Collective behavior of swarmalators on a ring

Strategy to control synchronized dynamics in swarmalator systems

Synchronization forms the basis of many coordination phenomena in natural systems, enabling them to function cohesively and support their fundamental operations. However, there are scenarios where synchronization disrupts a system's proper functioning, necessitating mechanisms to control or suppress it. While several methods exist for controlling synchronization in nonspatially embedded oscillators, to the best of our knowledge, no such strategies have been developed for swarmalators (oscillators that simultaneously move in space and synchronize in time). In this work, we address this gap by introducing a control strategy based on Hamiltonian control theory to suppress synchronization in a system of swarmalators confined to a one-dimensional space. The numerical investigations we performed demonstrate that the proposed control strategy effectively suppresses synchronized dynamics within the swarmalator population. We studied the impact of the number of controlled swarmalators and the strength of the control term in its original form and a simplified one.

Synchronization of wave-propelled capillary spinners

When a millimetric body is placed atop a vibrating liquid bath, the relative motion between the object and the interface generates outward-propagating waves with an associated momentum flux. Prior work has shown that isolated chiral objects, referred to as spinners, can thus rotate steadily in response to their self-generated wavefield. Here, we consider the case of two cochiral spinners held at a fixed spacing from one another but otherwise free to interact hydrodynamically through their shared fluid substrate. Two identical spinners are able to synchronize their rotation, with their equilibrium phase difference sensitive to their spacing and initial conditions, and even cease to rotate when the coupling becomes sufficiently strong. Nonidentical spinners can also find synchrony provided their intrinsic differences are not too disparate. A hydrodynamic wave model of the spinner interaction is proposed, recovering all salient features of the experiment. In all cases, the spatially periodic nature of the capillary wave coupling is directly reflected in the emergent equilibrium behaviors.

Synchronization and self-assembly of free capillary spinners

Chiral active particles are able to draw energy from the environment to self-propel in the form of rotation. We describe an experimental arrangement wherein chiral objects, spinners, floating on the surface of a vibrated fluid rotate due to emitted capillary waves. We observe that pairs of spinners can assemble at quantized distances via the mutually generated wavefield, phase synchronize and, in some circumstances, globally rotate about a point midway between them. A mathematical model based on wave-mediated interactions captures the salient features of the assembly and synchronization while a qualitative argument is able to rationalize global rotations based on interference and radiation stress associated with the wavefield. Extensions to larger collections are demonstrated, highlighting the potential for this tabletop system to be used as an experimental system capable of synchronizing and swarming.

Global synchronization theorem for coupled swarmalators

The global stability of oscillator networks has attracted much recent attention. Ordinarily, the oscillators in such studies are motionless; their spatial degrees of freedom are either ignored (e.g., mean field models) or inactive (e.g., geometrically embedded networks like lattices). Yet many real-world oscillators are mobile, moving around in space as they synchronize in time. Here, we prove a global synchronization theorem for a simple model of such swarmalators where the units move on a 1D ring. This can be thought of as a generalization from oscillators connected on random networks to oscillators connected on temporal networks, where the edges are determined by the oscillators' movements.

Statistical description of mobile oscillators in embryonic pattern formation

Synchronization of mobile oscillators occurs in numerous contexts, including physical, chemical, biological, and engineered systems. In vertebrate embryonic development, a segmental body structure is generated by a population of mobile oscillators. Cells in this population produce autonomous gene expression rhythms and interact with their neighbors through local signaling. These cells form an extended tissue where frequency and cell mobility gradients coexist. Gene expression kinematic waves travel through this tissue and pattern the segment boundaries. It has been shown that oscillator mobility promotes global synchronization. However, in vertebrate segment formation, mobility may also introduce local fluctuations in kinematic waves and impair segment boundaries. Here, we derive a general framework for mobile oscillators that relates local mobility fluctuations to synchronization dynamics and pattern robustness. We formulate a statistical description of mobile phase oscillators in terms of probability density. We obtain and solve diffusion equations for the average phase and variance, revealing the relationship between local fluctuations and global synchronization in a homogeneous population of oscillators. Analysis of the probability density for large mobility identifies a mean-field onset, where locally coupled oscillators start behaving as if each oscillator was coupled with all the others. We extend the statistical description to inhomogeneous systems to address the gradients present in the vertebrate segmenting tissue. The theory relates pattern stability to mobility, coupling, and pattern wavelength. The general approach of the statistical description may be applied to mobile oscillators in other contexts, as well as to other patterning systems where mobility is present.

