A review of swarmalators and their potential in bio-inspired computing
Kevin O'Keeffe Christian Bettstetter







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

Institute of Networked and Embedded Systems, University of Klagenfurt, 9020, Austria 



From fireflies to heart cells, many systems in Nature show the remarkable ability to spontaneously fall into synchrony. By imitating Nature's success at self-synchronizing, scientists have designed cost-effective methods to achieve synchrony in the lab, with applications ranging from wireless sensor networks to radio transmission. A similar story has occurred in the study of swarms, where inspiration from the behavior flocks of birds and schools of fish has led to 'low-footprint' algorithms for multi-robot systems. Here, we continue this 'bio-inspired' tradition, by speculating on the technological benefit of fusing swarming with synchronization. The subject of recent theoretical work, minimal models of so-called 'swarmalator' systems exhibit rich spatiotemporal patterns, hinting at utility in 'bottom-up' robotic swarms. We review the theoretical work on swarmalators, identify possible realizations in Nature, and discuss their potential applications in technology.
 
 
 
 
 

A review of swarmalators and their potential in bio-inspired computing
    Christian Bettstetter
    December 30, 2023
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§ INTRODUCTION

In 1967 Winfree proposed a model for coupled oscillators that spontaneously synchronize <cit.>. Beyond a critical coupling strength, the oscillators overcome the disordering effect of the dissimilarities in their natural frequencies and spontaneously lock their cycles. Kuramoto later simplified Winfree's model and solved it exactly <cit.>. Since then, the study of sync has matured into a vibrant field <cit.>. On the theoretical side, theorists have twisted Kuramoto's model in various ways resulting in rich phenomena like glassy behavior <cit.> and chimeras states <cit.>. On the applied side, coupled oscillators have found use in neurobiology <cit.>, cardiac dynamics <cit.>, and the bunching of school buses <cit.>. An interesting application of sync is the design of `bio-inspired' algorithms. Here, by aping the simplicity of coupled oscillator models, researchers have devised resource-efficient algorithms to procure synchrony in the lab, useful for networked computing and robotics <cit.>.


A story with many parallels to the sync story has evolved in the study of swarms. In 1995 Vicsek proposed a simple model of swarming agents <cit.>, which – like the sync transition of coupled oscillators – showed a transition from disorder to order: beyond a critical coupling strength, the agents switched from a gas-like, incoherent state to one in which the agents moved as a coherent flock. The novelty of the out-of-equilibrium nature of the flocking transition – the system is out of equilibrium since agents constantly consume energy to propel themselves – piqued the minds of physicists and other theorists, which in turn helped give rise to the field of active matter <cit.>, a field perhaps more vibrant than the field of synchronization. In a final mirroring of the sync story, the bio-inspired community has also mimicked the minimalism of Vicsek's and other models of swarming  to design novel algorithms for optimization <cit.> and robotic swarms <cit.>.


These stories demonstrate swarming and synchronization are intimately related. In a sense, the two effects are `spatiotemporal opposites': in synchronization the units self-organize in time but not in space; in swarming the units self-organize in space but not in time. Given this conceptual twin­ship, it is natural to wonder about the possibility of units which can self-organize in both space and time – that is, to wonder how swarming and synchronization might interact. And more pertinent to this work, to wonder how a mix of swarming and synchronization could be useful in technology.


Several researchers have started to address these questions by analyzing systems that both sync and swarm. Von Brecht and Unimsky have generalized swarming particles by endowing them with an internal polarization vector <cit.>, equivalent to an oscillator's phase. Others have attacked the problem from the other direction, by considering synchronizing oscillators able to move around in space <cit.>. In these studies, however, the oscillators' movements affect their phase dynamics, but not the other way around; thus, the interaction between swarming and synchronization is only one-way. Two-way interaction between swarming and synchronization has also been considered: The pioneering work is the Iwasa-Tanaka model of chemotactic oscillators <cit.>, i.e., oscillators which interact through a background diffusing chemical. More recent works were carried out by Starnini et al <cit.>, Belovs et al <cit.>, and O'Keeffe et al <cit.> who proposed `bottom-up' toy models without reference to a background medium. O'Keeffe et al called the elements of their systems `swarmalators', to capture their twin identities as swarming oscillators, and to distinguish them from the mobile oscillators mentioned above for which the coupling between swarming and synchronization is unidirectional. 





This paper reviews research on swarmalators and other systems that mix swarming and synchronization. Our main motivation is to identify if the interplay of sync and swarming can be useful for bio-inspired computing and related engineering domains. We outline the theoretical work on swarmalators, experimental realizations, and finally conjecture on their technological utility.







§ MODELS COMBINING SWARMING AND SYNCHRONIZATION

To combine synchronization with swarming, we first define what we mean by swarming. While to our knowledge there is no universally agreed on definition, swarming systems typically have at least one of two key features: (i) aggregation, arising from a balance between the units' mutual attraction and repulsion and (ii) alignment, referring to the units' tendency to align their orientation in space and move in a flock. Synchronization can thus be combined with either aggregation alone, alignment alone, or with both aggregation and alignment. In what follows, we present classes of model based on these three ways to combine sync and swarming.







 §.§ Aggregation and Synchronization

The paradigmatic model of biological aggregation has the form

    𝐱̇_i = 1/N∑_j ≠ i^N 𝐈_ att( 𝐱_𝐣 - 𝐱_𝐢) - 𝐈_ rep( 𝐱_𝐣 - 𝐱_𝐢 ),

where 𝐱_i ∈ℝ^d (usually with d ≤ 3) is the i-th particle's position, 𝐈_ att captures the attraction between particles, and 𝐈_ rep captures the repulsion between them. The competition between  𝐈_ att and 𝐈_ rep gives rise to congregations of particles with sharp boundaries, in accordance with many biological systems (see <cit.> for a review).

The paradigmatic model in synchronization is the Kuramoto model <cit.>:

    θ̇_̇i̇ = ω_i + K/N∑_j ≠ i^N sin(θ_j - θ_i).

Here θ_i ∈ S^1 and ω_i are the phase and natural frequency of the i-th oscillator. The sine term captures the oscillators' coupling, where K > 0 means oscillators tend to synchronize, and K < 0 means oscillators tend to desynchronize. As mentioned, beyond a critical coupling strength K_c, a fraction of oscillators overcome the disordering effects imposed by their distributed natural frequencies ω_i and spontaneously synchronize.

