists a region C in every subregion D of which A is abundant.

In [6], one can find several examples corresponding to Definition 2 and
Definition 3.

From Definition 3, it follows that a set is non-Baire if there is a region
(which is not necessarily unique) in every subregion of which both the set and
its complement are abundant. In this paper, we identify this special region as
a test region corresponding to the non-Baire set.

Below we establish two alternative characterizations for non-Baire sets in a
category base. We define

######  Definition 6.

A set A in a category base (X,$\mathcal{C}$) as having the ($\star$)-property
if there exists a region C such that for every subregion D of C and every set
E which essentially contains D , A$\cap$E is non-Baire.
(A set E contains D essentially means that D$-$E is meager [6])

######  Theorem 7.

In any category base, a set is non-Baire if and only if it satisfies the
($\star$)-property.

###### Proof.

Suppose A is a non-Baire set in a category base (X,$\mathcal{C}$) and C be a
test region for A. Let D be any subregion of C. It suffices to show that D can
be considered as a test region for A$\cap$P where P is a subset of X which
essentially contains D. To prove our contention, we choose any subregion E of
D and note that in E both A$\cap$P and its complement are abundant. This is so
because in E both E$\cap$A$\cap$P which can be expressed as
E$\cap$A$-$A$\cap$(E$-$P) and its complement which is
(E$-$A)$\cup$(A$\cap$(E$-$P)) are abundant and E being a subregion of D is
also a subregion of C.

Conversely, if A is a Baire set then every region C contains a subregion D in
which either A or its complement is meager. If D$\cap$A is meage, choose D$-$A
as our set E which essentially contains D althouth A$\cap$E=$\phi$ is a Baire
set. If D$-$A is meager, choose D$\cap$A as our set E which essentially
contains D but A$\cap$E=D$-$A is meager and hence Baire.

This proves the theorem. ∎

######  Definition 8.

In a category base (X,$\mathcal{C}$), a set A is called completely non-Baire
in a region D if for every Baire set B such that B$\cap$D is abundant, both
A$\cap$B and (D$-$A)$\cap$B are abundant.

The above definition is somewhat analogous to the notion of a completely
I-nonmeasurable set, given in [8].

######  Theorem 9.

In any Baire base, a set is non-Baire if and only if it is completely non-
Baire in some region.

###### Proof.

Let A be a non-Baire set in a Baire base (X,$\mathcal{C}$). Then according to
the definition, there is a region D in $\mathcal{C}$ in every subregion of
which both A and its complement (X$-$A) are abundant. Let B be any Baire set
such that B$\cap$D is abundant. Since by Theorem 4, every region is here a
Baire set, we may assume that B$\subseteq$D. From the Fundamental theorem,
there is a subregion C of D which is essentially contained in B. Consequently,
both B$\cap$A and B$\cap$(D$-$A) are abundant.

Conversely, suppose A is completely non-Baire in a region D. Since by
hypothesis every region is an abundant Baire set, both A and its complement
are abundant in every subregion of D. Hence A is non-Baire. ∎

Next we introduce

######  Definition 10.

A family $\mathcal{F}$ of non-Baire sets as an uniform non-Baire family in a
category base (X,$\mathcal{C}$) if there is a region in $\mathcal{C}$ which
can act as a common test region for all members of $\mathcal{F}$.

With reference to this definition, we prove the following decomposition
theorem.

######  Theorem 11.

If (X,$\mathcal{C}$) is a point-meager, Baire base where $\mathcal{C}$
satisfies CCC (countable chain condition) and every region has cardinality
$\aleph_{1}$, there exists an uniform non-Baire family which induces a
decomposition of X.

###### Proof.

Let S be a non-Baire set in (X,$\mathcal{C}$) and T=X$-$S. The existence of S
is ensured by Theorem 6, Ch 2,Sec II, [6]. Then there is a region D in every
subregion of which both S and T are abundant. Now by Grzegorek’s unification
of Sierpinski’s theorems [1], (see also Theorem 16, Ch 2, Sec II, [6]) both S
and T can be decomposed into families {$S_{\alpha}\\}_{\alpha<\Omega}$ and
{$T_{\alpha}\\}_{\alpha<\Omega}$ ($\Omega$ is the first ordinal of cardinality
$\aleph_{1}$) of mutually disjoint sets such that each $S_{\alpha}$ (resp.
$T_{\alpha}$) is abundant in every region in which S (resp. T) is abundant.
Since $X-S\subseteq X-S_{\alpha}$ (resp. X$-T\subseteq X-T_{\alpha}$), so
every Sα and its complement (resp. Tα and its complement ) are abundant in
every subregion of D proving that Sα (resp. Tα) are both non-Baire sets for
all $\alpha<\Omega$ and D can act as a common test region for the entire
family {$S_{\alpha}$, $T_{\alpha}$ : $\alpha<\Omega$}. Moreover, this family
constitutes a partition of X. Thus we find the existence of an uniform non-
Baire family inducing a decomposition of X. ∎

A generalization of the classical Banach-Mazur game [7] was presented in [3]
by Morgan through the introduction of the notion of $\mathcal{M}$-family of
sets, which was defined in the following manner :

######  Definition 12.

A family $\mathcal{C}$ of subsets of a non-empty set X is called an
$\mathcal{M}$-family if it satisfies the following set of axioms :

  1. (1)

The intersection of any descending sequence of $\mathcal{C}$-sets is non-
empty.

  2. (2)

Suppose x is a point in X, then
i) there is a $\mathcal{C}$-set containing x, i.e., X=$\cup$$\mathcal{C}$.
ii) for each $\mathcal{C}$-set A, there is a $\mathcal{C}$-set B$\subseteq$A
such that x$\notin$B.

  3. (3)

Let A be a $\mathcal{C}$-set and D be a nonempty family of disjoint
$\mathcal{C}$-sets having cardinality less than the cardinality of
$\mathcal{C}$.
i) If A$\cap$($\cup$$\mathcal{D}$) contains a $\mathcal{C}$-set , then there
is a $\mathcal{D}$-set D such that A$\cap$D contains a $\mathcal{C}$-set.
ii) If A$\cap$($\cup$$\mathcal{D}$) contains no $\mathcal{C}$-set, then there
is a $\mathcal{C}$-set B$\subseteq$A which is disjoint from every set in
$\mathcal{D}$.

Evidently, any $\mathcal{M}$-family is a special case of a category base, so
the definitions of meager, abundant and Baire sets are as usual. Moreover, any
$\mathcal{M}$-family is point-meager and every member of it is an abundant set
which follows directly from the axioms (1) and (2) in the above Definition.

If $\mathcal{C}$ is a non-empty family of subsets of a non-empty set X and S
is a subset of X, then as described in [3], the generalized Banach-Mazur game
$\Gamma$(S, $\mathcal{C}$) is played as follows : Two players I and II,
alternatively choose sets from $\mathcal{C}$ to define a descending sequence
of sets, player I selecting the sets in the sequence with odd indices and
player II selecting the sets with even indices. If the intersection of the
constructed sequence has atleast one point (which obviously exists if
$\mathcal{C}$ is an $\mathcal{M}$-family) in S, then player I wins; otherwise
player II wins. In the game $\Gamma$(S, $\mathcal{C}$), let us denote player
I, player II by using symbols $\langle$S$\rangle$, $\langle$X$-$S$\rangle$
respectively.

