Uniform Coupling Strength Research Articles

Interplay of inertia and adaptive couplings in the emergent dynamics of Kuramoto ensemble

We study the emergent dynamics of Kuramoto oscillators under the interplay between inertia and adaptive couplings. For the phase dynamics, we use the inertial Kuramoto model with time-dependent mutual coupling strengths. For the constant and uniform coupling strength, the inertial Kuramoto model can exhibit the slow relaxation dynamics toward phase-locked state under suitable conditions on system parameters and initial data. To model the time-evolution of mutual coupling strengths, we employ two types of coupling functions, namely “Hebbian coupling” and “anti-Hebbian coupling”. With these modeling spirit, the resulting coupled dynamics for the phase and mutual coupling strength becomes the coupled second-order and first-order systems of ordinary differential equations. For the proposed coupled system, we provide several sufficient frameworks for phase and frequency synchronization in terms of system parameters and initial data for Hebbian and anti-Hebbian coupling functions.

Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature

The synchronization of an ensemble of oscillators is a phenomenon present in systems of different fields, ranging from social to physical and biological systems. This phenomenon is often described mathematically by the Kuramoto model, which assumes oscillators of fixed natural frequencies connected by an equal and uniform coupling strength, with an analytical solution possible only for an infinite number of oscillators. However, most real-life synchronization systems consist of a finite number of oscillators and are often perturbed by external fields. This paper accommodates the perturbation using a time-dependent coupling strength K(t) in the form of a sinusoidal function and a step function using 32 oscillators that serve as a representative of finite oscillators. The temporal evolution of order parameter r(t) and phases θj(t), key indicators of synchronization, are compared between the uniform and time-dependent cases. The identical trends observed in the two cases give an important indication that the synchrony persists even under the influence of external factors, something very plausible in the context of real-life synchronization events. The occasional boosting of coupling strength is also enough to keep the assembly of oscillators in a synchronized state persistently.

The structure of uni-directional chain for the synchronization of networked chaotic systems

In this paper, we introduce the structure of uni-directional chain (UDC) as a benchmark to probe the synchronizability of networked chaotic system. By studying the relation between the length of UDC and its synchronizability, we identify a region of coupling strength, referred to as strong controllable region (SCR), where a UDC with arbitrary length could realize synchronization that is robust to non-local disturbance. More importantly, it is found that there exists a kind of chaotic oscillators where the region of negative master stability function (MSF) overlaps with the SCR. As MSF implements local stability analysis to provide the necessary condition for the appearance of synchronization, while SCR permits stable synchronization corresponding to non-local disturbance pertinent to sufficient condition, the role of MSF for such kind of oscillator may be equivalent to provide sufficient condition of network synchronization. Besides, we propose methods for the control of network synchronization based on the characteristic of UDC. For networks with fixed structure, by obtaining a minimal spanning tree, each node chooses one of its neighbors and follows its dynamics with uniform coupling strength to realize synchronization. For growing networks, by establishing directed connection from newly added nodes to existing nodes. An interacting structure which is similar to the combination of a group of UDCs is formed and the system is guaranteed to be controlled within the SCR.

Emergence of synchronous behavior in a network with chaotic multistable systems

In this work, we address the synchronization problem of complex dynamical networks with bidirectional, linear, and diffusive coupling, whose nodes have dynamics given by a particular class of piecewise linear (PWL) systems. The class of PWL systems are multistable unstable dissipative systems and exhibit chaotic behavior. The topologies that we consider are regular coupled network. Firstly, we consider that the complex network has a uniform coupling strength, and through the Lyapunov approach, we show that nodes can achieve complete or partial synchronization, where the synchronization solution depends on the inner coupling matrix and the initial conditions of each node. Secondly, we consider a weighted network, where the synchronization solution depends on the external coupling matrix, and even obtain a synchronous behavior type master-slave. Our theoretical results agree with the numerical simulations for a set of nodes in different topology networks.

