Partial synchronization in the second-order Kuramoto model: An auxiliary system method

Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.

Synchronization of nonlinearly coupled networks based on circle criterion

The problem of synchronization in networks of linear systems with nonlinear diffusive coupling and a connected undirected graph is studied. By means of a coordinate transformation, the system is reduced to the form of mean-field dynamics and a synchronization-error system. The network synchronization conditions are established based on the stability conditions of the synchronization-error system obtained using the circle criterion, and the results are used to derive the condition for synchronization in a network of neural-mass-model populations with a connected undirected graph. Simulation examples are presented to illustrate the obtained results.

Detecting coexisting oscillatory patterns in delay coupled Lur'e systems

This work addresses the problem of pattern analysis in networks consisting of delay-coupled identical Lur'e systems. We study a class of nonlinear systems, which, being isolated, are globally asymptotically stable. Assembling such systems into a network via time-delayed coupling may result in the change of network equilibrium stability under parameter variation in the coupling. In this work, we focus on cases where a Hopf bifurcation causes the change of stability of the network equilibrium and leads to the occurrence of oscillatory modes (patterns). Moreover, some of these patterns can co-exist for the same set of coupling parameters, which makes the analysis by means of common methods, such as the Lyapunov-Krasovskii method or the analysis of Poincaré maps, cumbersome. A numerically efficient algorithm, aiming at the computation of the oscillatory patterns occurring in such networks, is presented. Moreover, we show that our approach is able to deal with co-existing patterns, and both stable and unstable regimes can be simultaneously computed, which gives deep insight into the network dynamics. In order to illustrate the efficiency of the method, we present two examples in which the instability of the network equilibria is caused by a subcritical and a supercritical Hopf bifurcation. In addition, a bifurcation analysis of the subcritical case is performed in order to further explain the occurrence of the detected coexisting modes.

When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia

Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters. We extend our previous results on the bistability of synchronized clusters in a two-population network [I. V. Belykh et al., Chaos 26, 094822 (2016)CHAOEH1054-150010.1063/1.4961435] and demonstrate that the addition of a third population can induce chaotic intercluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd," which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the intercluster phase shifts can stably coexist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our stability results for predicting the emergence of chimeras and solitary states.

Synchronizability of directed networks: The power of non-existent ties

The understanding of how synchronization in directed networks is influenced by structural changes in network topology is far from complete. While the addition of an edge always promotes synchronization in a wide class of undirected networks, this addition may impede synchronization in directed networks. In this paper, we develop the augmented graph stability method, which allows for explicitly connecting the stability of synchronization to changes in network topology. The transformation of a directed network into a symmetrized-and-augmented undirected network is the central component of this new method. This transformation is executed by symmetrizing and weighting the underlying connection graph and adding new undirected edges with consideration made for the mean degree imbalance of each pair of nodes. These new edges represent "non-existent ties" in the original directed network and often control the location of critical nodes whose directed connections can be altered to manipulate the stability of synchronization in a desired way. In particular, we show that the addition of small-world shortcuts to directed networks, which makes "non-existent ties" disappear, can worsen the synchronizability, thereby revealing a destructive role of small-world connections in directed networks. An extension of our method may open the door to studying synchronization in directed multilayer networks, which cannot be effectively handled by the eigenvalue-based methods.

State-dependent effective interactions in oscillator networks through coupling functions with dead zones

The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have 'dead zones', that is the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories, the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Uptake kinetics and storage capacity of dissolved inorganic phosphorus and corresponding dissolved inorganic nitrate uptake in Saccharina latissima and Laminaria digitata (Phaeophyceae)