Now suppose C$\in$$\mathcal{C}$ such that S$\cap$C $\neq$ $\phi$ $\neq$
(X$-$S)$\cap$C. Then we can describe the game $\Gamma$(S,$\mathcal{C}_{|C}$)
in a manner similar as above where $\mathcal{C}_{|C}$={E$\in$$\mathcal{C}$ :
E$\subseteq$C}. As a consequence of the Theorem 2 [3], Theorem 11 may be
interpreted in terms of the generalized Banach-Mazur game in the following
manner :

######  Theorem 13.

Let $\mathcal{C}$ be an $\mathcal{M}$-family satisfying CCC (countable chain
condition) and every $\mathcal{C}$-set has cardinality $\aleph_{1}$. Then
there exist a $\mathcal{C}$-set D and a partition
X=$\bigcup\limits_{\alpha<\Omega}{X_{\alpha}}$ such that under the condition
that there exists a sequence
$\\{h_{n}\\}_{n=1}^{\infty}(h_{n}:\mathcal{C}\mapsto\mathcal{C})$ satisfying

  1. (1)

for every $\mathcal{C}$-set A, hn(A)$\subseteq$A ;

  2. (2)

for every sequence $\\{A_{n}\\}_{n=1}^{\infty}$ of $\mathcal{C}$-sets, if
$\\{h_{n}(A)\\}_{n=1}^{\infty}$ is descending, then
$\bigcap\limits_{n=1}^{\infty}h_{n}$(A) contains only one point;

no player $\langle X_{\alpha}\rangle$ ($\alpha<\Omega$) can have a winning
strategy in the game $\Gamma(X_{\alpha},\mathcal{C}_{|D}$).

## References

  * [1] E.Grzegorek,_On a paper of Karel Prikry concerning Ulam’s problem on families of measures_ ,Collaq. Math. 42,1979,197-208.
  * [2] J.C.MorganII, _Baire category from an abstract viewpoint_ , Fund. Math. 94(1976).
  * [3] ……..,_Infinite games and singular sets_ , Collaq. Math. 29,1974,7-17.
  * [4] ………,_On zero-one law_ ,Proc. Amer. Math. Soc,Vol 62, No.2,Feb1977,353-358.
  * [5] ………,On product bases, Pacific Journal of Mathematics, Vol. 99, No.1 (1982), 105-126.
  * [6] ………,_Point set theory_ , Marcel Dekker,Inc,1990.
  * [7] J.C.Oxtoby, _Measure and Category_ , Springer-Verlag, 1980.
  * [8] R.Ralowski & S.Zeberski, _Completely nonmeasurable unions_ , Cent. Eur. J. Math, 8(4), 2010, 683-687.
  * [9]


# Amplitude responses of swarmalators

Samali Ghosh  These authors contributed equally to this work  Physics and
Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road,
Kolkata 700108, India  Suvam Pal  These authors contributed equally to this
work  Physics and Applied Mathematics Unit, Indian Statistical Institute, 203
B. T. Road, Kolkata 700108, India  Gourab Kumar Sar  These authors contributed
equally to this work  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B. T. Road, Kolkata 700108, India  Dibakar Ghosh  Corresponding
to<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 B. T. Road, Kolkata 700108, India

###### Abstract

(Received XX MONTH XX; accepted XX MONTH XX; published XX MONTH XX)

Swarmalators are entities that swarm through space and sync in time and are
potentially considered to replicate the complex dynamics of many real-world
systems. So far, the internal dynamics of swarmalators have been taken as a
phase oscillator inspired by the Kuramoto model. Here, for the first time, we
examine the internal dynamics utilizing an amplitude oscillator capable of
exhibiting periodic and chaotic behaviors. To incorporate the dual interplay
between spatial and internal dynamics, we propose a general model that keeps
the properties of swarmalators intact. This adaptation calls for a detailed
study which we present in this paper. We establish our study with the Rössler
oscillator by taking parameters from both the chaotic and periodic regions.
While the periodic oscillator mimics most of the patterns in the previous
phase oscillator model, the chaotic oscillator brings some new fascinating
states.

DOI: XXXXXX

##  I introduction

Synchronization [1, 2, 3] is one of nature’s most fascinating and widespread
phenomena. It refers to the coordination or alignment of events or processes
in time, without necessarily influencing their spatial positions. This
phenomenon is observed in a wide range of scales, from the microscopic to the
macroscopic, and across various disciplines of science and nature. From
cardiac pacemaker cells [4], the metabolic cycle of yeast cells [5],
coherently flashing fireflies [6, 7] to power grid dynamics [8], Josephson
junction [9] and even the unexpected wobbling of London’s Millennium Bridge
[10], there are so many examples that highlight the occurrence of spontaneous
synchrony. Similarly, a complementary form of self-organization occurs in
swarming [11, 12, 13, 14] where individuals traverse through space, yet
overtly modify their internal states. As evidence, birds fly in flocks [15,
16], fish swim in schools [17], and bacteria aggregates [18, 19] in space, yet
do not always synchronize the timing of an internal state or rhythm. In a
sense, both phenomena are spatio-temporal but opposites. Research on swarming
such as the collective behavior of many body systems or self-propelled
particles [20], has garnered significant attention in recent decades. Studies
of synchronization and swarming jointly establish a fruitful connection
between physics and biology, delving into the influence of both spatial and
temporal dependencies of the agents.

In recent decades, these two domains have been explored autonomously.
Ultimately, the analysis of mobile oscillators bridged these two domains by
exploring the impact of oscillator movement on their internal state [21, 22,
23]. However, the reverse scenario was not investigated. Mobile oscillators
become the key element in modeling biological and robotic phenomena [24, 21,
25, 26]. Von Brecht and Uminsky [27] proposed an aggregation model where the
particles undergo an internal polarization vector comparable to the
oscillator’s phase. But in all these works, the relative distances between the
particles affected their internal phase. Nevertheless, their relative phases
did not significantly contribute to their actions. The theoretical exploration
of the systems that feel the combined effects of swarming and synchronization
started to garner interest about $17$ years ago. Tanaka [28], led the
initiative by proposing a model of chemotactic oscillators where oscillators
interact through a background diffusive chemical and produce diverse rich
phenomena. Following this, recently O’Keeffe et al. [29] introduced a toy
model of swarming and synchronization, without reference to any background
medium, called swarmalators where the bidirectional interaction was
considered, i.e., twin activities of the swarming oscillators can be tracked.
They formulated this model by using space-dependent generalized globally
coupled Kuramoto oscillators [30] that anticipate several rich spatiotemporal
patterns.

Since then there has been ongoing research to investigate further and
understand the dynamics of swarmalators in different system-interaction
configurations. The two-dimensional (2D) model was reduced to a one-
dimensional (1D) solvable ring model [31] which captures the behavior of swarm
in quasi-1D rings such as sperms or vinegar eels [32]. Using the Ott-Antonsen
(OA) ansatz [33], the first analytical description of these states and the
condition of their existence have become possible [34]. To understand the
influence of the surrounding environment, Hong et al. [35] introduced thermal
noise in a population of swarmalators and came across several distinct
collective patterns, some of which captivate the behavior of real-world
swarmalators. In another work, the impact of the external damping force on the
phase resulted in the phase transition from a nonforced model to full
synchrony via partial synchronization [36]. Several other aspects such as
time-delayed interactions [37], distributed coupling [38], finite cut-off
interaction distance [39], multiplexity [40] and time-varying competitive
phase interactions [41] among the swarmalators give rise to a plethora of
collective patterns. Sar et al. took the initiative to introduce random
pinning [42, 43, 44] subjected to the 1D ring model which delivers low-
dimensional chaos and abrupt transition to a synchronous state along with
phase wave and split phase wave states. Recently, swarmalators have been
investigated under different community structures, and in the purely repulsive
coupling, antiphase synchronization between the communities has been observed
[45]. Check out the review articles on swarmalators in [46, 47].