Optimal Synchronizability in Networks of Coupled Systems: Topological View

Many engineered and natural systems are modeled as networks of coupled systems. Synchronization is one of their crucial and well-studied behaviors. Uniform coupling strength has been the benchmark practice in the majority of the literature. This paper considers nonuniform coupling strength, and a modified approach to the problem of synchronizability optimization, enabling a reduction to a spectral radius minimization problem, which can reach a unique optimal point on the Pareto Frontier. It is established that adding any edge to a connected graph can only improve synchronizability in this optimal measure. This result is utilized for developing a hierarchy between topologies. It is shown that several proposed structural parameters, including betweenness centrality, do not have any simple relationship to the optimal synchronizability measure.

Deep-subwavelength spoof magnetic localized surface plasmon waveguiding over arbitrary bending angles.

A deep-subwavelength metal spiral structure (MSS) waveguide with arbitrary bending angles was proposed and demonstrated to propagate magnetic localized surface plasmons (MLSPs) in theoretical, simulated and experimental ways. The uniform coupling strengths and frequencies for adjacent MSSs with different azimuthal angles represent a significant advancement in the development of structures supporting MLSPs over arbitrary bending angles. The consistency among spectra, dispersion, and field distributions for five MSSs indicates that backward propagation of MLSPs over arbitrary bending angles is possible. In addition, a long S-chain consisting of adjacent MSSs at various angles holds promise for applications involving long-distance MLSPs waveguides.

Towards Generation of Cat States in Trapped Ions Set-Ups via FAQUAD Protocols and Dynamical Decoupling

The high fidelity generation of strongly entangled states of many particles, such as cat states, is a particularly demanding challenge. One approach is to drive the system, within a certain final time, as adiabatically as possible, in order to avoid the generation of unwanted excitations. However, excitations can also be generated by the presence of dissipative effects such as dephasing. Here we compare the effectiveness of Local Adiabatic and the FAst QUasi ADiabatic protocols in achieving a high fidelity for a target superposition state both with and without dephasing. In particular, we consider trapped ions set-ups in which each spin interacts with all the others with the uniform coupling strength or with a power-law coupling. In order to mitigate the effects of dephasing, we complement the adiabatic protocols with dynamical decoupling and we test its effectiveness. The protocols we study could be readily implemented with state-of-the-art techniques.

Optimizing synchronizability in networks of coupled systems

Of collective behaviors in networks of coupled systems, synchronization is of central importance and an extensively studied area. This is due to the fact that it is essential for the proper functioning of a wide variety of natural and engineered systems. Traditionally, uniform coupling strength has been the default choice and the synchronizability measure has been employed for analysis and enhancement of synchronizability. The main drawback of optimizing the synchronizability measure is that it can reach the Pareto frontier but not necessarily a unique point on the Pareto frontier. Additionally, the shortcoming of uniform coupling strength is that it can reach Pareto frontier in specific topologies including edge-transitive graphs. To achieve a unique optimal answer on the Pareto frontier, this paper takes a different approach and addresses the synchronizability in networks of coupled dynamical systems with nonuniform coupling strength and optimizing the synchronizability via maximizing the minimum distance between the nonzero eigenvalues of the Laplacian and the acceptable boundaries for the stability of the system. Furthermore, two solution methods, namely the concave–convex fractional programming and the Semidefinite Programming (SDP) formulations of the problem have been provided. The proposed solution methods have been compared over different topologies and branches of an arbitrary network, where the SDP based approach has shown to be less restricted and more suitable for a wider range of topologies.

Non-Markovianity over Ensemble Averages in Quantum Complex Networks

We consider bosonic quantum complex networks as structured finite environments for a quantum harmonic oscillator and investigate the interplay between the network structure and its spectral density, excitation transport properties and non-Markovianity. After a review of the formalism used, we demonstrate how even small changes to the network structure can have a large impact on the transport of excitations. We then consider the non-Markovianity over ensemble averages of several different types of random networks of identical oscillators and uniform coupling strength. Our results show that increasing the number of interactions in the network tends to suppress the average non-Markovianity. This suggests that tree networks are the random networks optimizing this quantity.

