Erosion of synchronization in networks of coupled oscillators

Impact of phase lag on synchronization in frustrated Kuramoto model with higher-order interactions

The study of first order transition (explosive synchronization) in an ensemble (network) of coupled oscillators has been the topic of paramount interest among the researchers for more than one decade. Several frameworks have been proposed to induce explosive synchronization in a network and it has been reported that phase frustration in a network usually suppresses first order transition in the presence of pairwise interactions among the oscillators. However, on the contrary, by considering networks of phase frustrated coupled oscillators in the presence of higher-order interactions (up to 2-simplexes) we show here, under certain conditions, phase frustration can promote explosive synchronization in a network. A low-dimensional model of the network in the thermodynamic limit is derived using the Ott-Antonsen ansatz to explain this surprising result. Analytical treatment of the low-dimensional model, including bifurcation analysis, explains the apparent counter intuitive result quite clearly.

Perfect synchronization in complex networks with higher-order interactions

Achieving perfect synchronization in a complex network, specially in the presence of higher-order interactions (HOIs) at a targeted point in the parameter space, is an interesting, yet challenging task. Here we present a theoretical framework to achieve the same under the paradigm of the Sakaguchi-Kuramoto (SK) model. We analytically derive a frequency set to achieve perfect synchrony at some desired point in a complex network of SK oscillators with higher-order interactions. Considering the SK model with HOIs on top of the scale-free, random, and small world networks, we perform extensive numerical simulations to verify the proposed theory. Numerical simulations show that the analytically derived frequency set not only provides stable perfect synchronization in the network at a desired point but also proves to be very effective in achieving a high level of synchronization around it compared to the other choices of frequency sets. The stability and the robustness of the perfect synchronization state of the system are determined using the low-dimensional reduction of the network and by introducing a Gaussian noise around the derived frequency set, respectively.

Amplification of explosive width in complex networks

We present an adaptive coupling strategy to induce hysteresis/explosive synchronization in complex networks of phase oscillators (Sakaguchi-Kuramoto model). The coupling strategy ensures explosive synchronization with significant explosive width enhancement. Results show the robustness of the strategy, and the strategy can diminish (by inducing enhanced hysteresis loop) the contrarian impact of phase frustration in the network, irrespective of the network structure or frequency distributions. Additionally, we design a set of frequency for the oscillators, which eventually ensure complete in-phase synchronization behavior among these oscillators (with enhanced explosive width) in the case of adaptive-coupling scheme. Based on a mean-field analysis, we develop a semi-analytical formalism, which can accurately predict the backward transition of the synchronization order parameter.

Development of structural correlations and synchronization from adaptive rewiring in networks of Kuramoto oscillators

Synchronization of non-identical oscillators coupled through complex networks is an important example of collective behavior, and it is interesting to ask how the structural organization of network interactions influences this process. Several studies have explored and uncovered optimal topologies for synchronization by making purposeful alterations to a network. On the other hand, the connectivity patterns of many natural systems are often not static, but are rather modulated over time according to their dynamics. However, this co-evolution and the extent to which the dynamics of the individual units can shape the organization of the network itself are less well understood. Here, we study initially randomly connected but locally adaptive networks of Kuramoto oscillators. In particular, the system employs a co-evolutionary rewiring strategy that depends only on the instantaneous, pairwise phase differences of neighboring oscillators, and that conserves the total number of edges, allowing the effects of local reorganization to be isolated. We find that a simple rule-which preserves connections between more out-of-phase oscillators while rewiring connections between more in-phase oscillators-can cause initially disordered networks to organize into more structured topologies that support enhanced synchronization dynamics. We examine how this process unfolds over time, finding a dependence on the intrinsic frequencies of the oscillators, the global coupling, and the network density, in terms of how the adaptive mechanism reorganizes the network and influences the dynamics. Importantly, for large enough coupling and after sufficient adaptation, the resulting networks exhibit interesting characteristics, including degree-frequency and frequency-neighbor frequency correlations. These properties have previously been associated with optimal synchronization or explosive transitions in which the networks were constructed using global information. On the contrary, by considering a time-dependent interplay between structure and dynamics, this work offers a mechanism through which emergent phenomena and organization can arise in complex systems utilizing local rules.

A universal order parameter for synchrony in networks of limit cycle oscillators

We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously prove that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids.

