Swarmalators with delayed interactions Nicholas Blum1Andre Li2Kevin OKeee3and Oleg Kogan1 1California Polytechnic State University San Luis Obispo CA 93407

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Swarmalators with delayed interactions

Nicholas Blum,1Andre Li,2Kevin O’Keeﬀe,3and Oleg Kogan1

1California Polytechnic State University, San Luis Obispo, CA 93407∗

2California Polytechnic State University, East Bay, CA 94542

3Senseable City Lab, Massachusetts Institute of Technology, Cambridge, MA 02139

We investigate the eﬀects of delayed interactions in a population of “swarmalators”, generaliza-

tions of phase oscillators that both synchronize in time and swarm through space. We discover two

steady collective states: a state in which swarmalators are essentially motionless in a disk arranged

in a pseudo-crystalline order, and a boiling state in which the swarmalators again form a disk, but

now the swarmalators near the boundary perform boiling-like convective motions. These states are

reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starﬁsh

embryos.

I. INTRODUCTION

Swarmalators are generalizations of phase oscillators that swarm around in space as they synchronize in time [1].

They are intended as prototypes for the many systems in which sync and swarming co-occur and interact, such as

biological microswimmers [2–9], forced colloids [10–17], magnetic domain walls [18, 19], robotic swarms [20–27] and

embryonic cells of starﬁsh [28] and zebraﬁsh [29].

Research on swarmalators is rising. Tanaka et. al. began the endeavour by introducing a universal model of

chemotactic oscillators with rich dynamics [30–33]. Later O’Keeﬀe, Hong, and Strogatz studied a mobile generalization

of the Kuramoto model [1]. This swarmalator model is currently being further studied. The eﬀects of phase noise [34],

local coupling [22, 35–37], external forcing [38], geometric conﬁnement [39–41], mixed sign interactions [42–44, 44], and

ﬁnite population sizes [45] have been studied. The well posedness of weak and strong solutions to swarmalator models

have also been addressed [46–48]. Reviews and potential application of swarmalators are provided here [49, 50]. Mobile

oscillators, where oscillators’ movements aﬀect their phases but not conversely, have also been studied [51–54].

This paper is about swarmalators with delayed interactions. Delays are, in this context, largely unstudied, although

they occur commonly in Nature and technology. In the case of microswimmers, the inter-swimmer coupling is mediated

by the surrounding ﬂuid and is therefore non-instantaneous. Delays are also an important factor to consider in

embryonic development. Here, they are a well established feature of gene expression and are believed to play a key

role in how cells, organs, and other agglomerations attain their shapes [29]. The authors of [29] write: “ Even though

cell coupling is local, involving cells which are in direct contact, cells require some time to synthesize and transport the

membrane ligands and receptors to their surface. Also, cells need time to integrate received information to its internal

gene expression dynamics, for example, by producing transcription factors. Each of these reaction processes takes a

diﬀerent time to be completed, and these times depend on cell type and cell state.” The continue: “This time delay

might be relevant for cell coupling because what cells acquire at the present time is the information of surrounding

cells some time ago. Thus, inherent delays in cell coupling are key to understanding information ﬂow in biological

tissues.” Time delay is also relevant to robotic swarms where digital communication comes with unavoidable lags and

may aﬀect both communication of the spatial or internal state of robots.

In short, delays are important for a broad class of swarmalators; in some cases delay aﬀects the communication

of internal state of particles, and in others it aﬀects the communication of both the internal and spatial state. He we

aim to take a ﬁrst step in understanding delay-coupled swarmalators theoretically, so we will focus on delays in just

the internal state of the original swarmalator model [1]. This model is a natural ﬁrst case-study because it captures

the behaviors of many natural swarmalators [4, 6, 12] yet is simple enough to analyze.

∗Electronic address: kevin.p.okeeﬀe@gmail.com, okogan@calpoly.edu

arXiv:2210.11417v1 [nlin.AO] 20 Oct 2022

II. SWARMALATORS WITH DELAY

We will introduce time delay into the swarmalator model proposed by O’Keeﬀe, Hong, and Strogatz (OHS) [1].

The equations describing the dynamics of such delayed swarmalators read

xi=vi+1

j6=i"xj−xi

|xj−xi|1 + Jcos(θj(t−τ)−θi(t))−xj−xi

|xj−xi|2#(1)

θi=ωi+K

j6=i

sin(θj(t−τ)−θi(t))

|xj−xi|.(2)

Here Nis the number of particles. All the spatial coordinates are evaluated without delay, at time t. The ﬁrst term in

Eq. (1) represents attraction - it causes the velocity of particle jto be directed towards particle iand vice versa. The

parameter Jcontrols the tendency of this attractive term to depend on internal phases; when J= 0 the attraction is

independent of internal phases. In order for the ﬁrst term to be attractive, |J|must be less than 1. The attractive

term has a magnitude that’s independent of the particle separation, i.e. it represents an all-to-all attraction (which

could also be called mean ﬁeld attraction) that is commonly used. For example, the same is done with phases in the

Kuramoto model.

