Explosive Synchronization Transitions Research Articles

Continuous and explosive synchronization transition in turbulent combustors

Thermoacoustic instabilities in turbulent combustors have disastrous consequences and present notorious challenges in their modeling, prediction, and control. Such instabilities are characterized by self-excited periodic oscillations, arising from a positive feedback between the acoustic pressure and heat release rate fluctuations. We present a mean-field approach to model thermoacoustic transitions. The nonlinear flame response is modeled using an ensemble of phase oscillators constrained to collectively evolve at the rhythm of acoustic fluctuations. Starting from the acoustic wave equation coupled with the phase oscillators, we derive the evolution equations for the amplitude and phase for acoustic oscillations. The model captures abrupt and continuous transitions to thermoacoustic instability observed in disparate combustors. We also discover that continuous and abrupt transitions happen through paradigmatic continuous and explosive synchronization, respectively. Importantly, our approach explains spatiotemporal synchronization and pattern formation underlying the transition to thermoacoustic instability. The versatility of the model in capturing different types of transitions suggests promising prospects for its extension to encompass a wide range of fluid dynamics phenomena.

Explosive synchronization in phase oscillator populations with attractive and repulsive adaptive interactions

The attractive and repulsive adaptive coupling schemes among dynamical agents are ubiquitous in systems ranging from physics, biology to neuroscience, which have attracted increasing interest and ample attention during recent years. Here, we extend the classical Kuramoto model to the coupling with mixed signs by considering a particular adaptive scheme in a system of globally coupled phase oscillators. The time-varying coupling weight between each pair of oscillators is correlated with the local coherence that involves according to a nonlinear differential equation. The studied model is capable of capturing the essential properties of the explosive synchronization induced by the adaptive interactions. We carry out extensive simulations to explore the collective dynamics in different perspectives. Remarkably, we reveal that the occurrence of the abrupt transitions of the macroscopic overall weights, as well as the emergence of the correlations between the microscopic structure features (including degrees and frequency dissasortativity) and local dynamics trigger the explosive synchronization, which manifests the suppression rule for the formation of small synchronized clusters. Furthermore, we provide an analytical treatment for the reduced system that allows us to grasp the underlying mechanism of the observed phenomena. Our study can deepen the understanding of the explosive synchronization transitions and other related abrupt dynamic phenomena occurring in networked oscillators with generic adaptive schemes.

Synchronization of phase oscillators on complex hypergraphs.

We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system's order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of two coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study the synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features.

Explosive transitions to synchronization in networks of frequency dipoles.

We reveal that an introduction of frequency-weighted inter-layer coupling term in networks of frequency dipoles can induce explosive synchronization transitions. The reason for explosive synchronization is that the oscillators with synchronization superiority are moderately suppressed. The findings show that a super-linear correlation induces explosive synchronization in networks of frequency dipoles, while a linear or sub-linear correlation excites chimera-like states. Clearly, the synchronization transition mode of networks of frequency dipoles is controlled by the power-law exponent. In addition, by means of the mean-field approximation, we obtain the critical values of the coupling strength within and between layers in two limit cases. The results of theoretical analysis are in good agreement with those of numerical simulation. Compared with the previous models, the model proposed in this paper retains the topological structure of network and the intrinsic properties of oscillators, so it is easy to realize pinning control.

Generic criterion for explosive synchronization in heterogeneous phase oscillator populations

Ordered and disordered behavior in large ensembles of coupled oscillators map to different functional states in a wide range of applications, e.g., active and resting states in the brain and stable and unstable power grid configurations. For this reason, explosive synchronization transitions, which facilitate fast, abrupt changes between these functional states, has recently seen significant interest from researchers. While previous work has identified properties of complex systems that support explosive synchronization, these investigations have been conducted largely on an ad-hoc basis. Here we provide an exact criterion for explosive synchronization in coupled phase oscillator ensembles by investigating the necessary relationship between dynamical and structural disorder. This result provides a critical step towards untangling the intertwined properties of complex systems responsible for abrupt phase transitions and bistability in biological and engineered systems.

Explosive synchronization induced by environmental coupling

The occurrence of explosive synchronization transition in a system of limit-cycle oscillators in the presence of two types of coupling; direct mean field diffusive and indirect environmental couplings, both operating simultaneously, is reported. The dynamics of coupled nonlinear Van der Pol and Rayleigh oscillators are studied in detail as a function of the distribution of intrinsic parameters of the oscillators. This explosive synchronization transition depends on the strength of indirect coupling and is irreversible giving rise to a characteristic hysteresis region. The different routes to synchronization observed in these coupled oscillators are studied in detail with the help of effective frequency and time series analysis. We have investigated the efficiency of the proposed scheme in various other topologies such as random, scale-free, and two–community networks as well.

Catalytic feed-forward explosive synchronization in multilayer networks.