To study the co-action of synchronization and aggregation, a natural strategy would be to stitch the aggregation model (<ref>) and the Kuramoto model (<ref>) together. This was the approach taken by O'Keeffe et al <cit.> who proposed the following swarmalator model:

    𝐱̇_i = 1/N∑_ j ≠ i^N [ 𝐱_j - 𝐱_i/|𝐱_j - 𝐱_i|( 1 + J cos(θ_j - θ_i)  ) -   𝐱_j - 𝐱_i/ | 𝐱_j - 𝐱_i|^2]
       θ̇_̇i̇ = ω_i + K/N∑_j ≠ i^N sin(θ_j - θ_i)/ |𝐱_j - 𝐱_i|  .

Equation (<ref>) models phase-dependent aggregation and Equation (<ref>) models position-dependent synchronization. The interaction between the space and phase dynamics is captured by the term  1 + J cos(θ_j - θ_i). If J > 0, “like attracts like”: swarmalators are preferentially attracted to other swarmalators with the same phase, while J < 0 indicates the opposite. For simplicity the authors considered identical swarmalators ω_i = ω, and by a change of reference they set ω = 0. 

The swarmalator model exhibits five long-term collective states. Figure <ref> showcases these states as scatter plots in the (x,y) plane, where swarmalators are represented by dots and the color of each dot represents the swarmalator's phase θ (color, recall, can be mapped to S^1 and so can be used to represent swarmalators' phases). The parameter dependence of these states are encapsulated in the phase diagram shown in Figure <ref>. The first three states, named the static sync, static async, and static phase wave, are – as their names suggest – static in the sense that the individual swarmalators are stationary in both position and phase. This stationarity allows the density of these swarmalators ρ(𝐱, θ) in these states to be constructed explicitly (the density ρ(𝐱, θ) is interpretted in the Eulerian sense, so that ρ(𝐱 + 𝐝𝐱, θ + d θ) gives the fraction of swarmalators with positions between 𝐱 and 𝐱 + d 𝐱 and phases between θ and θ + d θ). In the remaining splintered phase wave and active phase waves states swarmalators are no longer stationary. In the splintered phase wave state swarmalators execute small oscillation in both space and phase within each cluster. In the active phase wave the swarmalators split into counter-rotating groups – in both space and phase –  so that ⟨θ̇_̇i̇⟩ = ⟨ϕ̇_̇i̇⟩ = 0, where ϕ_i = arctan(y_i / x_i) is the spatial angle of the i-th oscillator and angle brackets denote population averages. The conservation of these two quantities follows from the governing equations; the pairwise terms are odd and thus cancel under summation. Three-dimensional analogues of the five collective states were also reported.

The stability properties of the static async state are unusual. Via a linear stability analysis in density space, an integral equation for the eigenvalues λ was derived. Yet numerical solutions to this integral equation, in the parameter regime where the static async state should be stable, produced a leading eigenvalue so small in magnitude that its sign could not be determined reliably. Thus, the stability of the state could not be analytically confirmed. There is however a parameter value at which the magnitude of λ increases sharply, which was used to derive a pseudo critical parameter value K_c ≈ - 1.2 J marking the apparent (i.e. as observed in simulations) destabilization of the state. Nevertheless, the true stability of the static async state remains a puzzle.



Some extensions of the work in <cit.> have been carried out. One addresses the conspicuously empty upper right quadrant in the (J,K) parameter plane in Figure <ref>, where only the trivial static sync state appears. By adding phase noise to (<ref>), Hong discovered <cit.> the active phase wave exists for K > 0. Unexpectedly, the splintered phase wave was not observed. O'Keeffe, Evers, and Kolokolnikov <cit.> have extended (<ref>) by allowing phase similarity to affect the spatial repulsion term, as well as the spatial attraction term (i.e. they multiply the second term in (<ref>) by a term 1+J_2 cos(θ_j - θ_i)). This led to the emergence of ring states, an example of which is depicted in Figure <ref>. They constructed and analyzed the stability of these ring states explicitly for a population of given size N. Analytic results for arbitrary N are potentially useful for robotic swarms, which presumably are realized in the small N limit. Another offshoot of this analysis was a heuristic method to predict the number of clusters which form in the splintered phase wave state; they viewed each cluster of synchronized swarmalators as one giant swarmalator which let them re-imagine the splintered phase wave state as a ring, allowing them to leverage their analysis. A precise description of the number of clusters formed is an open problem. 

Iwasa-Tanaka model. Iwasa and Tanaka proposed and studied a `swarm-oscillator' model in a series of papers <cit.>. The inspiration for their work comes from chemotactic oscillators, i.e., oscillators moving around in a diffusing chemical which mediates their interactions. They began with the general model

    𝐗̇_i(t)    = f(𝐗_i) + k g(S(𝐫_i,t)) 
    
        m r̈_̈ï(t)    = - γṙ_i - σ(𝐗_i) ∇ S 
    τ∂_τ S(𝐫, t)    = -S + d ∇^2 S + ∑_i h(𝐗) δ(𝐫 - 𝐫_i),

where 𝐗_i represents the internal state (which will later be identified as a phase), 𝐫 represents the position of the i-th oscillator, and S represents the concentration of the background chemical. By means of a center manifold calculation and a phase reduction technique they derived the simpler equations

    ψ̇_i(t)    =  ∑_j ≠ i e^- |𝐑_ji|sin( Ψ_ji - α |𝐑_ji| - c_1 ) 
    𝐫̇_i(t)    = c_3 ∑_j ≠ i𝐑̂_ji e^- |R_ji|sin( Ψ_ji - α |𝐑_ji| - c_2 )

where 𝐑_ji = 𝐑_j - 𝐑_i, Ψ_ji = ψ_j - ψ_i, and ψ_i is the i-th oscillator's phase. We call this Iwasa-Tanaka model. 
Notice the space-phase coupling in this model is somewhat peculiar; in contrast to the swarmalator model given by (<ref>) and (<ref>) the relative position 𝐑_ji and relative phase Φ_ji appear inside the sine terms in both the ṙ_i and ψ̇_i equations. Another difference between the two models is that 𝐫̇_i in (<ref>) has no hardshell repulsion term, which means the oscillators can occupy the same position in space. 