In most of the works done so far, the internal dynamics of swarmalators have
been explored in terms of phase oscillators through the lens of the Kuramoto
model. We envision swarmalators as a common playground of swarming and
synchronization for which the oscillatory dynamics need not be controlled by
the phase, and can also be rendered by the amplitude. In this work, we use one
such amplitude oscillator model that governs the internal dynamics of the
system. We choose parameters for the model such that the oscillations are
chaotic. Synchronization of chaotic oscillators, where two or more chaotic
systems evolve along similar chaotic trajectories despite starting from
different initial conditions, represents a highly intriguing dynamical
phenomenon that has been extensively investigated from various perspectives.
In the context of swarmalators, this chaotic behavior may arise from the
nonlinear interactions and feedback mechanisms among the individual components
of the swarmalator system. This scenario is an apt depiction of challenges
akin to the task coordination observed in swarming animals, which not only
synchronize their movements within a two-dimensional plane but also respond
collectively when subjected to threats or animal attacks. We study the
ramifications that arise when the internal dynamics of the swarmalators are
influenced by amplitude-mediated chaotic oscillators. We report our findings
of some of the existing states of swarmalators for chaotic dynamics. In Sec.
II, we represent the fundamental mathematical model which serves as the basis
for our study. We depict our results in Sec. III. Lastly, we sum up with
period one case and conclusions in Sec. IV and V, respectively.

##  II mathematical model

We consider $N$ number of swarmalators moving in a 2D region. The internal
dynamics ($\mathbf{x}_{i}\in\mathbb{R}^{d}$) of the swarmalators are governed
by a $d$-dimensional amplitude oscillator. The bidirectional interplay between
spatial and internal dynamics of the swarmalators is given by the following
set of equations,

$\displaystyle\dot{\mathbf{r}}_{i}=$
$\displaystyle\mathbf{v}_{i}+\dfrac{1}{N}\sum_{j\neq
i}^{N}\big{(}\mathbf{r}_{j}-\mathbf{r}_{i}\big{)}\bigg{[}\big{(}A+J_{1}e^{-E_{ij}}\big{)}-\dfrac{\big{(}B-J_{2}e^{-E_{ij}}\big{)}}{|\mathbf{r}_{j}-\mathbf{r}_{i}|^{2}}\bigg{]},$
(1)
$\displaystyle\dot{\mathbf{x}}_{i}=f(\mathbf{x}_{i})+\dfrac{K}{N}\sum_{j\neq
i}^{N}H(\mathbf{x}_{i},\mathbf{x}_{j},\mathbf{r}_{i},\mathbf{r}_{j}),$ (2)

for $i=1,2,\ldots,N$ where
$\mathbf{r}_{i}=(\xi_{i},\eta_{i})\in\mathbb{R}^{2}$ denotes the spatial
position of the $i$-th swarmalator in the 2D plane. The attractive force among
the swarmalators is represented by the 1st term inside the summation in Eq.
(1) as $I_{\text{att}}=\big{(}A+J_{1}e^{-E_{ij}}\big{)}$ and the second term
defines the repulsive interaction among the swarmalators as
$I_{\text{rep}}=\left(B-J_{2}e^{-E_{ij}}\right)/|\mathbf{r}_{j}-\mathbf{r}_{i}|^{2}$.
Here, $E_{ij}$ denotes the difference between internal dynamics of the $i$-th
and the $j$-th swarmalators and is defined as
$E_{ij}=|\mathbf{x}_{j}-\mathbf{x}_{i}|$, where $|\cdot|$ denotes the
Euclidean norm. $A$ and $B$ are the strengths of spatial attraction and
repulsion, respectively. $J_{1}$ and $J_{2}$ measure the influence of internal
dynamics of the swarmalators on spatial attraction and repulsion,
respectively.

$f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ in Eq. (2) denotes the uncoupled
identical internal dynamics of the swarmalators where
$\mathbf{x}_{i}\in\mathbb{R}^{d}$.
$H:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{d}$
stands for the effect of spatial configurations on the internal dynamics. In
our study, we employ the chaotic Rössler attractor to describe the internal
dynamics of the system, i.e.,

$\displaystyle f(\mathbf{x}_{i})=\begin{pmatrix}-y_{i}-z_{i}&\\\
x_{i}+ay_{i}&\\\ b+z_{i}(x_{i}-c)&\\\ \end{pmatrix},$ (3)

where $\mathbf{x}_{i}=(x_{i},y_{i},z_{i})\in\mathbb{R}^{3}$. The interacting
function $H$ defines the diffusive coupling between the $y$-components of the
Rössler oscillators, which is modulated by the spatial distance and is given
by

$H(\mathbf{x}_{i},\mathbf{x}_{j},\mathbf{r}_{i},\mathbf{r}_{j})=\left[0,\frac{y_{j}-y_{i}}{|\mathbf{r}_{j}-\mathbf{r}_{i}|^{\gamma}},0\right]^{\mathcal{T}},$
(4)

where $\mathcal{T}$ denotes the transpose of matrix and $K$ is the coupling
strength. We are concerned only about the interaction through the $y$
components in our study. While dealing with the diffusive coupling via $x$ or
$z$ components in the Rössler oscillator, the Master Stability Function
$(\Phi(K))$ [48] concerning the coupling strength $(K)$ is always positive,
which indicates that the coupled network of oscillators does not achieve any
stable synchronization state. When considering interactions among the $y$
components in the Rössler oscillator, $\Phi(K)$ is negative, i.e., the system
concludes to achieve synchrony [49].

In the rest parts of our investigation, we choose $N=200$ identical
swarmalators with velocities $\mathbf{v}_{i}=0$ and fix $\gamma=1$.

Figure 1: Measurement of the average synchronization error $\langle E\rangle$
of the Rössler oscillator (filled red circles) and the mean velocity of the
swarmalators $\langle v\rangle$ (open blue circles) as functions of the
coupling strength $K$. The dotted vertical lines are used to distinguish
distinct emerging states. (a) Async for $K=-0.068$, (b) splintered phase wave
for $K=0.012$, (c) phase wave for $K=0.072$, (d) chimera for $K=0.132$, and
(e) sync for $K=0.2$. We use the logarithmic scale to visualize $\langle
v\rangle$ to differentiate the states by their velocity. The color bar
signifies the $x$ component of the Rössler oscillator. We fix $J_{1}=2.0$ and
$J_{2}=0.5$. Heun’s method has been used to simulate $N=200$ swarmalators with
time-step $dt=0.01$ for $T=4000$ time units. Initially, the swarmalators are
placed uniformly randomly inside a square box of length $2$ centered at the
origin. All the resultant states are observed after a long transient. The
average synchronization error $\langle E\rangle$ and the mean velocity
$\langle v\rangle$ are calculated after wiping out the first $75\%$ data with
$10$ realizations.

##  III Results for chaotic system

Our primary objective in this study is to explore the swarmalator field
through the lens of chaotic oscillators. Hence we first focus on the chaotic
region of the Rössler oscillator. Later in Sec. IV, we also investigate the
collective patterns in the case of period one Rössler oscillator. Here we
choose $a=b=0.2$, and $c=5.7$ belonging to the chaotic regime. In our study,
every agent has three internal degrees of freedom ($x$, $y$, and $z$), and
their internal dynamics arise from an ensemble of globally connected Rössler
oscillators. In the subsequent study, we choose $A=1$ and $B=2$. The other
parameters $J_{1}$, $J_{2}$, and $K$ act as the primary control parameters.
The spatial attraction term, $A+J_{1}e^{-E_{ij}}$ increases as the relative
difference between the internal dynamics of the swarmalators $(E_{ij})$
decreases. In contrast, the repulsion force among the swarmalators
$(B-J_{2}e^{-E_{ij}})$ diminishes, when the term $(E_{ij})$ decreases and vice
versa. It can be readily comprehended in the following manner:

  1. (i)

When the difference of the internal dynamics among the swarmalators decreases,
the term $e^{-E_{ij}}$ increases exponentially and finally attains the value
unity. Hence, the attraction strength reduces to $A+J_{1}$ and that of
repulsion becomes $B-J_{2}$.

