Swarmalators Model

• Swarmalators are agents with positions and phase values that interact, merging swarming dynamics and oscillator synchronization into a unified model.
• Mathematical formulations extend classic models like Vicsek and Kuramoto, yielding diverse states such as static synchronization, phase waves, and chimera behaviors.
• Analytical, numerical, and topological methods reveal transitions influenced by noise, heterogeneity, and higher-order interactions, applicable to robotics and biological systems.

A swarmalator is an interacting agent whose state is described by both a position vector and an internal phase, with reciprocal dynamical coupling between its external motion and internal oscillation. Swarmalator models generalize classic swarming dynamics (as in the Vicsek or self-propelled particle frameworks) and phase oscillator synchronization (as in the Kuramoto model), yielding a hybrid system where agents aggregate, synchronize, and exhibit spatiotemporally complex collective states. Formally, for $N$ agents, each carries $x_i(t)\in\mathbb{R}^d$ (or on compact manifolds) and $\theta_i(t)\in\mathbb{S}^1$ (or, more generally, an orientation vector), whose evolution is governed by coupled ODEs with interaction kernels modulated by both spatial proximity and phase similarity (Sar et al., 2022). Since its introduction by O’Keeffe, Hong & Strogatz (2017), the swarmalator paradigm has produced rich dynamical behavior with deep implications for collective biological, chemical, and robotic systems.

1. General Formulation and Governing Equations

The canonical swarmalator dynamics in $d$ spatial dimensions are given by (Sar et al., 2022, Yadav et al., 2023): $$\dot{x}_i = v_i + \frac{1}{N}\sum_{j\ne i} \left[ I_{\rm att}(x_j-x_i) F_{\rm att}(\theta_j-\theta_i) - I_{\rm rep}(x_j-x_i) F_{\rm rep}(\theta_j-\theta_i) \right]$$

$$\dot\theta_i = \omega_i + \frac{K}{N}\sum_{j\ne i} H(\theta_j-\theta_i) G(x_j-x_i)$$

with variables and kernels:

• $v_i$ — self-propulsion velocity (can be set to zero; heterogeneity possible)
• $\omega_i$ — natural phase frequency; $K$ — overall phase coupling strength
• $I_{\rm att}$, $I_{\rm rep}$ — spatial attraction/repulsion, typically power-laws or normalized vectors
• $F_{\rm att}$, $F_{\rm rep}$ — phase-similarity modulations; $H$ — phase coupling function (e.g., $\sin$ or Winfree-type forms)
• $G$ — distance-dependent weighting for phase interactions

Analytical and numerical studies favor the two-dimensional instance: $$\dot{x}_i = \frac{1}{N}\sum_{j\ne i} \left[ \frac{x_j-x_i}{|x_j-x_i|}(1+J\cos(\theta_j-\theta_i)) - \frac{x_j-x_i}{|x_j-x_i|^2} \right]$$

$$\dot\theta_i = \frac{K}{N}\sum_{j\ne i} \frac{\sin(\theta_j-\theta_i)}{|x_j-x_i|}$$

$J$ tunes the phase-dependence of spatial attraction, while $K$ controls phase synchronization and desynchronization. Variations extend the model to $d$ spatial dimensions with vector orientations as "phases" (Yadav et al., 2023), ring geometries (O'Keeffe et al., 2022), and interaction kernels with explicit range or higher-harmonic structure (Sar et al., 22 Nov 2024, Smith, 2023).

2. Emergent Collective States and Order Parameters

Swarmalator models admit a diverse spectrum of collective states, summarized below (Sar et al., 2022, Hong et al., 2023, Sar et al., 12 Dec 2024):

• Static Synchronization: Agents lock phases ($R=1$) and aggregate in a spatial cluster. Achieved when $K>0$ for arbitrary $J$.
• Static Asynchrony: Phases and positions uniformly spread; no correlation. Occurs for $K$ sufficiently negative and subcritical $J$.
• Static Phase Wave: Positions on a ring/annulus with phases correlated to angular position; $S=1$ order parameter (spatial-phase correlation) quantifies this. Boundary: $K=0$, $J>0$.
• Splintered Phase Wave: Multiple phase-coherent clusters (spatial+phase splits), time-dependent but persistent for intermediate negative $K$.
• Active Phase Wave: Non-stationary, persistent circulation in physical and phase space; $K\ll-K_c$.
• Polarized/Splay State: For anti-aligned or mixed coupling (e.g., contrarians), spatial and phase splits into antipodal or equispaced clusters (Hao et al., 2023).
• Mixed/Chimera States: Partial locking in one variable (space or phase), drifting or incoherent in the other; often induced by distributed/higher-order couplings (O'Keeffe et al., 2022, Anwar et al., 2023, Hong et al., 2023).