The Iwasa-Tanaka model has rich collective states. An exhaustive catalogue of these states with respect to the model's four parameters is an ongoing effort <cit.>. Highlights include a family of clustered states <cit.> in which swarmalators collect in synchronous groups. The spatial distributions of these groups depend on their phase, similar to the splintered phase wave (Figure <ref>). The authors speculate this phase clustering is reminiscent of the chemotactic cell sorting during biological development <cit.>. The Iwasa-Tanaka model also produces ring states <cit.>, as well as an interesting `juggling' state <cit.> in which the population forms a “rotating triangular structure whose corers appear to `catch' and `throw' individual elements” – in other words, the population juggles the elements around the corners of a triangle (see Figure 1 in <cit.>). Aside from theoretical novelty, this juggling could conceivably be exploited in robotic swarms, potentially allowing some form of relay between the elements.






 §.§ Alignment and synchronization

In our proposed taxonomy, systems that combine alignment and synchronization are characterized by an internal phase θ and an orientation β without a dynamic spatial degree of freedom. In other words, the particles' position 𝐱 might affect their θ and β dynamics, but 𝐱 itself does not evolve in time. Although units characterized by just a phase θ and orientation β might seem odd, Leon and Liverpool studied a system with units which meet these criteria: a class of soft active fluids which constitute a `new type of nonequilibrium soft matter – a space-time liquid crystal' <cit.>. They developed a phenomenological theory of these soft fluids which allowed them to derive dynamical equations for order parameters quantifying the orientation order, phase order, and orientation-phase order. These revealed collective states which maximize each of these order parameters: aligned states with orientational order but no phase order, sync'd states with phase order but no orientational order, and states with both phase and orientational order. They were able to partially analyze these states, and conjectured the states could be realized in protein-filaments mixtures, such as cell cytoskeletons <cit.> or tissue-forming cells <cit.>. 





 §.§ Alignment, aggregation, and synchronization

Systems with units that align, aggregate, and synchronize – and therefore have dynamic state variables 𝐱, θ, β – are the least well studied. A small study was carried out in <cit.>, with the aim of checking if the swarmalator states reported are generic, i.e., robust to the inclusion of alignment dynamics (it was found they were). Beyond this preliminary study, the space of possible behaviors arising from alignment, aggregation, and synchronization is largely unexplored.







 §.§ Alignment and aggregation

While not strictly within our proposed taxonomy of swarmalator systems, we give a brief review of studies on aggregation and alignment. We do this because alignment can be viewed as a type of synchronization, where instead of units adjusting to a common phase θ, units instead adjust a common orientation β.  Or put another way, because an internal phase θ and orientation β are formally equivalent – both being circular variables – particles aligning their orientations is analogous to oscillators aligning their phases. 

Vicsek set the paradigm of this class of models with a beautiful and simple model:

    𝐫_i(t + δ t)= 𝐫_i(t) + (νδ t) n̂
       β_i(t + δ t) = ⟨β_j ⟩_|r_j - r_i|<r + η_i(t) ,

where n̂ = (cos(β), sin(β)), η_i(t) is a white noise, ν is the particles' speed, and r is the coupling range. For sufficiently strong noise this model exhibits a flocking transition, where particles switch from erratic incoherent movements to moving in a unidirectional flock.

Leibchen and Levis <cit.> modified the Vicsek model as follows:

    𝐫̇_i= νn⃗_i 
       β̇_̇i̇ = ω_i + K/π R^2∑_j ∈∂_i^N sin(θ_j - θ_i) + √(2D)η_i,

where ∂_i is the set of neighbours of the i-th particle, K is the phase coupling, and D is the noise strength. The key new feature is the ω_i term, signifying the particles have an intrinsic rotation. They find three collective states: a disordered state in which neither spatial order nor phase order exists, a chiral state in which macroscopic `droplets' of synchronized oscillators form in a sea of otherwise desynchronized oscillators, and finally a mutual flocking state in which oscillators with `opposite chirality cooperate to move coherently at a relative angle, forming non-rotating flocks'. The theoretical novelty of the latter state is that – in contrast to regular, non-moving, locally-coupled oscillators coupled in two or three dimensions – long-range synchronous clusters are observed, despite the fact that the oscillators are only locally coupled.






§ SWARMALATORS IN THE REAL-WORLD


To our knowledge, there are just two works which unequivocally realize swarmalators in the real world – by this we mean, precisely, a real-world system in which a bidirectional space-phase coupling is unambiguously exhibited. These works are: 

Swarmalatorbots. Bettstetter et al first realized the collective states of the swarmalator model (Figure <ref>) in the lab using small robots <cit.>. They programmed `swarmalatorbots' whose governing equations were derived from the swarmalator model (<ref>), (<ref>).

Magnetic domain walls. To our knowledge, the first realization of a natural swarmalator system was found by Hrabec et al when studying the magnetic domain walls <cit.>. Ordinarily, domain walls are described by a single spatial degree of freedom 𝐱. But as the authors note, the dynamics on the walls also depend on their internal structure. The authors minimally describe this internal state by a one dimensional polarization angle of the internal magnetic field θ. This allows the wall to be viewed as a point particle with position 𝐱 and phase θ – to be viewed as a swarmalator <cit.>. An experimental study of  coupling between two domain walls – usually just one wall is studied – revealed the walls can synchronize, which in turn affects the walls' velocity. Richer space-phase dynamics, such as families of Lissajous curves, are also reported. 



***



Beyond swarmalatorbots and magnetic domain walls, there are many systems in which a bidirectional coupling between swarming and synchronization might exist. We list these candidate swarmalator systems below:

Myxobacteria. Myxobacteria are bacteria commonly found in soil, and can produce interesting collective effects <cit.>. The phase variable of myxobacteria characterizes the internal growth cycle of the cell. Groups of cells interact through cell-to-cell contact, theorized to provide a channel through which swarming and synchronization can couple. Populations of myxobacteria can exhibit a `ripple phase', in which complex patterns of waves propagate through population <cit.>. In contrast to other wave phenomena in biological systems, such as the well-studied Dictyostelium discoideum, these rippling waves do not annihilate on collision. In <cit.> a Fokker-Planck type equation was used to analyze rippling waves. Realizing them in a microscopic swarmalator system is an open problem.