Effects of coupling range on the dynamics of swarmalators

We study a variant of the one-dimensional swarmalator model where the units' interactions have a controllable length scale or range. We tune the model from the long-range regime, which is well studied, into the short-range regime, which is insufficiently studied, and find diverse collective states: sync dots, where the swarmalators arrange themselves into k>1 delta points of perfect synchrony; q-waves, where the swarmalators form spatiotemporal waves with winding number q>1; and an active state where unsteady oscillations are found. We present the phase diagram and derive most of the threshold boundaries analytically. These states may be observable in real-world swarmalator systems with low-range coupling, such as biological microswimmers or active colloids.

Chimera states in a system of stationary and flying-through deterministic particles with an internal degree of freedom

We consider the effect of the emergence of chimera states in a system of coexisting stationary and flying-through in potential particles with an internal degree of freedom determined by the phase. All particles tend to an equilibrium state with a small number of potential wells, which leads to the emergence of a stationary chimera. An increase in the number of potential wells leads to the emergence of particles flying-through along the medium, the phases of which form a moving chimera. Further, these two structures coexist and interact with each other. In this case, an increase in the local synchronization degree of the chimera is observed in the areas of the synchronous cluster location.

Dynamics of swarmalators in the presence of a contrarian

Swarmalators are entities that combine the swarming behavior of particles with the oscillatory dynamics of coupled phase oscillators and represent a novel and rich area of study within the field of complex systems. Unlike traditional models that treat spatial movement and phase synchronization separately, swarmalators exhibit a unique coupling between their positions and internal phases, leading to emergent behaviors that include clustering, pattern formation, and the coexistence of synchronized and desynchronized states, etc. This paper presents a comprehensive analysis of a two-dimensional swarmalator model in the presence of a predatorlike agent that we call a contrarian. The positions and the phases of the swarmalators are influenced by the contrarian and we observe the emergence of intriguing collective states. We find that swarmalator phases are synchronized even with negative coupling strength when their interaction with the contrarian is comparatively strong. Through a combination of analytical methods and simulations, we demonstrate how varying these parameters can lead to transitions between different collective states.

Collective dynamics of swarmalators driven by a mobile pacemaker

Swarmalators, namely, oscillators with intrinsic frequencies that are able to self-propel to move in space, may undergo collective spatial swarming and meanwhile phase synchronous dynamics. In this paper, a swarmalator model driven by an external mobile pacemaker is proposed to explore the swarming dynamics in the presence of the competition between the external organization of the moving pacemaker and the intrinsic self-organization among oscillators. It is unveiled that the swarmalator system may exhibit a wealth of novel spatiotemporal patterns including the spindle state, the ripple state, and the trapping state. Transitions among these patterns and the mechanisms are studied with the help of different order parameters. The phase diagrams present systematic scenarios of various possible collective swarming dynamics and the transitions among them. The present study indicates that one may manipulate the formation and switching of the organized collective states by adjusting the external driving force, which is expected to shed light on applications of swarming performance control in natural and artificial groups of active agents.

Forced one-dimensional swarmalator model

We study a simple model of swarmalators subject to periodic forcing and confined to moving around a one-dimensional ring. This is a toy model for physical systems with a mix of sync, swarming, and forcing, such as colloidal micromotors. We find rich behavior: pinned states where the swarmalators are locked to the driving, sync states where their phases are either identical or have fixed differences, and unsteady states, such as swarmalator chimera where the population splits into two sync dots enclosed by a "train" of swarmalators that run around a peanut-shaped loop. We derive the stability thresholds for most of these states which give us a good approximation of the model's phase diagram.

Data-driven exploration of swarmalators with second-order harmonics

We explore the dynamics of a swarmalator population comprising second-order harmonics in phase interaction. A key observation in our study is the emergence of the active asynchronous state in swarmalators with second-order harmonics, mirroring findings in the one-dimensional analog of the model, accompanied by the formation of clustered states. Particularly, we observe a transition from the static asynchronous state to the active phase wave state via the active asynchronous state. We have successfully delineated and quantified the stability boundary of the active asynchronous state through a completely data-driven method. This was achieved by utilizing the enhanced image processing capabilities of convolutional neural networks, specifically, the U-Net architecture. Complementing this data-driven analysis, our study also incorporates an analytical stability of the clustered states, providing a multifaceted perspective on the system's behavior. Our investigation not only sheds light on the nuanced behavior of swarmalators under second-order harmonics, but also demonstrates the efficacy of convolutional neural networks in analyzing complex dynamical systems.