  2. (ii)

The opposite scenario occurs when $E_{ij}$ reaches a larger value so that the
term $e^{-E_{ij}}\rightarrow 0$ and the strength of attraction and repulsion
become independent of $J_{1}$ and $J_{2}$.

Depending on the choice of these control parameters, the swarmalators exhibit
various long-term emerging states, ranging from asynchronous to synchronous
states. Before explaining these states, we define some order parameters that
prove to be beneficial in quantifying various properties of the emerging
states.

Figure 2: Splintered phase wave and phase wave states for swarmalators with
chaotic Rössler oscillator. Snapshots of the splintered phase wave state are
shown at (a) $t=3000$, (b) $t=4000$, and (c) $t=5000$ time units for
$K=0.012$. (d), (e) and (f) depict the snapshots of the phase wave state at
$t=3000,\ 4000$, and $5000$ time units, respectively where $K=0.072$. In the
entire simulation, we fix $J_{1}=2.0$, and $J_{2}=0.5$. We integrate Eqs. (1)-
(2) by Heun’s method using $T=5000$ time units with step-size $dt=0.01$ for
$N=200$ swarmalators.

To measure the amount of internal disorder among the swarmalators, we define
the time average of the synchronization error of the Rössler oscillator as

$\langle
E\rangle=\left\langle\sum_{i,j=1}^{N}\dfrac{|\mathbf{x}_{j}(t)-\mathbf{x}_{i}(t)|}{N(N-1)}\right\rangle_{t},$
(5)

where $\langle\cdots\rangle_{t}$ stands for the time average, after discarding
the initial transients. The red curve (filled circles) in Fig. 1 refers to the
time average synchronization error $\langle E\rangle$ as a function of the
coupling strength $K$ delineated through the left $y$-axis. Our model reveals
both stationary and non-stationary states. To distinguish them, we measure the
mean velocity of the swarmalators as

$\langle
v\rangle=\Bigg{\langle}\frac{1}{N}\sum_{i=1}^{N}\sqrt{{{\Delta\xi_{i}}^{2}+{\Delta\eta_{i}}^{2}}}\Bigg{\rangle}_{t},$
(6)

where $(\Delta\xi_{i},\Delta\eta_{i})$ is the displacement of the $i$-th
swarmalator in the 2D plane in time interval $t$ to $t+dt$. The blue curve
(open circles) in Fig. 1 represents this quantity as a function of $K$ plotted
in the log-linear scale in the right $y$-axis. With the understanding of these
order parameters, we delve into examining the emerging collective states of
our model.

###  III.1 Emerging collective states

We explore the synchronization and spatial properties of the swarmalator
model. Based on numerical analysis, we find that the parameter $K$ plays a
pivotal role in our case study, as the interaction strength among the Rössler
oscillators controls the synchronization phenomena, eventually determining the
spatial pattern formation of the swarmalators. Consequently, we systematically
adjust $K$ across a defined range, leading to the identification of five
distinct long-term collective states: sync, async, splintered phase wave,
phase wave, and chimera. As the swarmalators diverge due to strong repulsion
for $K<-0.13$ and remain synchronized for $K>0.3$, we meticulously focus on
the region $K\in[-0.13,0.3]$ where we can objectify all the emerging states.
Figures 1(a)-(e) best illustrate these states where each of the swarmalators
is represented by the scatter plots in the $(\xi,\eta)$ plane and their colors
indicate the $x$ component of the Rössler oscillator. We also scrutinize the
pattern formation by coloring the swarmalators according to the other two
components $y$ and $z$ and find similar outcomes (see Fig. 1 in the
Supplementary Material [50] ).

Async: We start from the left panel in Fig. 1 where $K$ is negative. In the
range $[-0.12,0.0]$, the swarmalators are moving (notice the velocity profile
in Fig. 1 depicted by the blue curve) and they arrange themselves within a
circular disc. Given that the interaction strength $K$ is negative, it hinders
their ability to exhibit coherent behavior within the internal oscillations.
The corresponding internal dynamics remain desynchronized, exhibiting a wide
range of $x$ values seen in Fig. 1(a). $\langle E\rangle$ is notably elevated
in this region, suggesting a highly desynchronized behavior. We embellish this
as the async. Look at Movie 1 of the Supplementary Material [50] for the time
evolution of the state.

Splintered phase wave: Moving to the right of Fig. 1, as we increase the value
of $K$ from 0, we note a discernible trend where the swarmalators form
multiple clusters, each exhibiting motion confined within its respective
cluster, and their activity never dies off. The region between the black and
pink dotted vertical lines ($0.0<K<0.032$) illustrates the state specifically.
If we recall the splintered phase wave of the phase-influenced swarmalator
model [29], this state is the closest realization of that. For this, we refer
to this state as splintered phase wave (see Fig. 1(b)). Figures 2(a)-(c)
display the snapshots of this state at different times which demonstrates the
evolution over time. We further plot the $x$ components of the Rössler
oscillator as a function of the indices for this state in Fig. 3(a). The
clusters display distinct internal behaviors in this state, each isolated from
the others. Find supplementary [50] Movie 2 for a better visualization of this
state.

Phase wave: When we gradually increase $K$ from the splintered phase wave
state, swarmalators change their positions in a very rapid manner. They are
spatially attracted towards the ones having minimal synchronization error
$E_{ij}$ and the disjoint cluster formation like the splintered phase wave,
disappears. Compared to the other states, we observe that swarmalators move
with a higher velocity in this state which we will discuss in detail in the
subsequent sections. We refer to this state as the phase wave (look into Fig.
1(c)). They form a deformed disk-like structure, where the positions are
distributed inside the disk and they are attracted to the ones having nearby
internal dynamics (in the phase wave state reported with phase oscillators,
they were distributed uniformly inside an annular ring). In Figs. 2(d)-(f), we
capture the snapshots at different times to analyze their movement. Also, see
Movie 3 in Supplementary [50] for the time evolution of the phase wave state.
We specify this region between the pink and the yellow dotted vertical lines
($0.032<K<0.116$) shown in Fig. 1 where the phase wave is realized.

Chimera: On further increment of $K$, we observe a bunch of swarmalators
organize themselves within a cluster having similar internal dynamics
(coherent $x$ values), and the rest of the swarmalators form another cluster
with a wide range of $x$ values. All of them show feeble spatial movements
within their specific clusters. (Find Movie 4 of the Supplementary Material
[50]). The nature of this state is similar to the chimera state where coherent
and incoherent behaviors coexist [51, 52, 53]. We represent this state as
chimera state. Figure 1(d) illustrates this state by showing the snapshot in
the $(\xi,\eta)$ plane. We observe this state when $0.116<K<0.176$. The region
is highlighted by the yellow and the violet dotted vertical lines in Fig. 1.
For a clear picture, we plot the $x$ components of the Rössler oscillators
against their respective indices in Fig. 3(b). The presence of one coherent
group where the $x$ values are the same is observed along with an incoherent
group where the $x$ values are distributed.

Sync: When $K$ is positively large ($K>0.176$), the Rössler oscillators get
synchronized and $\langle E\rangle$ goes to zero. The spatial movement of the
swarmalators also diminishes, i.e., $\langle v\rangle=0$. On top of that, we
observe all the swarmalators organize themselves inside a circular disk. They
are synchronized at every instant, indicating that the collective internal
dynamics become identical for all the swarmalators (look at Movie 5 in
Supplementary [50]). We mark this state as sync which is represented by the
scatter plots in Fig. 1(e).

Therefore, with increasing coupling strength $K$ the entire route from the
async to the sync state can be depicted as async $\rightarrow$ splintered
phase wave $\rightarrow$ phase wave $\rightarrow$ chimera $\rightarrow$ sync.