Typical order parameters:

• $R e^{i\Phi} = \frac{1}{N}\sum_j e^{i\theta_j}$ — Kuramoto order (phase sync)
• $$S_\pm e^{i\Psi_\pm} = \frac{1}{N}\sum_j e^{i(\phi_j\pm\theta_j)}$$ — space-phase "rainbow" correlations
• $U$ — fraction completing full radial–phase winding

3. Analytical Stability and Bifurcation Structure

Key bifurcation thresholds identify transitions between states:

• The asynchronous (disk, ring) state loses stability at $K_c = -1.2J$ in 2D; for ring models, this generalizes to $J+K=0$ (Sar et al., 2022, O'Keeffe et al., 2022).
• Phase-wave regimes are stabilized for $K=0, J>0$ with perfect $x$–$\theta$ correlations.
• Critical population size for ring phase-waves: $N > N_{\max} = 8/(2-J_1)(1-J_2)$ (with modulated repulsion) (Sar et al., 2022).
• Synchronization can be promoted by higher-order or competitive interactions, e.g., $K_{ij}$ switching between attractive and repulsive values, yielding $\pi$-state clusters separated by $d_\pi = 1/(1-J)$ (Sar et al., 2022, Smith, 2023, Anwar et al., 23 Apr 2025).
• Non-reciprocal forces and multiplicative Winfree coupling yield new types of phase-wave and antiphase states not accessible by pure sine coupling (Ghosh et al., 21 Apr 2025).

Analysis techniques include linear stability (Fourier mode, Jacobian blocks), Poisson kernel Ott–Antonsen reductions (yielding algebraic amplitude equations for order parameters), and mean-field equations for continuum limits (Sar et al., 2022, O'Keeffe et al., 2022, Anwar et al., 2023).

4. Impact of Model Variations: Heterogeneity, Noise, Higher-order Terms

Heterogeneity: Disorder in velocities or phase frequencies splits the population into locked and drifting groups, leading to mixed or chimera states with explicit bifurcation boundaries. Distributed couplings can be delta, Gaussian, or multimodal; the existence and stability thresholds depend on averages $\langle J\rangle$, $\langle K\rangle$ (O'Keeffe et al., 2022, Yoon et al., 2022).

Thermal Noise: Adding Gaussian noises to spatial and phase variables blurs sharp transitions, produces mixed states, and shifts critical boundaries ($J_+ = 2(D_x+D_\theta)$ for static phase-wave emergence) (Hong et al., 2023). The Ott–Antonsen closure fails, requiring Fourier-mode expansions.

Higher-order Interactions: Three-body (triadic) couplings $K_2,J_2$ induce subcritical transitions, bistability, and abrupt (first-order) jumps in the order parameters; large $K_2$ can stabilize sync even for negative pairwise coupling (Anwar et al., 2023, Anwar et al., 23 Apr 2025). Analytical self-consistency equations are cubic or sextic polynomials in the relevant order parameters.

Coupling Range: Short-range kernels permit discrete multicluster "sync-dots" and multivalued phase-wave solutions with maximal winding number scaling with the kernel parameter $p$ (Sar et al., 22 Nov 2024). In 2D/3D, topological transitions, spatially coherent states, and gas-like active phases can emerge (Anwar et al., 23 Apr 2025, Yadav et al., 2023).

5. Topological States and Transitions

Recent work has shown that swarmalator states can be assigned continuous topological invariants:

• Topological charge $Q$ (winding number): Counts net phase winding around spatial contours, distinguishing vortex, anti-vortex, and trivial (sync) states (Louodop et al., 17 Nov 2025).
• Helicity $\gamma$: Local chirality measured by the angle of phase vectors relative to radial positions, with variance $V(\gamma)$ serving as a topological order parameter.
• Abrupt changes in $Q$ and decay of $V(\gamma)$ coincide with transitions to full synchronization and collapse of spatial phase patterns; they resolve exotic, chimera, and multi-layer synchronization regimes.

6. Generalizations and Multidimensional Extensions

The framework generalizes naturally to higher-dimensional spatial domains ($d>2$), internal vector orientations ($\sigma_i\in S^{d-1}$), and more complex feedback kernels (Yadav et al., 2023). In $d=3$ and beyond, new states—static and active shells, spiky/flower configurations, turning tubes, and chimera cores—appear, mirroring phenomena in schooling fish, cell sorting, microrobotics, and gene expression waves.

Ring, torus, and periodic boundary conditions allow for fully analytic bifurcation diagrams and extension to domains relevant for experiments in annular microchannels and confined colloid suspensions (O'Keeffe et al., 2023, O'Keeffe et al., 2022).

7. Open Problems and Directions

Key open questions include:

• Rigorous stability criteria for active and splintered states in higher dimensions (Sar et al., 2022, Yadav et al., 2023)
• Classification of mixed/chimera states under disorder and noise
• Effects of non-pairwise/higher-order interactions on phase diagrams and hysteresis (Anwar et al., 23 Apr 2025, Anwar et al., 2023)
• Realization of topological and vacillation states in biological and robotic platforms
• Analytical reductions for time-dependent, non-identical, and locally forced swarmalator systems.

The swarmalator paradigm thus provides a mathematically tractable, physically rich, and extensible platform for studying the entwined phenomena of collective motion and synchronization in active matter and networked multi-agent systems (Sar et al., 2022, O'Keeffe et al., 2023, Yadav et al., 2023).