Two mechanisms of remote synchronization in a chain of Stuart-Landau oscillators

Phase reduction explains chimera shape: When multibody interaction matters

We present an extension of the Kuramoto-Sakaguchi model for networks, deriving the second-order phase approximation for a paradigmatic model of oscillatory networks-an ensemble of nonidentical Stuart-Landau oscillators coupled pairwisely via an arbitrary coupling matrix. We explicitly demonstrate how this matrix translates into the coupling structure in the phase equations. To illustrate the power of our approach and the crucial importance of high-order phase reduction, we tackle a trendy setup of nonlocally coupled oscillators exhibiting a chimera state. We reveal that our second-order phase model reproduces the dependence of the chimera shape on the coupling strength that is not captured by the typically used first-order Kuramoto-like model. Our derivation contributes to a better understanding of complex networks' dynamics, establishing a relation between the coupling matrix and multibody interaction terms in the high-order phase model.

Inferring connectivity of an oscillatory network via the phase dynamics reconstruction

We review an approach for reconstructing oscillatory networks' undirected and directed connectivity from data. The technique relies on inferring the phase dynamics model. The central assumption is that we observe the outputs of all network nodes. We distinguish between two cases. In the first one, the observed signals represent smooth oscillations, while in the second one, the data are pulse-like and can be viewed as point processes. For the first case, we discuss estimating the true phase from a scalar signal, exploiting the protophase-to-phase transformation. With the phases at hand, pairwise and triplet synchronization indices can characterize the undirected connectivity. Next, we demonstrate how to infer the general form of the coupling functions for two or three oscillators and how to use these functions to quantify the directional links. We proceed with a different treatment of networks with more than three nodes. We discuss the difference between the structural and effective phase connectivity that emerges due to high-order terms in the coupling functions. For the second case of point-process data, we use the instants of spikes to infer the phase dynamics model in the Winfree form directly. This way, we obtain the network's coupling matrix in the first approximation in the coupling strength.

High-order phase reduction for coupled 2D oscillators

Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. The coupling should be small for such reduction, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework, allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction.

Collective behavior of identical Stuart-Landau oscillators in a star network with coupling asymmetry effects

In this study, we investigated the impact of the asymmetry of a coupling scheme on oscillator dynamics in a star network. We obtained stability conditions for the collective behavior of the systems, ranging from an equilibrium point over complete synchronization (CS) and quenched hub incoherence to remote synchronization states using both numerical and analytical methods. The coupling asymmetry factor α significantly influences and determines the stable parameter region of each state. For α ≠ 1, the equilibrium point can emerge when the Hopf bifurcation parameter a is positive, which is impossible for diffusive coupling. However, CS can occur even if a is negative under α < 1. Unlike diffusive coupling, we observe more behavior when α ≠ 1, including additional in-phase remote synchronization. These results are supported by theoretical analysis and validated through numerical simulations and independent of network size. The findings may offer practical methods for controlling, restoring, or obstructing specific collective behavior.

Phase frustration induced remote synchronization

Remote synchronization (RS) may take an important role in brain functioning and its study has attracted much attention in recent years. So far, most studies of RS are focused on the Stuart-Landau oscillators with mean-field coupling. However, realistic cases may have more complicated couplings and behaviors, such as the brain networks. To make the study of RS a substantial progress toward realistic situations, we here present a model of RS with phase frustration and show that RS can be induced for those systems where no RS exists when there is no phase frustration. By numerical simulations on both the Stuart-Landau and Kuramoto oscillators, we find that the optimal range of RS depends on the match of phase frustrations between the hub and leaf nodes and a fixed relationship of this match is figured out. While for the non-optimal range of RS, we find that RS exists only in a linear band between the phase frustrations of the hub and leaf nodes. A brief theoretical analysis is provided to explain these results.

Enlarged Kuramoto model: Secondary instability and transition to collective chaos

The emergence of collective synchrony from an incoherent state is a phenomenon essentially described by the Kuramoto model. This canonical model was derived perturbatively, by applying phase reduction to an ensemble of heterogeneous, globally coupled Stuart-Landau oscillators. This derivation neglects nonlinearities in the coupling constant. We show here that a comprehensive analysis requires extending the Kuramoto model up to quadratic order. This "enlarged Kuramoto model" comprises three-body (nonpairwise) interactions, which induce strikingly complex phenomenology at certain parameter values. As the coupling is increased, a secondary instability renders the synchronized state unstable, and subsequent bifurcations lead to collective chaos. An efficient numerical study of the thermodynamic limit, valid for Gaussian heterogeneity, is carried out by means of a Fourier-Hermite decomposition of the oscillator density.