Explosive synchronization with asymmetric frequency distribution

Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling

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.

Discontinuous phase transition switching induced by a power-law function between dynamical parameters in coupled oscillators

In biological or physical systems, the intrinsic properties of oscillators are heterogeneous and correlated. These two characteristics have been empirically validated and have garnered attention in theoretical studies. In this paper, we propose a power-law function existed between the dynamical parameters of the coupled oscillators, which can control discontinuous phase transition switching. Unlike the special designs for the coupling terms, this generalized function within the dynamical term reveals another path for generating the first-order phase transitions. The power-law relationship between dynamic characteristics is reasonable, as observed in empirical studies, such as long-term tremor activity during volcanic eruptions and ion channel characteristics of the Xenopus expression system. Our work expands the conditions that used to be strict for the occurrence of the first-order phase transitions and deepens our understanding of the impact of correlation between intrinsic parameters on phase transitions. We explain the reason why the continuous phase transition switches to the discontinuous phase transition when the control parameter is at a critical value.

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.

Frequency and Phase Characteristics of Candle Flame Oscillation

The combustion of candles exhibits a variety of dynamical behaviors. Binding several candles together will result in flickering of candle flames, which is generally described as a nonlinear oscillator. The impact on the frequency of the flame by several factors, such as the arrangement, the number and the asymmetry of the oscillators, is discussed. Experimental results show that the frequency gradually decreases as the number of candles increases in the case of an isolated oscillator, while alternation between the in-phase and the anti-phase synchronization appears in a coupled system of two oscillators. Moreover, envelopes in the amplitude of the oscillatory luminance are displayed when candles are coupled asymmetrically. Since the coupling between oscillators is dominated by thermal radiation, a “overlapped peaks model” is proposed to phenomenologically explain the relationship between temperature distribution, coupling strength and the collective behavior in coupled system of candle oscillators in both symmetric and asymmetric cases.

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.

Explosive death of conjugate coupled Van der Pol oscillators on networks

Explosive death phenomenon has been gradually gaining attention of researchers due to the research boom of explosive synchronization, and it has been observed recently for the identical or nonidentical coupled systems in all-to-all network. In this work, we investigate the emergence of explosive death in networked Van der Pol (VdP) oscillators with conjugate variables coupling. It is demonstrated that the network structures play a crucial role in identifying the types of explosive death behaviors. We also observe that the damping coefficient of the VdP system not only can determine whether the explosive death state is generated but also can adjust the forward transition point. We further show that the backward transition point is independent of the network topologies and the damping coefficient, which is well confirmed by theoretical analysis. Our results reveal the generality of explosive death phenomenon in different network topologies and are propitious to promote a better comprehension for the oscillation quenching behaviors.

Dynamics of phase oscillators with generalized frequency-weighted coupling

Heterogeneous coupling patterns among interacting elements are ubiquitous in real systems ranging from physics, chemistry to biology communities, which have attracted much attention during recent years. In this paper, we extend the Kuramoto model by considering a particular heterogeneous coupling scheme in an ensemble of phase oscillators, where each oscillator pair interacts with different coupling strength that is weighted by a general function of the natural frequency. The Kuramoto theory for the transition to synchronization can be explicitly generalized, such as the expression for the critical coupling strength. Also, a self-consistency approach is developed to predict the stationary states in the thermodynamic limit. Moreover, Landau damping effects are further revealed by means of linear stability analysis and resonance poles theory below the critical threshold, which turns to be far more generic. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

Intermittent Bellerophon state in frequency-weighted Kuramoto model

Recently, the Bellerophon state, which is a quantized, time dependent, clustering state, was revealed in globally coupled oscillators [Bi et al., Phys. Rev. Lett. 117, 204101 (2016)]. The most important characteristic is that in such a state, the oscillators split into multiple clusters. Within each cluster, the instantaneous frequencies of the oscillators are not the same, but their average frequencies lock to a constant. In this work, we further characterize an intermittent Bellerophon state in the frequency-weighted Kuramoto model with a biased Lorentzian frequency distribution. It is shown that the evolution of oscillators exhibits periodical intermittency, following a synchronous pattern of bursting in a short period and resting in a long period. This result suggests that the Bellerophon state might be generic in Kuramoto-like models regardless of different arrangements of natural frequencies.

Model bridging chimera state and explosive synchronization

Global synchronization and partial synchronization are the two distinctive forms of synchronization in coupled oscillators and have been well studied in recent decades. Recent attention on synchronization is focused on the chimera state (CS) and explosive synchronization (ES), but little attention has been paid to their relationship. Here we study this topic by presenting a model to bridge these two phenomena, which consists of two groups of coupled oscillators, and its coupling strength is adaptively controlled by a local order parameter. We find that this model displays either CS or ES in two limits. In between the two limits, this model exhibits both CS and ES, where CS can be observed for a fixed coupling strength and ES appears when the coupling is increased adiabatically. Moreover, we show both theoretically and numerically that there are a variety of CS basin patterns for the case of identical oscillators, depending on the distributions of both the initial order parameters and the initial average phases. This model suggests a way to easily observe CS, in contrast to other models having some (weak or strong) dependence on initial conditions.