Solvable model of driven matter with pinning

We present a simple model of driven matter in a 1D medium with pinning impurities, applicable to magnetic domains walls, confined colloids, and other systems. We find rich dynamics, including hysteresis, reentrance, quasiperiodicity, and two distinct routes to chaos. In contrast to other minimal models of driven matter, the model is solvable: we derive the full phase diagram for small N, and for large N, we derive expressions for order parameters and several bifurcation curves. The model is also realistic. Its collective states match those seen in the experiments of magnetic domain walls.

Order, chaos, and dimensionality transition in a system of swarmalators

Similarly to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, the system exhibits hyperchaos, maintaining spatial correlation under certain conditions and breaking it under others on what we interpret as a dimensionality transition.

A Cellular Potts Model of the interplay of synchronization and aggregation

We investigate the behavior of systems of cells with intracellular molecular oscillators ("clocks") where cell-cell adhesion is mediated by differences in clock phase between neighbors. This is motivated by phenomena in developmental biology and in aggregative multicellularity of unicellular organisms. In such systems, aggregation co-occurs with clock synchronization. To account for the effects of spatially extended cells, we use the Cellular Potts Model (CPM), a lattice agent-based model. We find four distinct possible phases: global synchronization, local synchronization, incoherence, and anti-synchronization (checkerboard patterns). We characterize these phases via order parameters. In the case of global synchrony, the speed of synchronization depends on the adhesive effects of the clocks. Synchronization happens fastest when cells in opposite phases adhere the strongest ("opposites attract"). When cells of the same clock phase adhere the strongest ("like attracts like"), synchronization is slower. Surprisingly, the slowest synchronization happens in the diffusive mixing case, where cell-cell adhesion is independent of clock phase. We briefly discuss potential applications of the model, such as pattern formation in the auditory sensory epithelium.

Directional synchrony among self-propelled particles under spatial influence

Synchronization is one of the emerging collective phenomena in interacting particle systems. Its ubiquitous presence in nature, science, and technology has fascinated the scientific community over the decades. Moreover, a great deal of research has been, and is still being, devoted to understand various physical aspects of the subject. In particular, the study of interacting active particles has led to exotic phase transitions in such systems which have opened up a new research front-line. Motivated by this line of work, in this paper, we study the directional synchrony among self-propelled particles. These particles move inside a bounded region, and crucially their directions are also coupled with spatial degrees of freedom. We assume that the directional coupling between two particles is influenced by the relative spatial distance which changes over time. Furthermore, the nature of the influence is considered to be both short and long-ranged. We explore the phase transition scenario in both the cases and propose an approximation technique which enables us to analytically find the critical transition point. The results are further supported with numerical simulations. Our results have potential importance in the study of active systems like bird flocks, fish schools, and swarming robots where spatial influence plays a pertinent role.

Order parameter dynamics in complex systems: From models to data

Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.

Swarmalators with delayed interactions

We investigate the effects of delayed interactions in a population of "swarmalators," generalizations 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 pseudocrystalline 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.

Smooth transformations and ruling out closed orbits in planar systems

This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth mapping. Using elementary techniques, we obtain a unified picture of different classes of dynamical systems, some of which are classically viewed as distinct. We specifically give two examples of Hamiltonian systems with first integrals, which are simultaneously gradient systems. Potential applications of this apparent duality are discussed. The second part of this study is concerned with ruling out closed orbits in steady planar systems. We reformulate Bendixson's criterion using the coordinate-independent Helmholtz decomposition derived in the first part, and we derive another, similar criterion. Our results allow for automated ruling out of closed orbits in certain regions of phase space and could be used in the future for efficient seeding of initial conditions in numerical algorithms to detect periodic solutions.

Attractive and repulsive interactions in the one-dimensional swarmalator model

We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.