Inhibitory couplings are crucial for the normal functioning of many real-world complex systems. Inhibition in one layer has been shown to induce explosive synchronization in another excitatory (or positive) layer of duplex networks. By extending this framework to multiplex networks, this article shows that inhibition in a single layer can act as a catalyst, leading to explosive synchronization transitions in the rest of the layers feed-forwarded through intermediate layer(s). Considering a multiplex network of coupled Kuramoto oscillators, we demonstrate that the characteristics of the transition emergent in a layer can be entirely controlled by the intra-layer coupling of other layers and the multiplexing strengths. The results presented here are essential to fathom the synchronization behavior of coupled dynamical units in multi-layer systems possessing inhibitory coupling in one of its layers, representing the importance of multiplexing.

Collective dynamics of heterogeneously and nonlinearly coupled phase oscillators

Coupled oscillators have been used to study synchronization in a wide range of social, biological, and physical systems, including pedestrian-induced bridge resonances, coordinated lighting up of firefly swarms, and enhanced output peak intensity in synchronizing laser arrays. Here we advance this subject by studying a variant of the Kuramoto model, where the coupling between the phase oscillators is heterogeneous and nonlinear. In particular, the quenched disorder in the coupling strength and the intrinsic frequencies are correlated, and the coupling itself depends on the amplitude of the mean-field of the system. We show that the interplay of these factors leads to a fascinatingly rich collective dynamics, including explosive synchronization transitions, hybrid transitions with hysteresis absence, abrupt irreversible desynchronization transitions, and tiered phase transitions with or without a vanishing onset. We develop an analytical treatment that enables us to determine the observed equilibrium states of the system, as well as to explore their asymptotic stability at various levels. Our research thus provides theoretical foundations for a number of self-organized phenomena that may be responsible for the emergence of collective rhythms in complex systems.

Explosive synchronization in a mobile network in the presence of a positive feedback mechanism

Synchronization is a process that describes the coherent dynamics of a large ensemble of interacting units. The study of explosive synchronization transition attracts considerable attention. Here, I report the explosive transition within the framework of a mobile network, while each oscillator is controlled by global-order parameters of the system. Using numerical simulation, I find that the explosive synchronization (ES) transition behavior can be controlled by simply adjusting the fraction of controlled oscillators. The influences of some parameters on explosive synchronization are studied. Moreover, due to the presence of the positive feedback mechanism, I prevent the occurrence of the synchronization of continuousphase transition and make phase transition of the system a first-order phase transition accompanied by a hysteresis loop.

Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes.

The higher-order interactions of complex systems, such as the brain, are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Explosive, continuous and frustrated synchronization transition in spiking Hodgkin–Huxley neural networks: The role of topology and synaptic interaction

Synchronization is an important collective phenomenon in interacting oscillatory agents. Many functional features of the brain are related to synchronization of neurons. The type of synchronization transition that may occur (explosive vs. continuous) has been the focus of intense attention in recent years, mostly in the context of phase oscillator models for which collective behavior is independent of the mean-value of natural frequency. However, synchronization properties of biologically-motivated neural models depend on the firing frequencies. In this study we report a systematic study of gamma-band synchronization in spiking Hodgkin–Huxley neurons which interact via electrical or chemical synapses. We use various network models in order to define the connectivity matrix. We find that the underlying mechanisms and types of synchronization transitions in gamma-band differs from beta-band. In gamma-band, network regularity suppresses transition while randomness promotes a continuous transition. Heterogeneity in the underlying topology does not lead to any change in the order of transition, however, correlation between number of synapses and frequency of a neuron will lead to explosive synchronization in heterogeneous networks with electrical synapses. Furthermore, small-world networks modeling a fine balance between clustering and randomness (as in the cortex), lead to explosive synchronization with electrical synapses, but a smooth transition in the case of chemical synapses. We also find that hierarchical modular networks, such as the connectome, lead to frustrated transitions. We explain our results based on various properties of the network, paying particular attention to the competition between clustering and long-range synapses.

Explosive synchronization through dynamical environment

We report the emergence of an explosive synchronization transition in the identical oscillators interacting indirectly through a network of dynamical agents. The transition from incoherent state to coherent state and vice–versa in these coupled oscillator exhibits an abrupt as well as irreversible. Such transition depends on the network topology as well as the interaction between the oscillators and dynamical agents rather than degree-frequency correlation in the network of oscillators. The occurrence of explosive synchronization is studied in details by using an appropriate order parameter for limit-cycle oscillators with respect to the different parameters like rewiring probability, average degree, and diffusion rate in dynamical agents.

Explosive synchronization transition in a ring of coupled oscillators

Explosive synchronization(ES), as one kind of abrupt dynamical transition in nonlinearly coupled systems, is currently a subject of great interests. Given a special frequency distribution, a mixed ES is observed in a ring of coupled phase oscillators which transits from partial synchronization to ES with the increment of coupling strength. The coupling weight is found to control the size of the hysteresis region where asynchronous and synchronized states coexist with each other. Theoretical analyses reveal that the transition varies from the mixed ES, to the ES and then to a continuous one with the increment of coupling weight. Our results are helpful to extend the understanding of the ES in homogenous networks.