Uptake rates of dissolved inorganic phosphorus and dissolved inorganic nitrogen under unsaturated and saturated conditions were studied in young sporophytes of the seaweeds Saccharina latissima and Laminaria digitata (Phaeophyceae) using a "pulse-and-chase" assay under fully controlled laboratory conditions. In a subsequent second "pulse-and-chase" assay, internal storage capacity (ISC) was calculated based on VM and the parameter for photosynthetic efficiency Fv /Fm . Sporophytes of S. latissima showed a VS of 0.80 ± 0.03 μmol · cm-2  · d-1 and a VM of 0.30 ± 0.09 μmol · cm-2  · d-1 for dissolved inorganic phosphate (DIP), whereas VS for DIN was 11.26 ± 0.56 μmol · cm-2  · d-1 and VM was 3.94 ± 0.67 μmol · cm-2  · d-1 . In L. digitata, uptake kinetics for DIP and DIN were substantially lower: VS for DIP did not exceed 0.38 ± 0.03 μmol · cm-2  · d-1 while VM for DIP was 0.22 ± 0.01 μmol · cm-2  · d-1 . VS for DIN was 3.92 ± 0.08 μmol · cm-2  · d-1 and the VM for DIN was 1.81 ± 0.38 μmol · cm-2  · d-1 . Accordingly, S. latissima exhibited a larger ISC for DIP (27 μmol · cm-2 ) than L. digitata (10 μmol · cm-2 ), and was able to maintain high growth rates for a longer period under limiting DIP conditions. Our standardized data add to the physiological understanding of S. latissima and L. digitata, thus helping to identify potential locations for their cultivation. This could further contribute to the development and modification of applications in a bio-based economy, for example, in evaluating the potential for bioremediation in integrated multitrophic aquacultures that produce biomass simultaneously for use in the food, feed, and energy industries.

Enhancing master-slave synchronization: The effect of using a dynamic coupling

This paper introduces a modified master-slave synchronization scheme for dynamical systems. In contrast to the standard configuration, the slave system does not receive any driving signal from the master, but rather the interaction is through a linear dynamical system. The key feature of the proposed coupling scheme is that it induces synchronization in certain systems that cannot be synchronized when using the classical static interconnection. Likewise, the dynamic coupling achieves synchronization for arbitrarily large coupling strength values in certain systems for which the classical configuration is applicable only within a narrow interval of coupling strength values. The performance of the synchronization scheme is illustrated in pairs of identical chaotic and mechanical oscillators.

Bistability of patterns of synchrony in Kuramoto oscillators with inertia

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.

Popov-Type Criterion for Consensus in Nonlinearly Coupled Networks

This paper addresses consensus problems in nonlinearly coupled networks of dynamic agents described by a common and arbitrary linear model. Interagent interaction rules are uncertain but satisfy the standard sector condition with known sector bounds; both the agent's model and interaction topology are time-invariant. A novel frequency-domain criterion for consensus is offered that is similar to and extends the classical Popov's absolute stability criterion.

Synchronization on complex networks of networks

In this paper, pinning synchronization on complex networks of networks is investigated, where there are many subnetworks with the interactions among them. The subnetworks and their connections can be regarded as the nodes and interactions of the networks, respectively, which form the networks of networks. In this new setting, the aim is to design pinning controllers on the chosen nodes of each subnetwork so as to reach synchronization behavior. Some synchronization criteria are established for reaching pinning control on networks of networks. Furthermore, the pinning scheme is designed, which shows that the nodes with very low degrees and large degrees are good candidates for applying pinning controllers. Then, the attack and robustness of the pinning scheme are discussed. Finally, a simulation example is presented to verify the theoretical analysis in this paper.

Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance

All edges in the classical Watts and Strogatz's small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength and several topological variations.

Spontaneous scale-free structure in adaptive networks with synchronously dynamical linking

Inspired by the anti-Hebbian learning rule in neural systems, we study how the feedback from dynamical synchronization shapes network structure by adding new links. Through extensive numerical simulations, we find that an adaptive network spontaneously forms scale-free structure, as confirmed in many real systems. Moreover, the adaptive process produces two nontrivial power-law behaviors of deviation strength from mean activity of the network and negative degree correlation, which exists widely in technological and biological networks. Importantly, these scalings are robust to variation of the adaptive network parameters, which may have meaningful implications in the scale-free formation and manipulation of dynamical networks. Our study thus suggests an alternative adaptive mechanism for the formation of scale-free structure with negative degree correlation, which means that nodes of high degree tend to connect, on average, with others of low degree and vice versa. The relevance of the results to structure formation and dynamical property in neural networks is briefly discussed as well.