Swarmalators on a ring with uncorrelated pinning

We present a case study of swarmalators (mobile oscillators) that move on a 1D ring and are subject to pinning. Previous work considered the special case where the pinning in space and the pinning in the phase dimension were correlated. Here, we study the general case where the space and phase pinning are uncorrelated, both being chosen uniformly at random. This induces several new effects, such as pinned async, mixed states, and a first-order phase transition. These phenomena may be found in real world swarmalators, such as systems of vinegar eels, Janus matchsticks, electrorotated Quincke rollers, or Japanese tree frogs.

Synchronization of Sakaguchi swarmalators

Swarmalators are phase oscillators that cluster in space, like fireflies flashing in a swarm to attract mates. Interactions between particles, which tend to synchronize their phases and align their motion, decrease with the distance and phase difference between them, coupling the spatial and phase dynamics. In this work, we explore the effects of inducing phase frustration on a system of swarmalators that move on a one-dimensional ring. Our model is inspired by the well-known Kuramoto-Sakaguchi equations. We find, numerically and analytically, the ordered and disordered states that emerge in the system. The active states, not present in the model without frustration, resemble states found previously in numerical studies for the two-dimensional swarmalators system. One of these states, in particular, shows similarities to turbulence generated in a flattened media. We show that all ordered states can be generated for any values of the coupling constants by tuning the phase frustration parameters only. Moreover, many of these combinations display multistability.

Generalized frustration in the multidimensional Kuramoto model

The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D=2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix K that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration that tends to hinder synchronization. In a recent paper we studied in detail the role of the coupling matrix for D=2. Here we extend this analysis to arbitrary dimensions. We show that, for identical particles, when the natural frequencies are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether D is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If D is odd the phase transition is discontinuous and active states can be suppressed for some distributions of natural frequencies.

Diverse behaviors in non-uniform chiral and non-chiral swarmalators

We study the emergent behaviors of a population of swarming coupled oscillators, dubbed swarmalators. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.

Pinning in a system of swarmalators

We study a population of swarmalators (swarming/mobile oscillators) which run on a ring and are subject to random pinning. The pinning represents the tendency of particles to stick to defects in the underlying medium which competes with the tendency to sync and swarm. The result is rich collective behavior. A highlight is low dimensional chaos which in systems of ordinary, Kuramoto-type oscillators is uncommon. Some of the states (the phase wave and split phase wave) resemble those seen in systems of Janus matchsticks or Japanese tree frogs. The others (such as the sync and unsteady states) may be observable in systems of vinegar eels, electrorotated Quincke rollers, or other swarmalators moving in disordered environments.

Exact potentials in multivariate Langevin equations

Systems governed by a multivariate Langevin equation featuring an exact potential exhibit straightforward dynamics but are often difficult to recognize because, after a general coordinate change, the gradient flow becomes obscured by the Jacobian matrix of the mapping. In this work, a detailed analysis of the transformation rules for Langevin equations under general nonlinear mappings is presented. We show how to identify systems with exact potentials by understanding their differential-geometric properties. To demonstrate the power of our method, we use it to derive exact potentials for broadly studied models of nonlinear deterministic and stochastic oscillations. In selected examples, we visualize the identified potentials. Our results imply a broad class of exactly solvable stochastic models, which can be self-consistently defined from given deterministic gradient systems.

Sync and Swarm: Solvable Model of Nonidentical Swarmalators

We study a model of nonidentical swarmalators, generalizations of phase oscillators that both sync in time and swarm in space. The model produces four collective states: asynchrony, sync clusters, vortexlike phase waves, and a mixed state. These states occur in many real-world swarmalator systems such as biological microswimmers, chemical nanomotors, and groups of drones. A generalized Ott-Antonsen ansatz provides the first analytic description of these states and conditions for their existence. We show how this approach may be used in studies of active matter and related disciplines.

Matrix coupling and generalized frustration in Kuramoto oscillators

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.

Swarmalators on a ring with distributed couplings

We study a simple model of identical "swarmalators," generalizations of phase oscillators that swarm through space. We confine the movements to a one-dimensional (1D) ring and consider distributed (nonidentical) couplings; the combination of these two effects captures an aspect of the more realistic two-dimensional swarmalator model. We discover several collective states which we describe analytically. These states imitate the behavior of vinegar eels, catalytic microswimmers, and other swarmalators which move on quasi-1D rings.