Exact explosive synchronization transitions in Kuramoto oscillators with time-delayed coupling

Synchronization commonly occurs in many natural and man-made systems, from neurons in the brain to cardiac cells to power grids to Josephson junction arrays. Transitions to or out of synchrony for coupled oscillators depend on several factors, such as individual frequencies, coupling, interaction time delays and network structure-function relation. Here, using a generalized Kuramoto model of time-delay coupled phase oscillators with frequency-weighted coupling, we study the stability of incoherent and coherent states and the transitions to or out of explosive (abrupt, first-order like) phase synchronization. We analytically derive the exact formulas for the critical coupling strengths at different time delays in both directions of increasing (forward) and decreasing (backward) coupling strengths. We find that time-delay does not affect the transition for the backward direction but can shift the transition for the forward direction of increasing coupling strength. These results provide valuable insights into our understanding of dynamical mechanisms for explosive synchronization in presence of often unavoidable time delays present in many physical and biological systems.

A small change in neuronal network topology can induce explosive synchronization transition and activity propagation in the entire network

We here study explosive synchronization transitions and network activity propagation in networks of coupled neurons to provide a new understanding of the relationship between network topology and explosive dynamical transitions as in epileptic seizures and their propagations in the brain. We model local network motifs and configurations of coupled neurons and analyze the activity propagations between a group of active neurons to their inactive neuron neighbors in a variety of network configurations. We find that neuronal activity propagation is limited to local regions when network is highly clustered with modular structures as in the normal brain networks. When the network cluster structure is slightly changed, the activity propagates to the entire network, which is reminiscent of epileptic seizure propagation in the brain. Finally, we analyze intracranial electroencephalography (IEEG) recordings of a seizure episode from a epilepsy patient and uncover that explosive synchronization-like transition occurs around the clinically defined onset of seizure. These findings may provide a possible mechanism for the recurrence of epileptic seizures, which are known to be the results of aberrant neuronal network structure and/or function in the brain.

Effects of Reduced Frequency on Network Configuration and Synchronization Transition**Supported by the Open Fund from Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing under Grant No 2015CSOBDP0101, and the National Natural Science Foundation of China under Grant No 11162019.

Synchronization of networked phase oscillators depends essentially on the correlation between the topological structure of the graph and the dynamical property of the elements. We propose the concept of ‘reduced frequency’, a measure which can quantify natural frequencies of each pair of oscillators. Then we introduce an evolving network whose linking rules are controlled by its own dynamical property. The simulation results indicate that when the linking probability positively correlates with the reduced frequency, the network undergoes a first-order phase transition. Meanwhile, we discuss the circumstance under which an explosive synchronization can be ignited. The numerical results show that the peculiar butterfly shape correlation between frequencies and degrees of the nodes contributes to an explosive synchronization transition.

Explosive synchronization in clustered scale-free networks: Revealing the existence of chimera state

The collective dynamics of Kuramoto oscillators with a positive correlation between the incoherent and fully coherent domains in clustered scale-free networks is studied. Emergence of chimera states for the onsets of explosive synchronization transition is observed during an intermediate coupling regime when degree-frequency correlation is established for the hubs with the highest degrees. Diagnostic of the abrupt synchronization is revealed by the intrinsic spectral properties of the network graph Laplacian encoded in the heterogeneous phase space manifold, through extensive analytical investigation, presenting realistic MC simulations of nonlocal interactions in discrete time dynamics evolving on the network.

Explosive transitions to synchronization in weighted static scale-free networks

The emergence of explosive synchronization transitions in networks of phase oscillators has become recently one of the most interesting topics. We simulate the Kuramoto model on top of a family of weighted static scale-free networks. It is found that when the strength of the network’s edge is linearly correlated with frequency gap of pair of oscillators at its ends, i.e., the microscopic correlation exponent β is equal to 1, the model with the degree distribution exponent γ > 3 undergoes a first-order phase transition, while the transition becomes second order at 2 < γ ≤ 3. We also find that in homogeneous networks (γ → ∞) the explosive synchronization is replaced by a continuous phase transition when the microscopic correlation exponent β is changed from positive to negative. This is a new discovery of explosive synchronization transitions in weighted complex networks, which provides a fresh angle and tool to understand this explosive behavior.

Criterion for the emergence of explosive synchronization transitions in networks of phase oscillators

The emergence of explosive synchronization transitions in networks of phase oscillators recently has become one of the most interesting topics. It is widely believed that the large frequency mismatch of a pair of oscillators (also known as disassortativity in frequency) is a direct cause of an explosive synchronization. It is found that, besides the disassortativity in frequency, the disassortativity in node degree also shows up in connection with the first-order synchronization transition. In this paper, we simulate the Kuramoto model on top of a family of networks with different degree-degree and frequency-frequency correlation patterns. Results show that only when the degrees and natural frequencies of the network's nodes are both disassortative can an explosive synchronization occur.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.