Chimera-like states in structured heterogeneous networks

Chimera-like states are manifested through the coexistence of synchronous and asynchronous dynamics and have been observed in various systems. To analyze the role of network topology in giving rise to chimera-like states, we study a heterogeneous network model comprising two groups of nodes, of high and low degrees of connectivity. The architecture facilitates the analysis of the system, which separates into a densely connected coherent group of nodes, perturbed by their sparsely connected drifting neighbors. It describes a synchronous behavior of the densely connected group and scaling properties of the induced perturbations.

Optimal phase synchronization in networks of phase-coherent chaotic oscillators

We investigate the existence of an optimal interplay between the natural frequencies of a group of chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function J(ω,L) linking the natural frequencies ω_i of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix L, the latter accounting for the specific organization of the network of interactions between oscillators. We use the classical Rössler system to show that the synchrony alignment function obtained for phase oscillators can be extended to phase-coherent chaotic systems. Finally, we carry out a series of experiments with nonlinear electronic circuits to show the robustness of the theoretical predictions despite the intrinsic noise and parameter mismatch of the electronic components.

Optimal synchronization of directed complex networks

We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between the network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also allows us to study the properties of synchrony-optimized networks, and in particular, investigate the role of directed network properties such as nodal in- and out-degrees. For instance, we find that in optimally rewired networks, the heterogeneity of the in-degree distribution roughly matches the heterogeneity of the natural frequency distribution, but no such relationship emerges for out-degrees. We also observe that a network's synchronization properties are promoted by a strong correlation between the nodal in-degrees and the natural frequencies of oscillators, whereas the relationship between the nodal out-degrees and the natural frequencies has comparatively little effect. This result is supported by our theory, which indicates that synchronization is promoted by a strong alignment of the natural frequencies with the left singular vectors corresponding to the largest singular values of the Laplacian matrix.

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

Erosion of synchronization: Coupling heterogeneity and network structure

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

We present an analysis of conditions under which the dynamics of a frustrated Kuramoto-or Kuramoto-Sakaguchi-model on sparse networks can be tuned to enhance synchronization. Using numerical optimization techniques, linear stability, and dimensional reduction analysis, a simple tuning scheme for setting node-specific frustration parameters as functions of native frequencies and degrees is developed. Finite-size scaling analysis reveals that even partial application of the tuning rule can significantly reduce the critical coupling for the onset of synchronization. In the second part of the paper, a codynamics is proposed, which allows a dynamic tuning of frustration parameters simultaneously with the ordinary Kuramoto dynamics. We find that such codynamics enhance synchronization when operating on slow time scales, and impede synchronization when operating on fast time scales relative to the Kuramoto dynamics.

Collective frequency variation in network synchronization and reverse PageRank

A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an outflow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.

SYNCHRONIZATION OF HETEROGENEOUS OSCILLATORS UNDER NETWORK MODIFICATIONS: PERTURBATION AND OPTIMIZATION OF THE SYNCHRONY ALIGNMENT FUNCTION

Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications-for which proper functionality depends sensitively on the extent of synchronization-there remains a lack of understanding for how systems can best evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.

Rheotaxis performance increases with group size in a coupled phase model with sensory noise: The effects of noise and group size on rheotaxis

Many fish exhibit rheotaxis, a behavior in which fish orient themselves relative to flow. Rheotaxis confers many benefits, including energetic cost savings and interception of drifting prey. Despite the fact that most species of fish school during at least some portion of their life, little is known about the importance of rheotactic behavior to schooling fish and, conversely, how the presence of nearby conspecifics affects rheotactic behavior. Understanding how rheotaxis is modified by social factors is thus of ecological importance. Here we present a mathematical model in the form of an all-to-all, coupled-oscillator framework over the non-Euclidean space of fish orientations to model group rheotactic behavior. Individuals in the model measure the orientation of their neighbors and the flow direction relative to their own orientation. These measures are corrupted by sensory noise. We study the effect of sensory noise and group size on internal (i.e., within the school) and external (i.e., with the flow) disagreement in orientation. We find that under noisy environmental conditions, increased group size improves rheotaxis. Results of this study have implications for understanding animal behavior, as well as for potential applications in bio-inspired engineering.

Control of coupled oscillator networks with application to microgrid technologies

The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.