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.

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Sensors can measure air quality, traffic congestion, and other aspects of urban environments. The fine-grained diagnostic information they provide could help urban managers to monitor a city’s health. Recently, a “drive-by” paradigm has been proposed in which sensors are deployed on third-party vehicles, enabling wide coverage at low cost. Research on drive-by sensing has mostly focused on sensor engineering, but a key question remains unexplored: How many vehicles would be required to adequately scan a city? Here, we address this question by analyzing the sensing power of a taxi fleet. Taxis, being numerous in cities, are natural hosts for the sensors. Using a ball-in-bin model in tandem with a simple model of taxi movements, we analytically determine the fraction of a city’s street network sensed by a fleet of taxis during a day. Our results agree with taxi data obtained from nine major cities and reveal that a remarkably small number of taxis can scan a large number of streets. This finding appears to be universal, indicating its applicability to cities beyond those analyzed here. Moreover, because taxis’ motion combines randomness and regularity (passengers’ destinations being random, but the routes to them being deterministic), the spreading properties of taxi fleets are unusual; in stark contrast to random walks, the stationary densities of our taxi model obey Zipf’s law, consistent with empirical taxi data. Our results have direct utility for town councilors, smart-city designers, and other urban decision makers.

Synchronization is a universal phenomenon, occurring in systems as disparate as Japanese tree frogs and Josephson junctions. Typically, the elements of synchronizing systems adjust the phases of their oscillations, but not their positions in space. The reverse scenario is found in swarming systems, such as schools of fish or flocks of birds; now the elements adjust their positions in space, but without (noticeably) changing their internal states. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed, and analyzed in the continuum limit. Here, we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density we calculate explicitly. These states may be observable in groups of Japanese tree frogs, ferromagnetic colloids, and other systems with an interplay between swarming and synchronization.