Robust adaptive hierarchical insensitive tracking control of a class of leader-follower agents

In this paper, the hierarchical insensitive tracking control problem is addressed for a class of uncertain leader-follower agents with disturbed networks and controller multiplicative coefficient variations. With a uniform coupling strength, the agents are supported to be coupled in a multi-layer network structure. Based on the structure, an adaptive compensation control strategy is constructed to ensure that the follower agents track the leader agent. Depending on the control strategy, the relationship among bounds of tracking errors, sizes of network disturbances and uncertainties of agents and controllers, as well as coupling strength is explicitly explored. On the basis of Lyapunov stability theory, it is shown that bounded tracking of the resulting adaptive networked agents can be reached, and tracking errors can be reduced as small as desired. The effectiveness of the proposed design is illustrated via a decoupled model of F-18 aircraft.

Network synchronization, diffusion, and the paradox of heterogeneity

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Transition to chaos in complex dynamical networks

The transition from a non-chaotic state to a chaotic state is a commonly concerned issue in the study of coupled dynamical networks. In this work, we consider a network consisting of nodes that are in non-chaotic states with parameters in non-chaotic regions before they are coupled together. We show that if these non-chaotic nodes are linked together through a suitable structural topology, positive Lyapunov exponents of the coupled network can be generated by choosing a certain uniform coupling strength, and the threshold for this coupling strength is determined by the complexity of the network topology. Moreover, we show that topological effects of scale-free and random networks, which are two basic types of complex network models, can be visualized based on their topological sensitivity to random failures and intentional attacks. Our simulation results on a 1000-node scale-free network and a 1000-node random network of the Logistic maps have verified that, during the transition from non-chaotic to chaotic states, if the topology is more heterogenous then the coupling strength required to achieve the transition can be decreased.

Clustering in neural networks with heterogeneous and asymmetrical coupling strengths

The existence and stability of phase-clustered states have been studied previously in networks of weakly coupled oscillators with uniform coupling strengths [Physica D 63 (1993) 424]. However, several studies have shown that if the coupling is uniform and repulsive, it is hard to obtain stable phase-clustered states in networks of realistic neural oscillators when noise is present [Neural Comput. 7 (1995) 307; Phys. Rev. E 57 (1998) 2150]. This problem was avoided by introducing heterogeneity in the distribution of coupling strengths [J. Phys. Soc. Jpn. 72 (2003) 443]. It has been shown that heterogeneous coupling strengths make the occurrence of stable clustered states possible in small networks of repulsively coupled neural oscillators of all kinds [J. Comput. Neurosci. 14 (2003) 139; SIAM J. Appl. Math., submitted for publication]. The present work extends these results to large networks of N identical neurons that are globally coupled with heterogeneous and asymmetrical coupling strengths. Conditions for the existence and stability of a state of n synchronized clusters at evenly distributed phases, called the state of n splay-phase clusters, are derived. Clusters of different sizes, i.e. containing different numbers of neurons, are studied. The existence of such a state is guaranteed if the strength of the coupling originating from one neuron to other neurons is inversely proportional to the size of the cluster to which it belongs. This condition is called the rule of inverse cluster-size. At the state of n splay-phase clusters, the N-neuron network behaves like a network of n “big neurons”. Stability of this state is determined by n eigenvalues of which only one determines the stability of intra-cluster phase differences. The remaining n−1 conditions determine the stability of inter-cluster phase differences, but only n h=(n− mod (n,2))/2 of them have distinct real parts due to symmetry. Heterogeneous coupling makes the stability conditions depend on coupling strengths. This analysis not only reveals how clustered states occur in more general kinds of networks, but also illustrates how the stability of clustered states can be achieved in networks of repulsively coupled neural oscillators. Results on clustered states with phases that are not evenly distributed in the phase space are also presented. Potential applications of these results are discussed.