The second term in Eq. (1) is a short-range repulsion - it causes the velocity of particle ito be directed away from

particle j, but this term decays away with distance. It is intended to prevent clumping of all particles at one point.

The form of the model is motivated in [1].

Eq. (2) describes dynamics of internal phases. If θjis lagging behind θi, the i−jterm in the sum contributes

to the velocity of θithat tends to bring θicloser to θj. In other words, oscillator j“pulls” the phase of oscillator i

closer to it. This is the usual Kuramoto interaction. The parameter Kis an overall scaling factor for the strength of

phase attraction (positive K) or repulsion (negative K). Here the strength of the interaction depends on the distance

- oscillators that are closer in physical space will experience stronger tendency to align or counter-align their phases

with their close neighbors.

Thus, the picture is this: the phase dynamics aﬀects the strength of spatial attraction, while the spatial position

of particles aﬀects the strength of phase interaction.

The new addition in this work is the time delay in phase dependence. The particle iat time tresponds to the

phase of the particle jas it was a time τago - at time t−τ. In this work, we only add this eﬀect to the phase

dynamics. Physically, the phase represents the internal state - for example, the phase of a gene expression cycle.

Communication of such internal variable often takes place via chemical signals, which is a type of interaction that is

much slower than the interaction that communicates positions of objects in physical space [29].

We investigate the role of delay in this model. OHS discovered that the system can be found in one of the ﬁve

collective states in the absence of a delay. In the present paper we work mostly in the region of (J, K) space that in

this delay-free model corresponds to what OHS called the “active phase wave”. The swarmalators in this state move

in circles around the center of the annulus-shaped cluster - some move clockwise, and some counterclockwise, while

the internal phases change as they move around the center of the annulus. It is in this region of the (J, K) space is

where we found the interesting collective behavior induced by a delay. It is possible that other new behaviors take

place in other regions of (J, K) space, but this would be a subject for a future work.

This plan of this paper is as follows. In Section III numerical results are presented. Two new collective states

are presented in Subsections III A and III B respectively. Subsection III C describes the dynamical phase transition

between these states, as well as properties of long-time behavior of transient oscillations. Theoretical treatment is

presented Section IV. We compare theoretical predictions with numerical results there. We summarize in Section

III. TWO NEW TYPES OF COLLECTIVE BEHAVIOR - NUMERICAL RESULTS

We begin by presenting a phenomenon that occurs at suﬃciently large τ. The meaning of “suﬃciently large” will

be made precise in Section III C.

A. Pseudo-static quasi-crystal

We placed particles at random positions xwithin a square with corners at (±1,±1), and assigned initial phases

θuniformly at random from [0,2π]. Following an initial complicated transient, when particles quickly organize into

a nearly-circular disk, it enters into a coherent, synchronized collective motion characterized by decaying oscillations

of the radius. Fig. 1 demonstrates velocity vectors of particles at two snapshots in time - one during expansion, and

another during contraction - while Fig. 2(a) demonstrates oscillations of the average radius of the cluster, R(t) =

NPN

n=1 Rn(t). Naturally, the average velocity and the average speed of particles also exhibit oscillations. Fig. 2(b)

depicts the average speed |v|(t) = 1

NPN

n=1 |v|n(t).

-4 -3 -2 -1 0 1 2 3

-3

-2

-1

-3 -2 -1 0 1 2

-2

-1

𝑦𝑦

𝑥 𝑥

𝑡 = 73.17 𝑡 = 79.11

FIG. 1: Velocity vectors and particle positions at two instants of time. These times are at latter stages of oscillations (see

Fig. 2 for corresponding speeds and the average radius (not maximum radius) at these times). (N, J, K, τ ) = (100,1,−0.75,8).

0 20 40 60 80 100 120

time

0.6

0.8

1.2

1.4

1.6

1.8

2.2

2.4

Average Radius

Average radius, !

𝑅 𝑡

Average speed, ̅

𝑣 𝑡

time, 𝑡time, 𝑡

0 20 40 60 80 100 120

time

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Average Speed

(a) (b)

FIG. 2: (a) R(t). (b) v(t). The parameters are (N, J, K, τ ) = (100,1,−0.75,8)

We will refer to this collective behavior as “breathing” of the cluster. Because the oscillations during the breathing

decay, this stage of the system dynamics can be thought of as the longer portion of the transient. At earlier times, the

transient is more complicated, and does not result in breathing motion. The dynamics at very earlier times is complex.