Biological microswimmers. `Microswimmers' is an umbrella term for self-propelled micro­organisms confined to fluids, such as celia, bacteria, and sperm <cit.>. Groups of micro­swimmers show swarming behavior as a result of cooperative goal seeking, such as searching for food or light. They can also synchronize: Here the phase variable is associated with the rhythmic beating of swimmers' tails which – through hydrodynamic interactions – can synchronize with the beatings of others swimmers' tails. Whether or not this hydrodynamics provides a bidirectional coupling between sync and swarming – as required of swarmalators – is unclear, but to us seems plausible. Researchers have developed models of sperm under this assumption <cit.> and found clusters of synchronized sperm consistent with real data <cit.>. Vortex arrays of sperm have also been reported <cit.>. Here, sperm self-organize into subgroups arranged in a lattice; within each subgroup, the sperm move in a vortex, wherein a correlation between the their angular velocity and their phase velocity is realized, reminiscent of the splintered phase wave state (Figure <ref>). Simulations of realistic models have been carried out <cit.> which show other interesting phenomena.  

Japanese tree frogs. During mating season, male Japanese tree frogs attract females by croaking rhythmically. The croaking of neighbouring frogs tend to anti-synchronize due to a precedence effect: croaking shortly after a rival makes a frog look less dominant. Researchers have theorized that this competition leads to mutual interaction between frogs' space and phase dynamics, coupling sync and swarming. Models based on these assumption produce ring-like states where frogs arrange themselves on the borders of fields with interesting space-phase patterns, some of which are consistent with data on real tree-frogs' behavior collected in the wild.

Magnetic colloids. Similar to magnetic domain walls, a colloid's constituent particles can synchronize the orientation of their magnetic dipole vector when sufficiently close. When in solution, they are free to move around in space, creating a feedback loop between their space and phase dynamics. In ferromagnetic colloids confined to liquid-liquid interfaces, Snezhko and Aranson <cit.> found this interaction leads to the formation of `asters' – star-like arrangements of particles whose spatial angles correlate with the orientations of their magnetic dipole vectors, equivalent to the static phase wave (Figure <ref>). Yan et al explored how synchronization is useful in colloids of Janus particles <cit.>. Janus particles are micrometer-sized spheres with one hemisphere covered with nickle which gives them non-standard magnetic properties. In particular when subject to a precessing magnetic field, they oscillate about their centers of mass. This oscillation creates a coupling between particles, giving rise to `synchronization-selected' self-assembly. For example, zig-zag chains of particles and microtubes of synchronized particles can be realized.





§ SWARMALATORS IN BIO-INSPIRED COMPUTING

The computer engineering community has been inspired by coupled oscillators to develop new techniques for synchronization in communication and sensor networks <cit.>. Here, instead of  Kuramoto-type oscillators (<ref>), models of pulse-coupled oscillators have been borrowed. As the name suggests, pulse-coupled oscillators communicate by exchanging short signals; this time-discrete variant of coupling is a more natural fit for application in technology, where smooth, continuous coupling – as exemplified by Kuramoto oscillators –  is costly to achieve.
The canonical model of pulse-coupled oscillators is the Peskin model <cit.>, defined by 
    ẋ_̇i̇ = S_0 - γ x_i

with S_0, γ > 0 and x_i is a voltage like state variable for the i-th oscillator. When the oscillators' voltage reach a threshold value they (i) fire a pulse which instantaneously raises the voltage of all the other oscillators (ii) reset their voltage to zero, along with any other oscillator whose voltage was raised above the threshold on account of receiving a pulse. The synchronization properties of the Peskin model are well-studied <cit.>. More recently, estimates for the convergence speed have been derived <cit.>.
 
The simplicity, distributed nature, adaptability, and scalability of pulse-coupled oscillators make them attractive from an engineering point of view, where temporal coordination is often a goal. For example, synchronization is required for many tasks in different layers of computing and communications systems <cit.> such as in the alignment of transmission slots for efficient medium access <cit.>, scheduling of sleep cycles for energy efficiency <cit.>, and coordination of sensor readings to capture a scene from different perspectives <cit.>. It is the relative synchrony of the networked units, not necessarily the absolute time, that is relevant here. The synchronization precision required is determined by the specific task. The achievable precision depends on the environment and hardware, and is influenced by deviations in delays and phase rates. Experiments in laboratory environments show that the precision is in the order of hundred microseconds with low-cost programmable sensor platforms <cit.> and a few microseconds with field-programmable gate array (FPGA)-based radio boards <cit.>. Further use cases can be found in acoustics (synchronizing multiple loudspeakers) and energy systems (synchronizing decentralized grids <cit.>), to give two examples. 

Synchronization is also important in robotics in order to perform coordinated movements – and this is where temporal and spatial coordination unite. As mentioned, Bettstetter and colleagues <cit.> extended the swarmalator model <cit.> so that it could be applied to mobile robots. They implemented the extended model in the Robot Operating System 2 (ROS 2) and experimentally demonstrated that the space-time patterns achieved in theory (Figure <ref>) can be reproduced in the real world. Beyond realizing these specific states, we conjecture that swarmalator-type models will enable novel self-assembly procedures in other robotics groups, in turn enabling collaborative actions in monitoring, exploration, and manipulation. As envisioned in <cit.>, underwater robots could be designed, which – in imitation of biological microswimmers – could both swim in formation and synchronize their fin movements, thereby enhancing their functional capabilities. 



More speculatively, the swarmalator concept could be useful in the design of autonomous transport systems; for example, when multiple vehicles driving in a convoy have to avoid collisions with other convoys due to crossings and for the purpose of overtaking. The model could also enable self-configuring distributed antenna arrays (or loudspeakers) in which multiple antenna elements (or sound sources) automatically arrange their positions and orientations to create specific radiation patterns used to send radio (audio) signals in a synchronous way. Another promising application field is the planning and replanning of processes in factories, where products and machines must follow a certain space-time order and where standard optimization methods reach their limits due to the high system complexity. A final, more playful, application is in art: Artistic aerial light shows created by drone swarms running the swarmalator model produce charming visual displays. 











§ DISCUSSION

The implicit strategy in the theoretical works we  reviewed is: given a model, which spatiotemporal patterns emerge? Future work could consider the inverse strategy: given a desired spatiotemporal pattern, which form should the model take on? – a question often considered in engineering contexts. From a more general perspective, one has to design the local rules and interactions that guide a self-organizing system toward a desired global state <cit.>. Von Brecht et al have studied this inverse question for swarming systems <cit.>; perhaps their tools could be extended to swarmalators. 