Partial synchronization in diffusively time-delay coupled oscillator networks

We study networks of diffusively time-delay coupled oscillatory units and we show that networks with certain symmetries can exhibit a form of incomplete synchronization called partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of oscillatory units that satisfy a semipassivity property and have convergent internal dynamics.

NETWORK COMPLEXITY AND SYNCHRONOUS BEHAVIOR — AN EXPERIMENTAL APPROACH

We discuss synchronization in networks of Hindmarsh-Rose neurons that are interconnected via gap junctions, also known as electrical synapses. We present theoretical results for interactions without time-delay. These results are supported by experiments with a setup consisting of sixteen electronic equivalents of the Hindmarsh-Rose neuron. We show experimental results of networks where time-delay on the interaction is taken into account. We discuss in particular the influence of the network topology on the synchronization.

Mesoscale and clusters of synchrony in networks of bursting neurons

We study the role of network architecture in the formation of synchronous clusters in synaptically coupled networks of bursting neurons. We give a simple combinatorial algorithm that finds the largest synchronous clusters from the network topology. We demonstrate that networks with a certain degree of internal symmetries are likely to have cluster decompositions with relatively large clusters, leading potentially to cluster synchronization at the mesoscale network level. We also address the asymptotic stability of cluster synchronization in excitatory networks of Hindmarsh-Rose bursting neurons and derive explicit thresholds for the coupling strength that guarantees stable cluster synchronization.

Synchrony in tritrophic food chain metacommunities

The synchronous behaviour of interacting communities is studied in this paper. Each community is described by a tritrophic food chain model, and the communities interact through a network with arbitrary topology, composed of patches and migration corridors. The analysis of the local synchronization properties (via the master stability function approach) shows that, if only one species can migrate, the dispersal of the consumer (i.e., the intermediate trophic level) is the most effective mechanism for promoting synchronization. When analysing the effects of the variations of demographic parameters, it is found that factors that stabilize the single community also tend to favour synchronization. Global synchronization is finally analysed by means of the connection graph method, yielding a lower bound on the value of the dispersion rate that guarantees the synchronization of the metacommunity for a given network topology.

Coordinate transformation and matrix measure approach for synchronization of complex networks

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Multivariable Harmonic Balance for Central Pattern Generators

The central pattern generator (CPG) is a nonlinear oscillator formed by a group of neurons, providing a fundamental control mechanism underlying rhythmic movements in animal locomotion. We consider a class of CPGs modeled by a set of interconnected identical neurons. Based on the idea of multivariable harmonic balance, we show how the oscillation profile is related to the connectivity matrix that specifies the architecture and strengths of the interconnections. Specifically, the frequency, amplitudes, and phases are essentially encoded in terms of a pair of eigenvalue and eigenvector. This basic principle is used to estimate the oscillation profile of a given CPG model. Moreover, a systematic method is proposed for designing a CPG-based nonlinear oscillator that achieves a prescribed oscillation profile.

A partial synchronization theorem

When synchronization sets in, coupled systems oscillate in a coherent way. It is possible to observe also some intermediate regimes characterized by incomplete synchrony which are referred to as partial synchronization. The paper focuses on analysis of partial synchronization in networks of linearly coupled oscillators.

Global synchronization in lattices of coupled chaotic systems

Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.

Application of partial observability for analysis and design of synchronized systems

Synchronization in identical drive-response systems is a problem that can be cast in an observer design framework. In this paper we extend this approach by studying the analysis/design of partial synchronization by means of observer theory. In doing so, we introduce the concept of partial observer--an observer to reconstruct a part of the system state vector. It is also shown how the observability condition can be utilized to analyze the dynamics in an array of coupled identical systems.

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.