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Oscillators that sync and swarm
Kevin P. O'Keeffe, Hyunsuk Hong, Steven H. Strogatz
TL;DR
The paper studies oscillators whose phase and spatial dynamics are coupled, addressing collective behavior that combines synchronization with swarming. Using simple swarmalator models, it predicts five collective states and finds patterns robust to several model modifications.
Problem
The paper asks how systems can exhibit synchronization and swarming together when oscillators’ phase and spatial dynamics are coupled.
Method
The authors study simple swarmalator models analytically and numerically, examining collective patterns under aggregation and synchronization and selected model modifications.
Results
Five collective states emerge, including static and moving states, and the resulting patterns remain robust to changes in dimension, natural-frequency distributions, noise, and alignment dynamics.
Takeaways & Limitations
The model maps part of a broad landscape of emergent behavior and suggests these states could occur in biological or technological systems, with possible locomotive utility for non-stationary waves.
Takeaways & Limitations
The model initially excludes orientation and does not exhaustively study orientable swarmalators, restricting the work to robustness under simple alignment dynamics.
Abstract
from arXiv · showhide
Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.
Results
The model couples phase and spatial dynamics, producing five collective states across the (J, K) parameter plane. Three states are stationary, while splintered and active phase waves involve motion and phase–space organization.
- Model: The model couples phase interaction to spatial attraction, with J controlling phase-dependent attraction and K controlling phase synchronization or anti-synchronization.Positive J favors proximity between similar phases, negative J favors opposite phases, and K favors minimizing phase differences when positive but maximizing them when negative.
- Collective states: Numerics identify five long-term states: static sync, static async, static phase wave, splintered phase wave, and active phase wave.Three states are ultimately static in space and phase, whereas the remaining two are moving states; all have time-independent density at the population level.
- Stationary states: Static sync forms a crystal-like, fully synchronized spatial distribution for K > 0, whereas static async has every phase at every location and occurs for K < 0 or within a J-dependent wedge.The static async region includes J < 0, K < 0 and also J > 0 when J < |Kc|.
- Stationary states: At K = 0 and J > 0, the static phase wave forms an annulus whose spatial angle is perfectly correlated with phase.The phase values remain at their initial values while like-phased swarmalators aggregate near one another around the ring.
- Non-stationary states: For K < 0, the static phase wave first splinters into phase-distinct clusters that quiver, then becomes an active phase wave with regular coupled cycles in angle, phase, and radius.The active phase wave exhibits shear flow and radial travel between the annulus’s inner and outer rims.
- State classification: The order parameters S± and γ distinguish the states: S± tracks spatial-angle–phase correlation, while γ separates splintered from active phase waves.S± equals zero for static async, one for static phase wave, and is nonzero in the splintered and active states; γ is zero for splintered and nonzero for active.
Discussion
Swarmalators combine mobile spatial dynamics with internal phase dynamics, whose coupling produces diverse collective patterns. The model’s states were robust to several modifications and suggest experimentally testable systems and future applications.
- Swarmalators are mobile particles or agents with coupled phase and spatial degrees of freedom, allowing them to synchronize and swarm.
- Aligned versions of the model’s states persist while their centers of mass move, making these states mobile but equivalent to corresponding static states.
- The model’s collective patterns remained robust in one, two, and three dimensions, with distributed natural frequencies, noisy interactions, and alignment dynamics.
- Magnetic colloids and active spinners are promising physical candidates, while sperm and other biological microswimmers may realize related swarmalator states.
- Future work should examine more realistic coupling and interaction functions, because the chosen Kuramoto-like phase interaction yields only the trivial static sync state when K > 0.
- The model’s splintered and active phase waves were described as unprecedented among previously considered models.
Methods
The methods derive self-consistent density and support conditions for static swarmalator states using continuity, velocity, divergence, and interaction-kernel calculations. The analysis obtains explicit forms for some states and numerical treatment for the unit-vector kernel.
- Properties of static sync and async state: For static async and sync states, the analysis imposes zero velocity and zero divergence, then uses the continuity equation to derive self-consistent density equations.
- Properties of static sync and async state: The static async state has uniform density inside a disk, with ρasync(r, φ, θ, t) = 1/(2π^2) for 0 ≤ r ≤ Rasync.
- Properties of static sync and async state: Substitution into the governing equations verifies that the static async and sync solutions satisfy the required zero-velocity condition when R = 1.
- Properties of static phase wave state: For the static phase wave, the density is concentrated on an annulus with phase linked to polar angle, and self-consistency determines its radial profile and inner and outer radii.
- Properties of static phase wave state: With the linear attraction kernel Iatt(x) = x, the radial density becomes uniform and the annulus radii follow from setting velocity coefficients to zero.