Model reduction for the Kuramoto-Sakaguchi model: The importance of nonentrained rogue oscillators

Synchronization transitions in Kuramoto networks with higher-mode interaction

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type.

Generalized frustration in the multidimensional Kuramoto model

The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D=2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix K that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration that tends to hinder synchronization. In a recent paper we studied in detail the role of the coupling matrix for D=2. Here we extend this analysis to arbitrary dimensions. We show that, for identical particles, when the natural frequencies are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether D is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If D is odd the phase transition is discontinuous and active states can be suppressed for some distributions of natural frequencies.

Matrix coupling and generalized frustration in Kuramoto oscillators

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.

Collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the timescale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.

Mesoscopic model reduction for the collective dynamics of sparse coupled oscillator networks

The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques that capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages, we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from O(N²) to O(1). We perform numerical simulations for Erdős-Rényi networks and for modified Barabási-Albert networks demonstrating remarkable quantitative agreement at and close to bifurcation.

Model reduction for the collective dynamics of globally coupled oscillators: From finite networks to the thermodynamic limit

Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here, we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.