Future work could also extend the reviewed models, which – recall – were designed to be as minimal as possible. A wealth of synchronization phenomena have been found by adorning the Kuramoto model with new features like mixed-sign coupling <cit.>, non-local coupling <cit.>, and delayed interactions <cit.>. Equipping swarmalators with these features would likely cure the poverty of phenomena in the first, third, and fourth quadrants of the (J,K) plane (Figure <ref>)  and produce new dynamics in Iswasa-tanaka and Vicesek type models, too. Phase models more sophisticated than the Kuramoto model could also be explored; the Winfree model <cit.> or the newly introduced Janus oscillators <cit.> would be exciting to experiment with. Swarmalators with discrete and spatially local coupling also merit study and are useful in computer and communication systems. Stitching the Peskin model (<ref>) to the aggregation equation (<ref>) or the Vicsek model (<ref>) seems the natural way to do this. Continuing to study finite systems of swarmalators <cit.> is also pertinent, since many robotic systems lie in the low N regime. And finally, beyond richer phase models, the spatial dynamics of swarmalators could also be generalized. Even within the framework of the aggregation equation (<ref>), a menagerie of spatial patterns have been catalogued <cit.>. Swarmalator counterparts to these states would be exciting to explore. 

We hope to have outlined that swarmalators have great potential in computing and other fields of technology. Here we see swarmalators only as a case study in a broader class of systems with large technological utility: systems whose units have both spatial and internal degrees of freedom. The one-dimensional phase θ of a swarmalator is perhaps the simplest instance of an internal degree of freedom, which can more generally be represented by a feature vector 𝐟. This vector could describe a particle's (three dimensional) magnetic or electric field; a person's political affiliation, mood, or health state; a bacteria's phase in a non-circular developmental cycle; and the mode of a robot, machine, or vehicle. The collective states arising when particles' features 𝐟 and positions 𝐱 defines a wide, scarcely investigated area. We anticipate an exploration of this area will bear rich fruit in both Nature and technology. 






§ ACKNOWLEDGEMENTS

The work of C. Bettstetter is supported by the Austrian Science Fund (FWF) [grant P30012] and the Popper Kolleg Networked Autonomous Aerial Vehicles.


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96
natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL 


[Winfree(2001)]winfree2001geometry
authorA. T. Winfree,
  titleThe geometry of biological time,
  vol. volume12 (publisherSpringer Science & Business
  Media, year2001).

[Kuramoto(1975)]kuramoto1975international
authorY. Kuramoto,
  journalLecture Notes in Physics volume30,
  pages420 (year1975).

[Strogatz(2004)]strogatz2004sync
authorS. Strogatz,
  titleSync: The emerging science of spontaneous order
  (publisherPenguin UK, year2004).

[Pikovsky et al.(2003)Pikovsky,
  Rosenblum, and Kurths]pikovsky2003synchronization
authorA. Pikovsky,
  authorM. Rosenblum,
  and authorJ. Kurths,
  titleSynchronization: a universal concept in nonlinear
  sciences, vol. volume12 (publisherCambridge
  University Press, year2003).

[Acebrón et al.(2005)Acebrón,
  Bonilla, Vicente, Ritort, and Spigler]acebron2005kuramoto
authorJ. A. Acebrón,
  authorL. L. Bonilla,
  authorC. J. P. Vicente,
  authorF. Ritort, and
  authorR. Spigler,
  journalReviews of Modern Physics volume77,
  pages137 (year2005).

[Daido(1992)]daido1992quasientrainment
authorH. Daido,
  journalPhysical Review Letters volume68,
  pages1073 (year1992).

[Bonilla et al.(1993)Bonilla, Vicente,
  and Rubi]bonilla1993glassy
authorL. Bonilla,
  authorC. P. Vicente,
  and authorJ. Rubi,
  journalJournal of Statistical Physics
  volume70, pages921 (year1993).

[Iatsenko et al.(2014)Iatsenko,
  McClintock, and Stefanovska]iatsenko2014glassy
authorD. Iatsenko,
  authorP. V. McClintock,
  and
  authorA. Stefanovska,
  journalNature Communications volume5,
  pages4118 (year2014).

[Ottino-Löffler and
  Strogatz(2018)]ottino2018volcano
authorB. Ottino-Löffler
  and authorS. H.
  Strogatz, journalPhysical Review Letters
  volume120, pages264102
  (year2018).

[Kuramoto and
  Battogtokh(2002)]kuramoto2002coexistence
authorY. Kuramoto and
  authorD. Battogtokh,
  journalarXiv preprint cond-mat/0210694  (year2002).

[Abrams and Strogatz(2004)]abrams2004chimera
authorD. M. Abrams and
  authorS. H. Strogatz,
  journalPhysical Review Letters volume93,
  pages174102 (year2004).

[Abrams et al.(2008)Abrams, Mirollo,
  Strogatz, and Wiley]abrams2008solvable
authorD. M. Abrams,
  authorR. Mirollo,
  authorS. H. Strogatz,
  and authorD. A. Wiley,
  journalPhysical Review Letters volume101,
  pages084103 (year2008).

[Montbrió et al.(2015)Montbrió,
  Pazó, and Roxin]montbrio2015macroscopic
authorE. Montbrió,
  authorD. Pazó, and
  authorA. Roxin,
  journalPhysical Review X volume5,
  pages021028 (year2015).

[Pazó and Montbrió(2014)]pazo2014low
authorD. Pazó and
  authorE. Montbrió,
  journalPhysical Review X volume4,
  pages011009 (year2014).

[O'Keeffe and Strogatz(2016)]o2016dynamics
authorK. P. O'Keeffe and
  authorS. H. Strogatz,
  journalPhysical Review E volume93,
  pages062203 (year2016).

[Luke et al.(2013)Luke, Barreto, and
  So]luke2013complete
authorT. B. Luke,
  authorE. Barreto, and
  authorP. So,
  journalNeural Computation volume25,
  pages3207 (year2013).

[Peskin(1975)]peskin1975mathematical
authorC. S. Peskin,
  journalCourant Inst. Math  (year1975).

[Grudziński and
  Żebrowski(2004)]grudzinski2004modeling
authorK. Grudziński
  and authorJ. J.
  Żebrowski, journalPhysica A: Statistical
  Mechanics and its Applications volume336,
  pages153 (year2004).

[Gois and Savi(2009)]gois2009analysis
authorS. R. Gois and
  authorM. A. Savi,
  journalChaos, Solitons & Fractals volume41,
  pages2553 (year2009).

[Saw et al.(2018)Saw, Chung, Quek, Pang,
  and Chew]saw2018bus
authorV.-L. Saw,
  authorN. N. Chung,
  authorW. L. Quek,
  authorY. E. I. Pang,
  and authorL. Y. Chew,
  journalarXiv preprint arXiv:1812.00609  (year2018).

