Exotic swarming dynamics of high-dimensional swarmalators

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. Swarmalator models employed so far in the literature comprise only one-dimensional phase variables to represent the intrinsic dynamics of the natural collectives. Nevertheless, the latter can indeed be represented more realistically by high-dimensional phase variables. For instance, the alignment of velocity vectors in a school of fish or a flock of birds can be more realistically set up in three-dimensional space, while the alignment of opinion formation in population dynamics could be multidimensional, in general. We present a generalized $D$-dimensional swarmalator model, which more accurately captures self-organizing behaviors of a plethora of real-world collectives by self-adaptation of high-dimensional spatial and phase variables. For a more sensible visualization and interpretation of the results, we restrict our simulations to three-dimensional spatial and phase variables. Our model provides a framework for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the maneuvers of a school of fish, qualitatively and quantitatively, by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect the proposed high-dimensional swarmalator model to be potentially useful in describing swarming systems and programmable and reconfigurable collectives in a wide range of disciplines, including the physics of active matter, developmental biology, sociology, and engineering.

Figure 1

3D phase and its visualization. (a) Orientations ${\mathit{\sigma }}_i$ and ${\mathit{\sigma }}_j$ of $i\text{th}$ and $j\text{th}$ swarmalators, respectively, can be represented as vectors pointing on a unit sphere. (b) Deriving the analogs of $sin\left({\rho }_{ij}\right)$ and $cos\left({\rho }_{ij}\right)$ in terms of orientation vectors ${\mathit{\sigma }}_i$ and ${\mathit{\sigma }}_j$. Here ${\rho }_{ij}$ is the angle of inclination of orientation vectors, $cos\left({\rho }_{ij}\right)$ is simply the projection of ${\mathit{\sigma }}_j$ on ${\mathit{\sigma }}_i$, and $sin\left({\rho }_{ij}\right)$ is the $y$-component of ${\mathit{\sigma }}_j$. (c) Each swarmalator is represented by a cone with its apex pointing along its orientation vector. (d) The heat map, encoding the degree of orientation of the vectors, which in turn is determined by the polar phase $\left(\theta \right)$ and the azimuthal phase $\left(\varphi \right)$, is used to color the cones in accordance with the distribution of the initial conditions, which facilitates the identification of distinct collective states even when the cones are masked behind a dense set of cones.

Figure 2

Dynamical transition using order parameters. The dynamical transitions leading to distinct self-organizing collective behaviors as a function of the vision radius $R$ for (a) $J=-0.9$, (b) $J=0.1$, and (c) $J=0.9$. The synchronization order parameter $S$, hollowness $H$, kinetic energy $E$, number of clusters $N_c$, and orientation parameter $\mathrm{\Lambda }$ are used to characterize and distinguish distinct dynamical states. The parameters are fixed as ${\varepsilon }_a={\varepsilon }_r=0.5$ and $N=100$. See the main text for a more detailed explanation.

Figure 3

The synchronization order parameter $S$ is depicted in the two-parameter $(J,R)$ phase diagram for competitive attractive and repulsive interactions for ${\varepsilon }_a={\varepsilon }_r=0.5$. The left panel depicts the evolution of a chimera state as the vision radius $R$ increases from $R=1.4$ to 1.8. The snapshots of the swarmalators depicting distinct collective dynamical states are shown in the top and bottom panels. These states are characterized using order parameters ($S,\mathrm{\Lambda },N_c,E,H$).

Figure 4

(a) Synchronization order parameter $S$ is depicted as the heat map for ${\varepsilon }_a=0.5$ and ${\varepsilon }_r=0$ in the $(J,R)$ space. Refer to the text for details. (b) Swarmalators with orthogonal angular frequencies. Simulations are performed for $N=100,{\varepsilon }_a=0.9,{\varepsilon }_r=0.1$. The pumping state is a dynamic state in which swarmalators compress and expand in a rhythmic pattern.

Figure 5

Dynamics of two swarmalators. (a) Cluster separation of a two-cluster state with varying $J$. The solid red line is the analytical prediction, and the open circles are the simulation result. (b) Dynamics of the swarmalator model for $N=2$ for competitive interaction among the orientation vectors. The heat map illustrates the synchronization order parameter estimated from the evolution of the orientation vectors. The sufficient condition for synchronization was shown to be $R>|{\mathit{x}}_{\mathit{12}}|=1/\sqrt{1+J({\mathbf{\sigma }}_{\mathit{1}}·{\mathbf{\sigma }}_{\mathit{2}})}$. The analytical critical curve $R=1/\sqrt{1\pm J}$, depicted in the figure, agrees well with the simulation results.

Figure 6

Real-world parallels. (a) Experimentally observed [54] polarized (crystal) state for 300 golden shiners in shallow water at $t=330$. (b) Polarized state in our swarmalator model for $N=300, R=1.0, J=0.7, {\varepsilon }_a=0.9, {\varepsilon }_r=0$. (c) Experimentally observed milling state at $t=1450$. (d) Milling state observed in the proposed swarmalator model for $N=300, R=0.2, J=0.3, {\varepsilon }_a=0.7, {\varepsilon }_r=0$. (e), (f) Time evolution of synchronization $\left(S\right)$ and spatial vorticity $\left({\mathrm{\Gamma }}_x\right)$ order parameters characterizing polarized and milling states, respectively, from both experimental $\left[S\right(\text{expt})$ and ${\mathrm{\Gamma }}_x\left(\text{expt}\right)]$ and model $\left[S\right(\text{model})$ and ${\mathrm{\Gamma }}_x\left(\text{model}\right)]$ data with noise strength ($d_{x_k}=0.005,d_{{\sigma }_k}=0.005$).