Renormalization of oscillator lattices with disorder

Stability of twisted states on lattices of Kuramoto oscillators

Real world systems comprised of coupled oscillators have the ability to exhibit spontaneous synchronization and other complex behaviors. The interplay between the underlying network topology and the emergent dynamics remains a rich area of investigation for both theory and experiment. In this work, we study lattices of coupled Kuramoto oscillators with non-local interactions. Our focus is on the stability of twisted states. These are equilibrium solutions with constant phase shifts between oscillators resulting in spatially linear profiles. Linear stability analysis follows from studying the quadratic form associated with the Jacobian matrix. Novel estimates on both stable and unstable regimes of twisted states are obtained in several cases. Moreover, exploiting the "almost circulant" nature of the Jacobian obtains a surprisingly accurate numerical test for stability. While our focus is on 2D square lattices, we show how our results can be extended to higher dimensions.

Oscillators that sync and swarm

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.

Collective self-organized behavior can be observed in a variety of systems such as colloids and microswimmers. Here O’Keeffe et al. propose a model of oscillators which move in space and tend to synchronize with neighboring oscillators and outline five types of collective self-organized states.

Frequency spirals

We study the dynamics of coupled phase oscillators on a two-dimensional Kuramoto lattice with periodic boundary conditions. For coupling strengths just below the transition to global phase-locking, we find localized spatiotemporal patterns that we call "frequency spirals." These patterns cannot be seen under time averaging; they become visible only when we examine the spatial variation of the oscillators' instantaneous frequencies, where they manifest themselves as two-armed rotating spirals. In the more familiar phase representation, they appear as wobbly periodic patterns surrounding a phase vortex. Unlike the stationary phase vortices seen in magnetic spin systems, or the rotating spiral waves seen in reaction-diffusion systems, frequency spirals librate: the phases of the oscillators surrounding the central vortex move forward and then backward, executing a periodic motion with zero winding number. We construct the simplest frequency spiral and characterize its properties using analytical and numerical methods. Simulations show that frequency spirals in large lattices behave much like this simple prototype.

Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions

Recent experiments in one- and two-dimensional microfluidic arrays of droplets containing Belousov-Zhabotinsky reactants show a rich variety of spatial patterns [M. Toiya et al., J. Phys. Chem. Lett. 1, 1241 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest-neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2π/3. We report the existence of domains with either clockwise or anticlockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.

Vortices and the entrainment transition in the two-dimensional Kuramoto model

We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling K is larger than a threshold K{E} , there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at K{E} . We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase (K>K{E}) , vortices move in fixed paths around clusters, while in the unentrained phase (K<K{E}) , vortices sometimes wander off. These deviant vortices are responsible for the disappearance of the macroscopic cluster. The regularity of vortex motion is determined by whether clusters behave as single effective oscillators. The unentrained phase is also characterized by time-dependent cluster structure and the presence of chaos. Thus, the entrainment transition is actually an order-chaos transition. We present an analytical argument for the scaling K{E}∼K{L} for small lattices, where K{L} is the threshold for phase locking. By also deriving the scaling K{L}∼log N , we thus show that K{E}∼log N for small N , in agreement with numerics. In addition, we show how to use the linearized model to predict where vortices are generated.