[Werner-Allen et al.(2005)Werner-Allen,
  Tewari, Patel, Welsh, and Nagpal]werner-allen05
authorG. Werner-Allen,
  authorG. Tewari,
  authorA. Patel,
  authorM. Welsh, and
  authorR. Nagpal, in
  booktitleProc. ACM Conf. Embedded Networked Sensor Systems
  (SenSys) (addressSan Diego, CA, year2005).

[Hong and Scaglione(2005)]hong05
authorY.-W. Hong and
  authorA. Scaglione,
  journalIEEE J. Select. Areas Commun.
  volume23, pages1085 (year2005).

[Babaoglu et al.(2007)Babaoglu, Binci,
  Jelasity, and Montresor]babaoglu2007firefly
authorO. Babaoglu,
  authorT. Binci,
  authorM. Jelasity, and
  authorA. Montresor, in
  booktitleProc. Intern. Conf. on Self-Adaptive and
  Self-Organizing Systems (SASO) (organizationIEEE,
  year2007), pp. pages77–86.

[Tyrrell et al.(2010)Tyrrell, Auer, and
  Bettstetter]tyrrell10
authorA. Tyrrell,
  authorG. Auer, and
  authorC. Bettstetter,
  journalIEEE Trans. Mobile Comput. volume9,
  pages719 (year2010).

[Vicsek et al.(1995)Vicsek, Czirók,
  Ben-Jacob, Cohen, and Shochet]vicsek1995novel
authorT. Vicsek,
  authorA. Czirók,
  authorE. Ben-Jacob,
  authorI. Cohen, and
  authorO. Shochet,
  journalPhysical Review Letters volume75,
  pages1226 (year1995).

[Ramaswamy(2010)]ramaswamy2010mechanics
authorS. Ramaswamy,
  journalAnnu. Rev. Condens. Matter Phys.
  volume1, pages323 (year2010).

[Marchetti et al.(2013)Marchetti, Joanny,
  Ramaswamy, Liverpool, Prost, Rao, and Simha]marchetti2013hydrodynamics
authorM. C. Marchetti,
  authorJ.-F. Joanny,
  authorS. Ramaswamy,
  authorT. B. Liverpool,
  authorJ. Prost,
  authorM. Rao, and
  authorR. A. Simha,
  journalReviews of Modern Physics volume85,
  pages1143 (year2013).

[Sanchez et al.(2012)Sanchez, Chen,
  DeCamp, Heymann, and Dogic]sanchez2012spontaneous
authorT. Sanchez,
  authorD. T. Chen,
  authorS. J. DeCamp,
  authorM. Heymann, and
  authorZ. Dogic,
  journalNature volume491,
  pages431 (year2012).

[Dorigo and Birattari(2010)]dorigo2010ant
authorM. Dorigo and
  authorM. Birattari,
  titleAnt colony optimization
  (publisherSpringer, year2010).

[Meng et al.(2014)Meng, Liu, Gao, and
  Zhang]meng2014new
authorX. Meng,
  authorY. Liu,
  authorX. Gao, and
  authorH. Zhang, in
  booktitleProc. Intern. Conf. in Swarm Intelligence
  (organizationSpringer, year2014), pp.
  pages86–94.

[Meng et al.(2016)Meng, Gao, Lu, Liu, and
  Zhang]meng2016new
authorX.-B. Meng,
  authorX. Z. Gao,
  authorL. Lu,
  authorY. Liu, and
  authorH. Zhang,
  journalJournal of Experimental & Theoretical Artificial
  Intelligence volume28, pages673
  (year2016).

[Mirjalili et al.(2017)Mirjalili,
  Gandomi, Mirjalili, Saremi, Faris, and Mirjalili]mirjalili2017salp
authorS. Mirjalili,
  authorA. H. Gandomi,
  authorS. Z. Mirjalili,
  authorS. Saremi,
  authorH. Faris, and
  authorS. M. Mirjalili,
  journalAdvances in Engineering Software
  volume114, pages163 (year2017).

[Yang et al.(2013)Yang, Cui, Xiao,
  Gandomi, and Karamanoglu]yang2013swarm
authorX.-S. Yang,
  authorZ. Cui,
  authorR. Xiao,
  authorA. H. Gandomi,
  and
  authorM. Karamanoglu,
  titleSwarm intelligence and bio-inspired computation: theory
  and applications (publisherNewnes, year2013).

[Shi et al.(2001)]shi2001particle
authorY. Shi et al., in
  booktitleProc. Congress on Evolutionary Computation
  (organizationIEEE, year2001),
  vol. volume1, pp. pages81–86.

[Hamann(2018)]hamann2018swarm
authorH. Hamann,
  titleSwarm Robotics: A Formal Approach
  (publisherSpringer, year2018).

[Schmickl and Hamann(2011)]schmickl2011beeclust
authorT. Schmickl and
  authorH. Hamann,
  journalBio-inspired computing and communication networks pp.
  pages95–137 (year2011).

[Savkin(2004)]savkin2004coordinated
authorA. V. Savkin,
  journalIEEE Transactions on Automatic Control
  volume49, pages981 (year2004).

[Merkle et al.(2007)Merkle, Middendorf,
  and Scheidler]merkle2007swarm
authorD. Merkle,
  authorM. Middendorf,
  and
  authorA. Scheidler, in
  booktitleProc. IEEE Swarm Intelligence Symp.
  (year2007), pp. pages242–249.

[von Brecht and Uminsky(2016)]von2016anisotropic
authorJ. H. von Brecht
  and authorD. T.
  Uminsky, journalNonlinearity
  volume30, pages225 (year2016).

[Uriu et al.(2013)Uriu, Ares, Oates, and
  Morelli]uriu2013dynamics
authorK. Uriu,
  authorS. Ares,
  authorA. C. Oates, and
  authorL. G. Morelli,
  journalPhysical Review E volume87,
  pages032911 (year2013).

[Sevilla et al.(2014)Sevilla, Dossetti,
  and Heiblum-Robles]sevilla2014synchronization
authorF. J. Sevilla,
  authorV. Dossetti, and
  authorA. Heiblum-Robles,
  journalJournal of Statistical Mechanics: Theory and Experiment
  volume2014, pagesP12025
  (year2014).

[Stilwell et al.(2006)Stilwell, Bollt,
  and Roberson]stilwell2006sufficient
authorD. J. Stilwell,
  authorE. M. Bollt, and
  authorD. G. Roberson,
  journalSIAM Journal on Applied Dynamical Systems
  volume5, pages140 (year2006).