Figure 3: Scatter plots of the Rössler $x$ components against their
corresponding indices at $T=4000$ time units for $N=200$ swarmalators. (a)
Splintered phase wave for $K=0.012$ where distinct clusters are observed. (b)
Chimera for $K=0.072$ where two clusters are visible among which one is
synchronized and in the other one swarmalators show asynchronous behavior.
Heun’s method has been used to integrate Eqs 1 and 2 with step-size $dt=0.01$
to generate both the figures.

###  III.2 Radii of the sync and async states

Sync: Here, the agents do not exhibit any spatial movements. Moreover, each of
them embarks on identical internal dynamics at every instant of time. One can
capture this information regarding this state in terms of the velocity field,
which contains all necessary details, i.e., their spatial positions and
relative internal errors. The corresponding field can be illustrated with the
following form $\mathbf{v}=\mathbf{v}(\mathbf{r},\tilde{E}(\mathbf{r}))$.
Since the spatial movement of the agents is absent, we can conclude the
velocity field $\mathbf{v}=0$.

When examining the specifications of steady-state patterns, an insightful
approach is involved in investigating the continuity equation. In this state,
where the density $\rho(\mathbf{r},\tilde{E}(\mathbf{r}))$ represents the
stationary distribution of swarmalators with positions $\mathbf{r}$ in
$(\xi,\eta)$ plane and internal relative error $\tilde{E}(\mathbf{r})$, the
divergence of the velocity field must be zero. This requirement is underscored
by the normalizing property, stating that
$\int\rho(\mathbf{r},\tilde{E}(\mathbf{r}))d\mathbf{r}=1$. Finally, we end up
with a pair of simultaneous equations,

$\displaystyle\mathbf{v}$ $\displaystyle\equiv 0,$ (7)
$\displaystyle\nabla\cdot\mathbf{v}$ $\displaystyle\equiv 0.$ (8)

In Cartesian coordinates, the velocity field reads

$\displaystyle\mathbf{v}\left(\mathbf{r},\tilde{E}(\mathbf{r})\right)=\int\Bigg{(}\left(\mathbf{\tilde{r}}-\mathbf{r}\right)\left(A+J_{1}e^{-\tilde{E}(\mathbf{\tilde{r}})}\right)$
$\displaystyle-\dfrac{\mathbf{\tilde{r}}-\mathbf{r}}{|\mathbf{\tilde{r}}-\mathbf{r}|^{2}}\left(B-J_{2}e^{-\tilde{E}(\mathbf{\tilde{r}})}\right)\Bigg{)}\rho\left(\mathbf{\tilde{r}},\tilde{E}(\mathbf{\tilde{r}})\right)~{}d\mathbf{\tilde{r}}.$
(9)

In the sync state, the internal dynamics of the swarmalators become identical
at every instant of time. When the relative internal error boils down, i.e.,
$\tilde{E}\rightarrow 0$, the integral part of Eq. (III.2) no longer depends
on the relative error, rather it depends only on the spatial profile, which
allows us to treat Eq. (III.2) by excluding $\tilde{E}$. Remembering this
fact, Eq. (III.2) can be rewritten as,

$\displaystyle\mathbf{v}\left(\mathbf{r}\right)=\int\left(\left(\mathbf{\tilde{r}}-\mathbf{r}\right)\left(A+J_{1}\right)-\dfrac{\mathbf{\tilde{r}}-\mathbf{r}}{|\mathbf{\tilde{r}}-\mathbf{r}|^{2}}\left(B-J_{2}\right)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho\left(\mathbf{\tilde{r}}\right)~{}d\mathbf{\tilde{r}}.$
(10)

Recalling Eq. (8), by taking the divergence of Eq. (III.2) we get the
following equality

$\displaystyle\nabla\cdot\mathbf{v}\left(\mathbf{r}\right)=\int\biggl{(}2(A+J_{1})-2\pi\delta(\mathbf{\tilde{r}}-\mathbf{r})(B-J_{2})\biggr{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho(\mathbf{\tilde{r}})d\mathbf{\tilde{r}}=0.$
(11)

Simplifying Eq. (III.2), we obtain

$\int
2\pi\delta\left(\mathbf{r}-\mathbf{\tilde{r}}\right)\rho(\mathbf{\tilde{r}})d\mathbf{\tilde{r}}=\dfrac{2(A+J_{1})}{B-J_{2}},$
(12)

which gives,

$\rho(\mathbf{r})=\dfrac{1}{\pi}\dfrac{A+J_{1}}{B-J_{2}}.$ (13)

In this state, the swarmalators form a disc in the $(\xi,\eta)$ plane. The
center of position is conserved by the symmetric pairs of changes of Eq. (1).
So due to the rotational symmetry, the density of steady state can be written
as $\rho_{s}\left(\mathbf{r}\right)=1/(\pi R_{s}^{2})$ for $|\mathbf{r}|\leq
R_{s}$. Comparing Eq. (13) with this, we get the expression of the radius in
the sync state as

$R_{s}=\sqrt{\dfrac{B-J_{2}}{A+J_{1}}}.$ (14)

In Fig. 4(a), we explore the dependence of the radius of the sync state on
$J_{1}$. Furthermore, we collocate our analytical finding, as expressed in Eq.
(14), with the numerical outcome, revealing a strong concurrence between them.

Figure 4: (a) Radius of the sync state as a function of $J_{1}$ for $K=0.25$.
(b) The radius of the async state against $J_{1}$, where $K=-0.05$. The red
dots correspond to the simulation results and the black curve represents the
analytical measurement. Data were generated for $T=4000$ time units by
integrating Eqs 1 and 2 with Heun’s method for $(dt,N)=(0.01,200)$.

Async: In this segment, we analyze the async state, where the relative error
between the oscillators $(\tilde{E})$ takes a tremendous nonzero positive
value. As a result, the influence of $J_{1}$ and $J_{2}$ becomes
insignificant, and therefore, their effects can be disregarded in this
scenario. Hence, the term $e^{-\tilde{{E}}}\rightarrow 0$ and the Eq. (III.2)
can be rewritten in the following form,

$\mathbf{v}\left(\mathbf{r}\right)=\int\Bigg{(}\left(\mathbf{\tilde{r}}-\mathbf{r}\right)A-\dfrac{\mathbf{\tilde{r}}-\mathbf{r}}{|\mathbf{\tilde{r}}-\mathbf{r}|^{2}}B\Bigg{)}~{}\rho\left(\mathbf{\tilde{r}}\right)~{}d\mathbf{\tilde{r}}.$
(15)

Performing a comparable calculation with the sync state yields the form of the
probability density function for the async state as follows,

$\rho(\mathbf{r})=\dfrac{1}{\pi}\dfrac{A}{B}.$ (16)

Therefore, the radius for the async state takes the form as,

$R_{a}=\sqrt{\dfrac{B}{A}}.$ (17)

We validate our analytical finding given by Eq. (17) with the numerical
simulation in Fig. 4(b). This emphasizes that the radius of the async state is
independent of $J_{1}$ (also $J_{2}$, from the expression in Eq. (17)).

Figure 5: (a) Variation of $\langle E\rangle$ in the $J_{1}$-$K$ parameter
space for $J_{2}=0.5$, and in (b) $J_{2}-K$ parameter space for $J_{1}=2.0$.
The model is integrated with $N=200$ swarmalators using Heun’s method for
step-size $dt=0.01$ and $T=4000$ time units. After discarding the transients,
the average synchronization error $\langle E\rangle$ is calculated with the
last $25\%$ of data.

