Bellerophon state in the Kuramoto model with gravitation rules

The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.

[1] Large scale snow-atmosphere coupling strength (the degree to which the atmosphere responds to anomalies in snow cover) is quantitatively investigated using the Community Land Model (CLM 3.5) and its advanced snow scheme coupled to the Community Atmospheric Model (CAM). An index based on model ensemble statistics has been used to quantify the coupling strength. Multi-seasonal experiments reveal the spatial and temporal distribution of coupling strength during the snow depletion phase (March through June) over the Northern Hemisphere. The snow coupling strength was found to evolve in three distinct stages: before snowmelt, during snowmelt and after snowmelt. The coupling strength is strongest during snowmelt. The coupling strength after snowmelt, mainly due to hydrological effect (soil moisture impact), is stronger than coupling strength due to albedo effect (radiative impact) before snowmelt. A conceptual model is proposed to explain the mechanisms behind the spatial distribution of coupling strength.

We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist–contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties.

Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model

Multiband coupling effect on density of states and tunneling conductance spectra of ferromagnetic material

This paper aims at investigating the topology identification problem of complex dynamical networks with varying node dynamics parameters and fixed inner coupling matrices. In particular, by employing the unified chaotic system as node dynamics, this work further explores the influence of continuously changing node dynamics parameters on topology identification of complex dynamical networks with different coupling strengths. Results show that for sufficiently small or large coupling strengths, the performance of topology identification is not affected by the change of node parameters. Specifically, for small enough coupling strengths, the topological structure can be completely identified regardless of the change of node parameters, while for sufficiently large coupling strengths, the connectivity (presence and absence of connections) cannot be successfully identified. Furthermore, for certain coupling strengths, with the increase of node dynamics parameters, the topology identification varies from completely unidentifiable to partially or event completely identifiable. Therefore, the synchronization-based topology identification depends on node dynamics. Even for the same node dynamical model, different parameters can have a significant impact on identification results. Furthermore, for networks consisting of chaotic oscillators defining node dynamics, small coupling strengths are conducive to topology identification. A broader conclusion is that projective synchronization, rather than just complete synchronization, is an obstacle to the network topology identification. The findings in this paper will add to our understanding of conditions for identifying topologies of complex networks.

Introduction. Recent studies into the properties of spintronic oscillators have led to broadening their scope of practical application as devices for generating and processing signals. The practical implementation of spintronic oscillators is, however, significantly limited by their low power capacity, thus requiring synchronization between devices.Aim. Determination of conditions for the implementation of the synchronous regime of two antiferromagnetic spintronic oscillators coupled by a common current.Materials and methods. To simplify the numerical simulation of a system of coupled resistively antiferromagnetic oscillators, the method of multiple-time-scale analysis was used. This allowed a system of Kuramoto equations to be considered instead of the original system. To determine the locking band of the Kuramoto model, the homoclinic trajectory approximation method was applied.Results. A system of Kuramoto equation for the phases of partial oscillators under the influence of the inertial term and phase shift was obtained. Expressions describing the locking and synchronization band as functions of the system parameters (bias currents and sizes) were derived. The numerically simulated Kuramoto model was used to determine the bands of the synchronous and asynchronous regimes.Conclusion. The results of numerical simulations of the system of Kuramoto equations and the Adler equation for two coupled spintronic oscillators agree well with the theoretically calculated values of locking and synchronization ranges. The scheme for reducing the model of antiferromagnetic oscillators to a Kuramoto model can be further extended to the case of a larger number of coupled oscillators, which will simplify computational experiments and significantly reduce the time required for numerical simulations.