[Frasca et al.(2008)Frasca, Buscarino,
  Rizzo, Fortuna, and Boccaletti]frasca2008synchronization
authorM. Frasca,
  authorA. Buscarino,
  authorA. Rizzo,
  authorL. Fortuna, and
  authorS. Boccaletti,
  journalPhysical Review Letters volume100,
  pages044102 (year2008).

[Tanaka(2007)]tanaka2007general
authorD. Tanaka,
  journalPhysical Review Letters volume99,
  pages134103 (year2007).

[Iwasa et al.(2010)Iwasa, Iida, and
  Tanaka]iwasa2010hierarchical
authorM. Iwasa,
  authorK. Iida, and
  authorD. Tanaka,
  journalPhysical Review E volume81,
  pages046220 (year2010).

[Starnini et al.(2016)Starnini, Frasca,
  and Baronchelli]starnini2016emergence
authorM. Starnini,
  authorM. Frasca, and
  authorA. Baronchelli,
  journalScientific reports volume6,
  pages31834 (year2016).

[Belovs et al.(2017)Belovs,
  Livanovičs, and Cēbers]belovs2017synchronized
authorM. Belovs,
  authorR. Livanovičs,
  and
  authorA. Cēbers,
  journalPhysical Review E volume96,
  pages042408 (year2017).

[O'Keeffe et al.(2017)O'Keeffe, Hong, and
  Strogatz]o2017oscillators
authorK. P. O'Keeffe,
  authorH. Hong, and
  authorS. H. Strogatz,
  journalNature Communications volume8,
  pages1504 (year2017).

[Topaz and Bertozzi(2004)]topaz2004swarming
authorC. M. Topaz and
  authorA. L. Bertozzi,
  journalSIAM Journal on Applied Mathematics
  volume65, pages152 (year2004).

[Mogilner and Edelstein-Keshet(1999)]mogilner1999non
authorA. Mogilner and
  authorL. Edelstein-Keshet,
  journalJournal of Mathematical Biology
  volume38, pages534 (year1999).

[Kuramoto(1991)]kuramoto91
authorY. Kuramoto,
  journalPhysica B volume50,
  pages15 (year1991).

[O'Keeffe et al.(2018)O'Keeffe, Evers,
  and Kolokolnikov]o2018ring
authorK. P. O'Keeffe,
  authorJ. H. Evers, and
  authorT. Kolokolnikov,
  journalPhysical Review E volume98,
  pages022203 (year2018).

[Hong(2018)]hong2018active
authorH. Hong,
  journalChaos: An Interdisciplinary Journal of Nonlinear Science
  volume28, pages103112
  (year2018).

[Iwasa et al.(2011)Iwasa, Iida, and
  Tanaka]iwasa2011juggling
authorM. Iwasa,
  authorK. Iida, and
  authorD. Tanaka,
  journalPhysical Review E volume83,
  pages036210 (year2011).

[Iwasa and Tanaka(2017)]iwasa2017mechanism
authorM. Iwasa and
  authorD. Tanaka,
  journalPhysics Letters A volume381,
  pages3054 (year2017).

[Iwasa and Tanaka(2010)]iwasa2010dimensionality
authorM. Iwasa and
  authorD. Tanaka,
  journalPhysical Review E volume81,
  pages066214 (year2010).

[Iwasa et al.(2012)Iwasa, Iida, and
  Tanaka]iwasa2012various
authorM. Iwasa,
  authorK. Iida, and
  authorD. Tanaka,
  journalPhysics Letters A volume376,
  pages2117 (year2012).

[Leoni and Liverpool(2014)]leoni2014synchronization
authorM. Leoni and
  authorT. Liverpool,
  journalPhysical Review Letters volume112,
  pages148104 (year2014).

[Toner and Tu(1995)]toner1995long
authorJ. Toner and
  authorY. Tu,
  journalPhysical Review Letters volume75,
  pages4326 (year1995).

[Ahmadi et al.(2006)Ahmadi, Marchetti,
  and Liverpool]ahmadi2006hydrodynamics
authorA. Ahmadi,
  authorM. C. Marchetti,
  and authorT. B.
  Liverpool, journalPhysical Review E
  volume74, pages061913
  (year2006).

[Sachs(2005)]sachs2005pattern
authorT. Sachs,
  titlePattern formation in plant tissues,
  vol. volume25 (publisherCambridge University Press,
  year2005).

[Liebchen and Levis(2017)]liebchen2017collective
authorB. Liebchen and
  authorD. Levis,
  journalPhysical Review Letters volume119,
  pages058002 (year2017).

[Levis et al.(2018)Levis, Pagonabarraga,
  and Liebchen]levis2018activity
authorD. Levis,
  authorI. Pagonabarraga,
  and authorB. Liebchen,
  journalarXiv preprint arXiv:1802.02371  (year2018).

[Gniewek et al.(2019)Gniewek, Barciś,
  and Bettstetter]gniewek19
authorA. Gniewek,
  authorM. Barciś, and
  authorC. Bettstetter,
  journalarXiv preprint arXiv:1903.06440  (year2019).

[Hrabec et al.(2018)Hrabec,
  Křižáková, Pizzini, Sampaio, Thiaville, Rohart, and
  Vogel]hrabec2018velocity
authorA. Hrabec,
  authorV. Křižáková,
  authorS. Pizzini,
  authorJ. Sampaio,
  authorA. Thiaville,
  authorS. Rohart, and
  authorJ. Vogel,
  journalPhysical Review Letters volume120,
  pages227204 (year2018).

[Igoshin et al.(2001)Igoshin, Mogilner,
  Welch, Kaiser, and Oster]igoshin2001pattern
authorO. A. Igoshin,
  authorA. Mogilner,
  authorR. D. Welch,
  authorD. Kaiser, and
  authorG. Oster,
  journalProceedings of the National Academy of Sciences
  volume98, pages14913 (year2001).

[Elgeti et al.(2015)Elgeti, Winkler, and
  Gompper]elgeti2015physics
authorJ. Elgeti,
  authorR. G. Winkler,
  and authorG. Gompper,
  journalReports on progress in physics
  volume78, pages056601
  (year2015).

[Yang et al.(2008)Yang, Elgeti, and
  Gompper]yang2008cooperation
authorY. Yang,
  authorJ. Elgeti, and
  authorG. Gompper,
  journalPhysical Review E volume78,
  pages061903 (year2008).