The ﬁrst few breaths are also complicated - they are not purely radial, and can be accompanied by other types of

dynamics - including particle rearrangements and time-averaged expansion of the cluster (this is why the speed doesn’t

go to zero at maximum and minimum radius). Eventually, breathing motion becomes simpler - it consists of only

radial oscillations around the inﬁnite-time equilibrium radius value, and there are no particle rearrangements in this

latter stage. In Fig. 2 this happens around t= 80.

After the breathing transient dies down, it appears that a static pseudo-crystal is formed. Examples of these

pseudo-crystals are shown in Fig. 3 for three system sizes. Note that the radii of these pseudo-crystals depend on

N, i.e. the radii of the three clusters in Fig. 3 are not equal (see Fig. 7) - they have been scaled in Fig. 3. But the

inter-particle spacing relative to the cluster radius clearly decreases with larger N.

-3 -2 -1 0 1 2 3 4 5 6

-5

-4

-3

-2

-1

491.0188

-3 -2 -1 0 1 2 3 4 5 6 7

-4

-3

-2

-1

693.6204

𝑁 = 100 𝑁 = 500 𝑁 = 1000

-2 -1 0 1 2 3 4 5 6 7

-5

-4

-3

-2

-1

988.9714

FIG. 3: N= 100 (left), N= 500 (center) and N= 1000 (right) particle systems after the breathing transient. The values

of τwere 1.5τc, where τcis a critical delay at which the long-time behavior becomes boiling motion, as described in the next

Subsection (see Fig. 16 for values of τcat diﬀerent N). Here J= 1, K=−0.7.

However, a careful examination of the tail of R(t) plot demonstrates that there is some residual motion left. This

is seen on the logarithmic plot below.

0 100 200 300 400 500 600 700 800 900 1000 1100

time

-12

-10

-8

-6

-4

-2

Average Speed

Ln(average speed)

FIG. 4: (N, J, K, τ ) = (100,1,−0.75,8). Natural logarithm of the average speed versus time. A transition from decaying

oscillatory motion to creeping motion is very clear.

Around t= 150 we clearly see that breathing motion gives way to a diﬀerent type of motion with very small

velocities (we can call it creeping motion). The magnitude of these velocities continues to decay with time, but much

slower than during the breathing stage. We can deﬁne the transition to this creeping motion as an intersection of the

straight-line ﬁt on the logarithmic plot to the envelope of |v|(t) during breathing (red dashed line in Fig. 4) with the

function itself. There is no single deﬁning feature of this post-breathing velocity pattern - its character changes with

time and with respect to parameters. The only deﬁnite feature of these post-breathing creeping dynamics is that it

is rather disordered. We give an example of such a pattern in Fig. 5. Additional examples can be found in Figs. 29

and 30 in Appendix A.

-6 -5 -4 -3 -2 -1 0 1 2 3

-5

-4

-3

-2

-1

4988.1544

FIG. 5: Creeping particle motion for t≈5000, which is approximately 10 times the time at which the breathing ceased, as

deﬁned above. Here N= 400, and the values of τwas 1.5τc(see Fig. 16 for values of τc), J= 1, and K=−0.7. The vectors

have been automatically re-scaled to be visible. Thus, while the arrows appear to have the length comparable to those in Fig. 1,

this is because of the up-scaling.

There is no indication, given the range of our computational capabilities, that the post-breathing creeping motion

is a ﬁnite size eﬀect. We come to this conclusion by measuring the dependence of the average speed on Nat three

instants of time that follow the breathing. In Fig. 6 we plot the average speed versus Nmeasured at three instants

in time: the ﬁrst, labeled “time 1” is immediately after the end of the breathing motion as just deﬁned. The second,

or “time 2” is around 500 time units after end of the breathing motion, and the third, or “time 3” is 1000 time units

after the end of the breathing motion. The data in Fig. 6 suggests that there is no indication (at least in the range

of Ns that were studied) that the long-time average speed decreases with increasing system size.

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SwarmalatorswithdelayedinteractionsNicholasBlum,1AndreLi,2KevinO'Keee,3andOlegKogan11CaliforniaPolytechnicStateUniversity,SanLuisObispo,CA934072CaliforniaPolytechnicStateUniversity,EastBay,CA945423SenseableCityLab,MassachusettsInstituteofTechnology,Cambridge,MA02139Weinvestigatetheeectsofdelayedi...

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Swarmalators with delayed interactions Nicholas Blum1Andre Li2Kevin OKeee3and Oleg Kogan1 1California Polytechnic State University San Luis Obispo CA 93407

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