The standard Kuramoto model has been instrumental in explaining synchronization and desynchronization, two emergent phenomena often observed in biological, neuronal, and physical systems. While the Kuramoto model has turned out effective with one-dimensional oscillators, real-world systems often involve high-dimensional interacting units, such as biological swarms, necessitating a model of multidimensional oscillators. However, existing high-dimensional generalizations of the Kuramoto model commonly rely on a scalar-valued coupling strength, which limits their ability to capture the full complexity of high-dimensional interactions. This work introduces a matrix, A, to couple the interconnected components of the oscillators in a d-dimensional space, leading to a matrix-coupled multidimensional Kuramoto model that approximates a prototypical swarm dynamics by its first-order Fourier harmonics. Moreover, the matrix A introduces an inter-dimensional higher-order interaction that partly accounts for the emergence of 2 d system modes in a d-dimensional population, where each dimension can either be synchronized or desynchronized, represented by a set of almost binary order parameters. The binary system modes capture characteristic swarm behaviors such as fish milling or polarized schooling. Additionally, our findings provides a theoretical analogy to cerebral activity, where the resting state and the activated state coexist unihemispherically. It also suggests a new possibility for information storage in oscillatory neural networks.

Inferring the connectivity of coupled oscillators and anticipating their transition to synchrony through lag-time analysis

This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators. Stability for both all-to-all and random one-to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, DeltaT. For a desired system performance, DeltaT imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the DeltaT deadline. Multi-hopping is optimal if time permits, but one-to-all broadcasting is more energy efficient when M/DeltaT is sufficiently large.

This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles with the energy efficiency of communications in mind. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators over a sequence of logical graphs. Stability for both all-to-all and random one- to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, DeltaT. For desired system performance characteristics, DeltaT imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the DeltaT deadline. Due to the tight time deadline, a one-to-all single- hop broadcasting scheme is shown to be more energy efficient for practical choices of M/DeltaT. The proposed modularization is illustrated via a set of simulations where the overall communication energy to reach alignment is optimized.

Emergence of phase concentration for the Kuramoto–Sakaguchi equation

Beryllium has recently been discovered to harbor a Dirac nodal line (DNL) in its bulk phase and the DNL-induced nontrivial surface states (DNSSs) on its (0001) surface, rationalizing several already-existing historic puzzles [Phys. Rev. Lett. 117, 096401 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.096401]. However, to date the underlying mechanism as to why its (0001) surface exhibits an anomalously large electron-phonon coupling effect (λ_{e-ph}^{s}≈1.0) remains unresolved. Here, by means of first-principles calculations, we show that the coupling of the DNSSs with the phononic states mainly contributes to its novel surface e-ph enhancement. Besides the fact that the experimentally observed λ_{e-ph}^{s} and the main Eliashberg coupling function (ECF) peaks are reproduced well in our current calculations, we decompose the ECF α^{2}F(k,q;v) and the e-ph coupling strength λ(k,q;v) as a function of each electron momentum (k), each phonon momentum (q), and each phonon mode (v), evidencing the robust connection between the DNSSs and both α^{2}F(k,q;v) and λ(k,q;v). The results reveal the strong e-ph coupling between the DNSSs and the phonon modes, which contributes over 80% of the λ_{e-ph}^{s} coefficient on the Be (0001) surface. It highlights that the anomalously large e-ph coefficient on the Be (0001) surface can be attributed to the presence of its DNL-induced DNSSs, clarifying the long-debated mechanism.

On the double sphere model of synchronization

We performed numerical simulations with the Kuramoto model and experiments with oscillatory nickel electrodissolution to explore the dynamical features of the transients from random initial conditions to a fully synchronized (one-cluster) state. The numerical simulations revealed that certain networks (e.g., globally coupled or dense Erdős–Rényi random networks) showed relatively simple behavior with monotonic increase of the Kuramoto order parameter from the random initial condition to the fully synchronized state and that the transient times exhibited a unimodal distribution. However, some modular networks with bridge elements were identified which exhibited non-monotonic variation of the order parameter with local maximum and/or minimum. In these networks, the histogram of the transients times became bimodal and the mean transient time scaled well with inverse of the magnitude of the second largest eigenvalue of the network Laplacian matrix. The non-monotonic transients increase the relative standard deviations from about 0.3 to 0.5, i.e., the transient times became more diverse. The non-monotonic transients are related to generation of phase patterns where the modules are synchronized but approximately anti-phase to each other. The predictions of the numerical simulations were demonstrated in a population of coupled oscillatory electrochemical reactions in global, modular, and irregular tree networks. The findings clarify the role of network structure in generation of complex transients that can, for example, play a role in intermittent desynchronization of the circadian clock due to external cues or in deep brain stimulations where long transients are required after a desynchronization stimulus.