[Hayashi(1996)]hayashi1996insemination
authorF. Hayashi,
  journalJournal of Insect Physiology
  volume42, pages859 (year1996).

[Riedel et al.(2005)Riedel, Kruse, and
  Howard]riedel2005self
authorI. H. Riedel,
  authorK. Kruse, and
  authorJ. Howard,
  journalScience volume309,
  pages300 (year2005).

[Snezhko and Aranson(2011)]snezhko2011magnetic
authorA. Snezhko and
  authorI. S. Aranson,
  journalNature Materials volume10,
  pages698 (year2011).

[Yan et al.(2012)Yan, Bloom, Bae,
  Luijten, and Granick]yan2012linking
authorJ. Yan,
  authorM. Bloom,
  authorS. C. Bae,
  authorE. Luijten, and
  authorS. Granick,
  journalNature volume491,
  pages578 (year2012).

[Mathar and Mattfeldt(1996)]mathar96
authorR. Mathar and
  authorJ. Mattfeldt,
  journalSIAM J. Appl. Math. volume56,
  pages1094 (year1996).

[Simeone et al.(2008)Simeone,
  Spagnolini, Bar-Ness, and Strogatz]simeone08
authorO. Simeone,
  authorU. Spagnolini,
  authorY. Bar-Ness,
  and authorS. H.
  Strogatz, journalIEEE Signal Processing Mag.
  volume25, pages81 (year2008).

[Bottani(1995)]bottani1995pulse
authorS. Bottani,
  journalPhysical Review Letters volume74,
  pages4189 (year1995).

[Mirollo and
  Strogatz(1990)]mirollo1990synchronization
authorR. E. Mirollo and
  authorS. H. Strogatz,
  journalSIAM Journal on Applied Mathematics
  volume50, pages1645 (year1990).

[O'Keeffe et al.(2015)O'Keeffe,
  Krapivsky, and Strogatz]o2015synchronization
authorK. P. O'Keeffe,
  authorP. L. Krapivsky,
  and authorS. H.
  Strogatz, journalPhysical Review Letters
  volume115, pages064101
  (year2015).

[O'Keeffe(2016)]o2016transient
authorK. P. O'Keeffe,
  journalPhysical Review E volume93,
  pages032203 (year2016).

[Bregni(2002)]bregni02
authorS. Bregni,
  titleSynchronization of Digital Telecommunications
  Networks (publisherWiley, addressNew York, NY, USA,
  year2002).

[Roberts(1975)]Roberts:1975:APS
authorL. G. Roberts,
  journalSIGCOMM Comput. Commun. Rev. volume5,
  pages28 (year1975).

[Ye et al.(2004)Ye, Heidemann, and
  Estrin]ye04
authorW. Ye,
  authorJ. Heidemann,
  and authorD. Estrin,
  journalIEEE/ACM Trans. Netw. volume12,
  pages493 (year2004).

[Aghajan and Cavallaro(2009)]aghajan:book
authorH. Aghajan and
  authorA. Cavallaro,
  titleMulti-Camera Networks: Principles and Applications
  (publisherAcademic Press, year2009).

[Pagliari and Scaglione(2011)]pagliari11
authorR. Pagliari and
  authorA. Scaglione,
  journalIEEE Trans. Mobile Comput.
  volume10, pages392 (year2011).

[Brandner et al.(2016)Brandner,
  Schilcher, and Bettstetter]brandner16:cn
authorG. Brandner,
  authorU. Schilcher,
  and
  authorC. Bettstetter,
  journalComputer Networks volume97,
  pages74 (year2016).

[Rohden et al.(2012)Rohden, Sorge, Timme,
  and Witthaut]rohden12
authorM. Rohden,
  authorA. Sorge,
  authorM. Timme, and
  authorD. Witthaut,
  journalPhysical Review Letters volume109,
  pages064101 (year2012).

[Skardal and Arenas(2015)]skardal15
authorP. S. Skardal and
  authorA. Arenas,
  journalScience Advances volume1,
  pagese1500339 (year2015).

[Prehofer and Bettstetter(2005)]prehofer05:commag
authorC. Prehofer and
  authorC. Bettstetter,
  journalIEEE Commun. Mag. volume43,
  pages78 (year2005).

[von Brecht and Uminsky(2012)]von2012soccer
authorJ. H. von Brecht
  and authorD. Uminsky,
  journalJournal of Nonlinear Science
  volume22, pages935 (year2012).

[Hong and Strogatz(2011)]hong2011kuramoto
authorH. Hong and
  authorS. H. Strogatz,
  journalPhysical Review Letters volume106,
  pages054102 (year2011).

[Maistrenko et al.(2014)Maistrenko,
  Penkovsky, and Rosenblum]maistrenko2014solitary
authorY. Maistrenko,
  authorB. Penkovsky,
  and
  authorM. Rosenblum,
  journalPhysical Review E volume89,
  pages060901 (year2014).

[Hong et al.(2016a)Hong,
  O'Keeffe, and Strogatz]hong2016phase
authorH. Hong,
  authorK. P. O'Keeffe,
  and authorS. H.
  Strogatz, journalPhysical Review E
  volume93, pages022219
  (year2016a).

[Hong et al.(2016b)Hong,
  O'Keeffe, and Strogatz]hong2016correlated
authorH. Hong,
  authorK. P. O'Keeffe,
  and authorS. H.
  Strogatz, journalChaos: An Interdisciplinary
  Journal of Nonlinear Science volume26,
  pages103105 (year2016b).

[Yeung and Strogatz(1999)]yeung1999time
authorM. S. Yeung and
  authorS. H. Strogatz,
  journalPhysical Review Letters volume82,
  pages648 (year1999).

[Winfree(1967)]winfree67
authorA. T. Winfree,
  journalJ. Theoretical Biology volume16,
  pages15 (year1967).

[Nicolaou et al.(2019)Nicolaou, Eroglu,
  and Motter]nicolaou2019multifaceted
authorZ. G. Nicolaou,
  authorD. Eroglu, and
  authorA. E. Motter,
  journalPhysical Review X volume9,
  pages011017 (year2019).

[Kolokolnikov et al.(2011)Kolokolnikov,
  Sun, Uminsky, and Bertozzi]kolokolnikov2011stability
authorT. Kolokolnikov,
  authorH. Sun,
  authorD. Uminsky, and
  authorA. L. Bertozzi,
  journalPhysical Review E volume84,
  pages015203 (year2011).