###  III.3 Dependence on system parameters

We explore the impact of $J_{1}$ and $J_{2}$ on the dynamics of the
swarmalators. For this, we choose to vary one of $J_{1}$ and $J_{2}$ with the
coupling strength $K$, keeping the other fixed. In Fig. 5(a), simultaneously
varying $J_{1}$ with $K$, while fixing $J_{2}$ at 0.5, we observe the system’s
dynamical behavior. In Fig. 5(b), we vary $J_{2}$ over a range of $K$ by
keeping $J_{1}=2.0$. The color bar signifies the time average synchronization
error $\langle E\rangle$ of the Rössler oscillator. From the $J_{1}-K$
parameter space, it is evident that the collective behavior of the system is
independent of $J_{1}$ and primarily depends on $K$. From Fig. 5(b), we
observe a similar scenario except in the upper right corner, where the system
becomes highly unstable and collapses for a high $J_{2}$ value. It is worth
reminding ourselves again that the spatial attraction and repulsion terms are
given by $\left(A+J_{1}e^{-E_{ij}}\right)$ and
$\left(B-J_{2}e^{-E_{ij}}\right)/|\mathbf{r}_{j}-\mathbf{r}_{i}|^{2}$,
respectively. When the value of $K$ is large enough to attain synchrony among
the Rössler oscillators, the repulsion term eventually becomes $B-J_{2}$. We
should have $J_{2}<B$, or else the repulsive term acts like an attraction
term. In that case, some swarmalators might collide with each other which is
not feasible for a swarmalator system. Since we have fixed $B=2$ in our work,
we see that when $K$ is sufficiently large and $J_{2}\rightarrow 2$, there is
an unbounded region (the white region in Fig. 5(b)).

###  III.4 Velocity profiles of the states

In the preceding sections, we studied several emerging patterns exhibited by
our model and thoroughly examined their dynamics. The most remarkable outcome
is the discovery of numerous states that have already been reported in the
phase-influenced swarmalator model. In addition to the amplitude dynamics of
the underlying Rössler oscillator, we notice the swarmalators exhibiting
various spatial movements in the emerging states. One can explore and
distinguish these profiles in terms of the velocity of the center of positions
$(v)$ for each state. Previously in Fig. 1, we have captured the essence of
nonzero time-averaged velocity over a range of coupling strength $K$. We
showcase the corresponding temporal evolution of the ensemble displacement for
each state in Fig. 6. The swarmalators show very small displacement over time
in the async state as observed in Fig. 6(a), whereas, in the sync state,
swarmalators attain steady positional configuration, i.e., $v=0$. Look at Fig.
6(b) for the steady behavior of $v$ in the sync state for $K=0.2$. In the
remaining states, the overall structure pulsates over time. Notice the spiking
behavior in the velocity profiles for these states (see Figs. 6(c)-(e)). When
the swarmalators form the phase wave, they frequently undergo large
compression and expansion. Swarmalators try to minimize the difference between
their internal states when $K$ is positive. They get spatially attracted
towards each other when this difference decreases. On the other hand, when
they come too close to each other, the spatial repulsion function disperses
them away which gives a sudden jump in the velocity and the loop continues.
This gives rise to some kind of positional instability in the system. We
observe the spiking tails are significantly higher in the phase wave (Fig.
6(d)) compared to both the splintered phase wave (Fig. 6(c)) and chimera state
(Fig. 6(e)).

Figure 6: The temporal evolution of the velocity of the center of positions
$(v)$ regarding the emerging states. (a) Async $(K=-0.068)$, (b) sync
$(K=0.2)$, (c) splintered phase wave $(K=0.012)$ , (d) phase wave $(K=0.072)$,
and (e) chimera $(K=0.132)$. The model is integrated with $N=200$ swarmalators
using Heun’s method with step-size $dt=0.01$ for $T=4000$ time units. We show
the velocity time series for the last $10\%$ data for clarity. Figure 7:
Variation of average synchronization error $\langle E\rangle$ (filled red
circle) of period one Rössler oscillator and the mean positional velocity of
the swarmalators $\langle v\rangle$ (open blue circle) with the coupling
strength $K$. We choose the parameter values as $a=b=0.1,\ c=4.0$ and fix
$J_{1}=2.0,\ J_{2}=0.5$, $A=1.0$ and $B=2.0$. Dotted vertical lines separate
the regimes between different emerging states. Snapshots are shown in (a)
async for $K=-0.12$, (b) active phase wave for $K=-0.05$, (c) splintered phase
wave for $K=-0.02$, (d) static phase wave for $K=0.0$, and (e) sync for
$K=0.04$. The color bar signifies the $x$ component of the Rössler oscillator.
We integrate Eq. (1) and Eq. (2) by Heun’s method for
$(dt,T,N)=(0.01,4000,200)$. In all cases, swarmalators are initially placed
inside a box with dimension $[-1,1]\times[-1,1]$ uniformly at random. All the
resultant states are observed after a long transient. $\langle E\rangle$ and
$\langle v\rangle$ have been calculated with the last $25\%$ data after
discarding the transients with $10$ realizations of initial conditions.

##  IV Analysis on periodic oscillation

After investigating the swarmalators embedded with chaotic dynamics of the
Rössler oscillator, we explore them with the period one behavior. For this, we
fix the parameters as $a=0.1$, $b=0.1$, and $c=4.0$ where the Rössler
oscillator shows one periodic behavior.

Similar to the chaotic region, here also, we consistently adjust the
oscillators’ interaction strength $K$ over a finite range and observe the
variation of average synchronization error $\langle E\rangle$ and the mean
velocity $\langle v\rangle$ of the swarmalators to distinguish different
emerging collective behaviors. In Fig. 7, we demonstrate the variation of
$\langle E\rangle$ and $\langle v\rangle$ by filled red circles and open blue
circles, respectively. We again observe the async and sync states for extreme
values of $K$. Both the async and sync states (see Figs. 7(a) and (e))
resonate similarly to the case of the chaotic oscillator. The async state is
found when $-0.12<K<-0.096$. What is novel about the periodic region is the
discovery of three types of phase wave states, akin to those observed in the
phase-influenced swarmalator model. Inside the region highlighted by the black
and pink dotted vertical lines in Fig. 7 (where $-0.096<K<-0.038$), we observe
the swarmalators coordinating themselves within a disk and each one of them
tries to form clusters with another having coherent internal dynamics,
characterized by the nearest $x$ values. As a whole, they follow a circular-
like movement in the $(\xi,\eta)$ plane. Fig. 7(b) depicts this state. We
denote this state as the active phase wave due to the continuous movements of
swarmalators from one portion to another inside the disk. When we increase $K$
further from $-0.038$, we capture the swarmalators to form disjoint clusters
and move in the $(\xi,\eta)$ plane inside their respective groups. We identify
this state as the splintered phase wave (notice Fig.7(c)) as before. The
extent of this state is delineated by pink and yellow dotted vertical lines
($-0.038<K<0.0$). The difference between the active and the splintered phase
wave is best understood through Movie 6 of the Supplementary Material [50].
The first stationary state ($\langle v\rangle\approx 0$) occurs for the case
when $K=0$. The swarmalators become static by their position within the
regions highlighted by the yellow and violet dotted vertical lines where
$0.0<K<0.012$. They arrange themselves into small groups where they are
situated nearby to the ones having minimal relative error. We refer to this
state as static phase wave which is depicted by Fig. 7(d). Finally, we achieve
synchrony at $K=0.012$, after which all the swarmalators become completely
static by their positions ($\langle v\rangle=0$) and resonate with similar
internal dynamics at an instant of time. Therefore the overall dynamical route
in this case becomes async $\rightarrow$ active phase wave $\rightarrow$
splintered phase wave $\rightarrow$ static phase wave $\rightarrow$ sync.

