Dynamics of the Kuramoto model in the presence of correlation between distributions of frequencies and coupling strengths

Self-consistent autocorrelation of a disordered Kuramoto model in the asynchronous state

The Kuramoto model has provided deep insights into synchronization phenomena and remains an important paradigm to study the dynamics of coupled oscillators. Yet, despite its success, the asynchronous regime in the Kuramoto model has received limited attention. Here, we adapt and enhance the mean-field approach originally proposed by Stiller and Radons [Phys. Rev. E 58, 1789 (1998)1063-651X10.1103/PhysRevE.58.1789] to study the asynchronous state in the Kuramoto model with a finite number of oscillators and with disordered connectivity. By employing an iterative stochastic mean field approximation, the complex N-oscillator system can effectively be reduced to a one-dimensional dynamics, both for homogeneous and heterogeneous networks. This method allows us to investigate the power spectra of individual oscillators as well as of the multiplicative "network noise" in the Kuramoto model in the asynchronous regime. By taking into account the finite system size and disorder in the connectivity, our findings become relevant for the dynamics of coupled oscillators that appear in the context of biological or technical systems.

Route to synchronization in coupled phase oscillators with frequency-dependent coupling: Explosive or continuous?

Interconnected dynamical systems often transition between states of incoherence and synchronization due to changes in system parameters. These transitions could be continuous (gradual) or explosive (sudden) and may result in failures, which makes determining their nature important. In this study, we abstract dynamical networks as an ensemble of globally coupled Kuramoto-like phase oscillators with frequency-dependent coupling and investigate the mechanisms for transition between incoherent and synchronized dynamics. The characteristics that dictate a continuous or explosive route to synchronization are the distribution of the natural frequencies of the oscillators, quantified by a probability density function g(ω), and the relation between the coupling strength and natural frequency of an oscillator, defined by a frequency-coupling strength correlation function f(ω). Our main results are conditions on f(ω) and g(ω) that result in continuous or explosive routes to synchronization and explain the underlying physics. The analytical developments are validated through numerical examples.

Dynamics in the Sakaguchi-Kuramoto model with bimodal frequency distribution

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

Characterizing nonstationary coherent states in globally coupled conformist and contrarian oscillators

For decades, the description and characterization of nonstationary coherent states in coupled oscillators have not been available. We here consider the Kuramoto model consisting of conformist and contrarian oscillators. In the model, contrarians are chosen from a bimodal Lorentzian frequency distribution and flipped into conformists at random. A complete and systematic analytical treatment of the model is provided based on the Ott-Antonsen ansatz. In particular, we predict and analyze not only the stability of all stationary states (such as the incoherent, the π, and the traveling-wave states), but also that of the two nonstationary states: the Bellerophon and the oscillating-π state. The theoretical predictions are fully supported by extensive numerical simulations.

Rhythmic synchronization and hybrid collective states of globally coupled oscillators

Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators’ phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.

Synchronous dynamics in the Kuramoto model with biharmonic interaction and bimodal frequency distribution

In this work, we study the Kuramoto model with biharmonic interaction and bimodal frequency distribution. Rich synchronous dynamics, such as standing wave states, stationary partial synchronous dynamics, and multiplicity of singular synchronous dynamics, are found. Notably, we find a symmetry-breaking synchronous dynamics when the peaks in frequency distribution are not well separated. We present the phase diagrams for two cases: the peaks in the frequency distribution are well separated and the peaks are not well separated. We find that reducing peak distance tends to make the transition between standing wave states and stationary partial synchronous states to be continuous when the multiplicity of singular synchronous state is present or to be discontinuous when the multiplicity is absent.

Multistable states in a system of coupled phase oscillators with inertia

We investigate the generalized Kuramoto model of globally coupled oscillators with inertia, in which oscillators with positive coupling strength are conformists and oscillators with negative coupling strength are contrarians. We consider the correlation between the coupling strengths of oscillators and the distributions of natural frequencies. Two different types of correlations are studied. It is shown that the model supports multistable synchronized states such as different types of travelling wave states, π state and another type of nonstationary state: an oscillating π state. The phase distribution oscillates in a confined region and the phase difference between conformists and contrarians oscillates around π periodically in the oscillating π state. The different types of travelling wave state may be characterized by the speed of travelling wave and the effective frequencies of oscillators. Finally, the bifurcation diagrams of the model in the parameter space are presented.

Synchronization and Bellerophon states in conformist and contrarian oscillators

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency-dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean-field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.

Glassy states and super-relaxation in populations of coupled phase oscillators

Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe these systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problems, however, remain unclarified for decades, such as the existence and properties of the oscillator glass state. Here we present a detailed analysis of a very general form of the Kuramoto model. In particular, we find the conditions when it can exhibit glassy behaviour, which represents a kind of synchronous disorder in the present case. Furthermore, we discover a new and intriguing phenomenon that we refer to as super-relaxation where the oscillators feel no interaction at all while relaxing to incoherence. Our findings offer the possibility of creating glassy states and observing super-relaxation in real systems.