To measure the correlation between swarmalators’ spatial angle
$\phi_{j}=\tan^{-1}(\eta_{j}/\xi_{j})$ and the oscillators’ phase
$\theta_{j}=\tan^{-1}(y_{j}/x_{j})$, we define the following order parameters
as:

$S_{\pm}e^{i\psi\pm}=\frac{1}{N}\sum_{j=1}^{N}e^{i(\theta_{j}\pm\phi_{j})},$
(18)

which quantifies the correlation between the spatial and internal dynamics of
the swarmalators. We take the maximum of $S_{\pm}$ and define
$S=\max\\{S_{+},S_{-}\\}$. A non-zero value of $S$ signifies the presence of a
correlation between the swarmalators’ spatial position to the oscillators’
internal dynamics. First, we observe the time evolution of $S$ at different
values of $K$. For the $K<0$ region, we notice the chaotic nature of $S$
depicted by the black curve in Fig. 8(a). In the positive $K$ region, the
oscillation of $S$ completely dies and does not vary with time (see the blue
line in Fig. 8(a)). For $K=0$, when the dynamics of the Rössler oscillators do
not affect the spatial positions, we observe the periodic nature of $S$ with
time (look at the red curve in Fig. 8(a)). We have also looked at the power
spectrum to validate the chaotic (Fig. 8(b)), periodic (Fig. 8(c)), and
constant (Fig. 8(d)) behaviors of $S$ for $K=-0.1,0$, and $0.1$, respectively.

Figure 8: (a) Time series of $S$ for $K=-0.1$ (black), $K=0.0$ (red), and
$K=0.1$ (blue). (b), (c) and (d) are the normalized power spectrum for
$K=-0.1,0$, and $0.1$, respectively. We take $N=200$ swarmalators and run
simulations for $T=4000$ time units with step-size $dt=0.01$ using Heun’s
method.

##  V Discussions

It is worth mentioning that till now, swarmalators have been studied through
the vision of phase oscillators. The combined effect of phase-dependent
spatial aggregation and position-dependent phase synchronization defines a
novel phenomenon and leads to various rich spatio-temporal patterns. Some of
these stationary and non-stationary patterns can be found in many real-world
systems like Japanese tree frogs [54], magnetic domain walls [55], Janus
matchsticks [56], robotic swarms [57, 58] etc. Nowadays, researchers
investigate this system by applying suitable interacting functions, coupling
schemes, network structures, etc. Our work’s novelty lies in exploring the
system from the perspective of the amplitude oscillators.

In this article, we have studied the trade-off between the spatial dynamics
over the chaotic (and periodic) internal dynamics of the swarmalators and
vice-versa. We have modeled swarmalators so that their interactions are all-
to-all, and their internal dynamics are governed by the Rössler system. We
here claim that not only the phase oscillator, we can visualize the
swarmalator field from the perspective of the amplitude oscillator also. We
have encountered the presence of most of the states akin to those observed in
the phase-inspired swarmalator model [29]. Our model comprises three
parameters $K$, $J_{1}$, and $J_{2}$ which mainly regulate the long-term
behavior of the swarmalator system. We analytically derived the radius of the
sync and async states which confirms our numerical findings and emphasize the
originality of our work. We are unable to solve the radius of the other states
as their structure continuously changes with time. Subsequently, we explore
the system within the non-chaotic domain (period one), leading to the
emergence of five distinct long-term collective states.

In summary, our model aims to provide a fresh outlook on swarmalator dynamics.
One can think about how these internal dynamics can be further modified using
the higher-order coupling scheme for various parameter values. We expect there
might be other possible emerging states. Also, we anticipate that using a
hyperchaotic system (characterized by more than one positive Lyapunov
exponents) as swarmalators’ internal dynamics, or complex switching mechanisms
between multistable states, our system might exhibit a rich variety of complex
behaviors and diverse emerging states, surpassing the limitations associated
with a basic chaotic system. Implementing the vision range of each entity will
open a new avenue for study on the swarmalator. On top of that, we can
implement the angular dynamics under the influence of inertia on the bare
swarmalator system[29]. A pertinent future goal to explore would involve
examining alternative physically feasible structures for both spatial and
internal interactions. The analytical feasibility of our research offers a
platform for investigating these avenues in the coming times.

Figure 9: Temporal evolution of the five long-term emerging states for chaotic
Rössler attractor $(a=b=0.2,~{}c=5.7)$. The states in each row are depicted
concerning the $x$, $y$, and $z$ components of the Rössler attractor,
represented by the color bar. The model is integrated by Heun’s method with
$(dt,T,N)=(0.01,4000,200)$.

## Supplementary materials

## Movies

We explore the swarmalator system from a new perspective. We investigate the
two-dimensional swarmalator system through the lens of amplitude-mediated
oscillatory dynamics. We keep $K$ as the free parameter and vary across a wide
range to observe different emerging patterns and their behavior. To best
visualize the states, we make movies of them to show how they evolve with
time. To do this, we integrate the governing equations by Heun’s method and
fix $(dt,T,N)=(0.01,4000,200)$. Initially, the swarmalators are placed inside
a square of dimension $[-1,1]\times[-1,1]$, uniformly at random and their
internal dynamics are governed by the $x$ component of the Rössler attractor.

In all the movies, we fix $J_{1}=2.0$ and $J_{2}=0.5$. From Movie 1 to Movie
5, we show the time evolution of the states for which we choose the Rössler
parameters from the chaotic region such as $a=b=0.2$ and $c=5.7$.

In Movie 1, we show the async state where the swarmalators move and their
internal dynamics remain desynchronized. They form a disc and are uniformly
distributed in the $(\xi,\eta)$ plane.

Movie 2 illustrates the splintered phase wave, where we observe a noticeable
pattern whereby the swarmalators assemble into numerous clusters, each
displaying motion confined within its respective cluster, and their activity
persists indefinitely.

In Movie 3, we demonstrate the time evolution of the phase wave state, where
the swarmalators move with a high velocity and are spatially attracted towards
the one having minimal synchronization error $E_{ij}$.

Movie 4 depicts the chimera state beautifully, where coherence and incoherence
clusters coexist. The coherence is determined by analogous internal dynamics
among the swarmalators.

In Movie 5, we capture the time evolution of the sync state, i.e., the
swarmalators become static by their position, and their internal behavior
becomes identical at every instant.

The parameter values used are listed below:

  * •

Async: $K_{3}=-0.068$

  * •

Splinterded phase wave: $K_{3}=0.012$

  * •

Phase wave: $K_{3}=0.072$

  * •

Chimera: $K_{3}=0.132$

  * •

Sync: $K_{3}=0.2$

Movie 6 portrays the difference between the active phase wave and the
splintered phase wave state, which we find in the period-1 region of the
Rössler system. We fix the Rössler system’s parameters at $a=b=0.1$ and
$c=4.0$.

Please see the online version in journal page to see the movies.

## Collective five states with respect to $x$, $y$ and $z$ variables

As stated in the manuscript, we utilized the $x$ component of the Rössler
attractor to color the swarmalators, and we observed that similar outcomes
arise when using the $y$ and $z$ components. In Fig. 9, we have shown the time
evolution of the five emerging states by coloring the swarmalators according
to $x$ (upper row), $y$ (middle row), and $z$ (bottom row) components of the
chaotic Rössler attractor.

## References

  * Pikovsky et al. [2001] A. Pikovsky, M. Rosenblum, and J. Kurths, _A universal concept in nonlinear sciences_ , vol. 2 (2001).
  * Winfree [1980] A. T. Winfree, _The geometry of biological time_ , vol. 2 (Springer, 1980).
  * Winfree [1967] A. T. Winfree, Journal of Theoretical Biology 16, 15 (1967).
  * Peskin [1975] C. S. Peskin, Courant Inst. Math (1975).
  * Aldridge and Pye [1976] J. Aldridge and E. K. Pye, Nature 259, 670 (1976).
  * Buck [1988] J. Buck, The Quarterly Review of Biology 63, 265 (1988).
  * Buck [1938] J. B. Buck, The Quarterly Review of Biology 13, 301 (1938).
  * Motter et al. [2013] A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Nature Physics 9, 191 (2013).
  * Wiesenfeld et al. [1996] K. Wiesenfeld, P. Colet, and S. H. Strogatz, Physical Review Letters 76, 404 (1996).
  * Strogatz et al. [2005] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Nature 438, 43 (2005).
  * Couzin et al. [2002] I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, Journal of Theoretical Biology 218, 1 (2002).
  * Couzin [2007] I. Couzin, Nature 445, 715 (2007).
  * Sumpter [2010] D. J. Sumpter, _Collective animal behavior_ (Princeton University Press, 2010).
  * Mogilner and Edelstein-Keshet [1999] A. Mogilner and L. Edelstein-Keshet, Journal of Mathematical Biology 38, 534 (1999).
  * Bialek et al. [2012] W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, and A. M. Walczak, Proceedings of the National Academy of Sciences 109, 4786 (2012).
  * Ballerini et al. [2008] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, et al., Proceedings of the National Academy of Sciences 105, 1232 (2008).
  * Katz et al. [2011] Y. Katz, K. Tunstrøm, C. C. Ioannou, C. Huepe, and I. D. Couzin, Proceedings of the National Academy of Sciences 108, 18720 (2011).
  * Levy and Requeijo [2008] D. Levy and T. Requeijo, Bulletin of Mathematical Biology 70, 1684 (2008).
  * Chavy-Waddy and Kolokolnikov [2016] P.-C. Chavy-Waddy and T. Kolokolnikov, Nonlinearity 29, 3174 (2016).
  * Pal et al. [2024] S. Pal, G. K. Sar, D. Ghosh, and A. Pal, Chaos: An Interdisciplinary Journal of Nonlinear Science 34 (2024).
  * Frasca et al. [2008] M. Frasca, A. Buscarino, A. Rizzo, L. Fortuna, and S. Boccaletti, Physical Review Letters 100, 044102 (2008).
  * Majhi et al. [2019a] S. Majhi, D. Ghosh, and J. Kurths, Physical Review E 99, 012308 (2019a).
  * Majhi and Ghosh [2017] S. Majhi and D. Ghosh, Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (2017).
  * Stilwell et al. [2006] D. J. Stilwell, E. M. Bollt, and D. G. Roberson, SIAM Journal on Applied Dynamical Systems 5, 140 (2006).
  * Fujiwara et al. [2011] N. Fujiwara, J. Kurths, and A. Díaz-Guilera, Physical Review E 83, 025101 (2011).
  * Buscarino et al. [2016] A. Buscarino, L. Fortuna, M. Frasca, and S. Frisenna, Chaos: An Interdisciplinary Journal of Nonlinear Science 26 (2016).
  * von Brecht and Uminsky [2016] J. H. von Brecht and D. T. Uminsky, Nonlinearity 30, 225 (2016).
  * Tanaka [2007] D. Tanaka, Physical Review Letters 99, 134103 (2007).
  * O’Keeffe et al. [2017] K. P. O’Keeffe, H. Hong, and S. H. Strogatz, Nature Communications 8, 1504 (2017).
  * Ott and Antonsen [2008a] E. Ott and T. M. Antonsen, Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 037113 (2008a).
  * O’Keeffe et al. [2018] K. P. O’Keeffe, J. H. Evers, and T. Kolokolnikov, Physical Review E 98, 022203 (2018).
  * Peshkov et al. [2022] A. Peshkov, S. McGaffigan, and A. C. Quillen, Soft Matter 18, 1174 (2022).
  * Ott and Antonsen [2008b] E. Ott and T. M. Antonsen, Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 037113 (2008b).
  * Yoon et al. [2022] S. Yoon, K. O’Keeffe, J. Mendes, and A. Goltsev, Physical Review Letters 129, 208002 (2022).
  * Hong et al. [2023] H. Hong, K. P. O’Keeffe, J. S. Lee, and H. Park, Physical Review Research 5, 023105 (2023).
  * Lizarraga and de Aguiar [2020] J. U. Lizarraga and M. A. de Aguiar, Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 053112 (2020).
  * Blum et al. [2024] N. Blum, A. Li, K. O’Keeffe, and O. Kogan, Physical Review E 109, 014205 (2024).
  * O’Keeffe and Hong [2022] K. O’Keeffe and H. Hong, Physical Review E 105, 064208 (2022).
  * Lee et al. [2021] H. K. Lee, K. Yeo, and H. Hong, Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 033134 (2021).
  * Kongni et al. [2023] S. J. Kongni, V. Nguefoue, T. Njougouo, P. Louodop, F. F. Ferreira, R. Tchitnga, and H. A. Cerdeira, Physical Review E 108, 034303 (2023).
  * Sar et al. [2022] G. K. Sar, S. N. Chowdhury, M. Perc, and D. Ghosh, New Journal of Physics 24, 043004 (2022).
  * Sar et al. [2023a] G. K. Sar, D. Ghosh, and K. O’Keeffe, Physical Review E 107, 024215 (2023a).
  * Sar et al. [2023b] G. K. Sar, K. O’Keeffe, and D. Ghosh, Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (2023b).
  * Sar et al. [2024] G. K. Sar, D. Ghosh, and K. O’Keeffe, Phys. Rev. E 109, 044603 (2024).
  * Ghosh et al. [2023] S. Ghosh, G. K. Sar, S. Majhi, and D. Ghosh, Physical Review E 108, 034217 (2023).
  * O’Keeffe and Bettstetter [2019] K. O’Keeffe and C. Bettstetter, Micro-and Nanotechnology Sensors, Systems, and Applications XI 10982, 383 (2019).
  * Sar and Ghosh [2022] G. K. K. Sar and D. Ghosh, Europhysics Letters 139, 53001 (2022).
  * Wolf et al. [1985] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D: Nonlinear Phenomena 16, 285 (1985).
  * Huang et al. [2009] L. Huang, Q. Chen, Y.-C. Lai, and L. M. Pecora, Physical Review E 80, 036204 (2009).
  * [50] See Supplementary Material for the movies depicting the time-evolving emerging states, along with a supporting figure illustrating the invariant nature achieved by utilizing different components of the Rossler system as the color bar.
  * Abrams et al. [2008] D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Physical Review Letters 101, 084103 (2008).
  * Abrams and Strogatz [2004] D. M. Abrams and S. H. Strogatz, Physical Review Letters 93, 174102 (2004).
  * Majhi et al. [2019b] S. Majhi, B. K. Bera, D. Ghosh, and M. Perc, Physics of Life Reviews 28, 100 (2019b).
  * Aihara et al. [2014] I. Aihara, T. Mizumoto, T. Otsuka, H. Awano, K. Nagira, H. G. Okuno, and K. Aihara, Scientific Reports 4, 3891 (2014).
  * Hrabec et al. [2018] A. Hrabec, V. Křižáková, S. Pizzini, J. Sampaio, A. Thiaville, S. Rohart, and J. Vogel, Physical Review Letters 120, 227204 (2018).
  * Chaudhary et al. [2014] K. Chaudhary, J. J. Juárez, Q. Chen, S. Granick, and J. A. Lewis, Soft Matter 10, 1320 (2014).
  * Esam et al. [2021] E.-A. Esam, K. Vana, D. Pianini, L. Frédéric, P. Barry, G. Sona, N. Ingrid, B. Mohamed, T. Sven, et al. (2021).
  * Barciś and Bettstetter [2020] A. Barciś and C. Bettstetter, IEEE Access 8, 218752 (2020).