Explosive synchronization coexists with classical synchronization in the Kuramoto model

Synchronization transitions in phase oscillator populations with partial adaptive coupling

The adaptation underlying many realistic processes plays a pivotal role in shaping the collective dynamics of diverse systems. Here, we untangle the generic conditions for synchronization transitions in a system of coupled phase oscillators incorporating the adaptive scheme encoded by the feedback between the coupling and the order parameter via a power-law function with different weights. We mathematically argue that, in the subcritical and supercritical correlation scenarios, there exists no critical adaptive fraction for synchronization transitions converting from the first (second)-order to the second (first)-order. In contrast to the synchronization transitions previously deemed, the explosive and continuous phase transitions take place in the corresponding regions as long as the adaptive fraction is nonzero, respectively. Nevertheless, we uncover that, at the critical correlation, the routes toward synchronization depend crucially on the relative adaptive weights. In particular, we unveil that the emergence of a range of interrelated scaling behaviors of the order parameter near criticality, manifesting the subcritical and supercritical bifurcations, are responsible for various observed phase transitions. Our work, thus, provides profound insights for understanding the dynamical nature of phase transitions, and for better controlling and manipulating synchronization transitions in networked systems with adaptation.

Topologically induced suppression of explosive synchronization

Nowadays, explosive synchronization is a well-documented phenomenon consisting in a first-order transition that may coexist with classical synchronization. Typically, explosive synchronization occurs when the network structure is represented by the classical graph Laplacian, and the node frequency and its degree are correlated. Here, we answer the question on whether this phenomenon can be observed in networks when the oscillators are coupled via degree-biased Laplacian operators. We not only observe that this is the case but also that this new representation naturally controls the transition from explosive to standard synchronization in a network. We prove analytically that explosive synchronization emerges when using this theoretical setting in star-like networks. As soon as this star-like network is topologically converted into a network containing cycles, the explosive synchronization gives rise to classical synchronization. Finally, we hypothesize that this mechanism may play a role in switching from normal to explosive states in the brain, where explosive synchronization has been proposed to be related to some pathologies like epilepsy and fibromyalgia.

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.

Explosive synchronization and chimera in interpinned multilayer networks

This Letter investigates the nature of synchronization in multilayered and multiplexed populations in which the interlayer interactions are randomly pinned. First, we show that a multilayer network constructed by setting up all-to-all interlayer connections between the two populations leads to explosive synchronization in the two populations successively, leading to the coexistence of coherent and incoherent populations forming chimera states. Second, a multiplex formation of the two populations in which only the mirror nodes are interconnected espouses explosive transitions in the two populations concurrently. The occurrence of both explosive synchronization and chimera are substantiated with rigorous theoretical mean-field analysis. The random pinning in the interlayer interactions concerns the practical problems where the impact of dynamics of one network on that of other interconnected networks remains elusive, as is the case for many real-world systems.

Amplification of explosive width in complex networks

We present an adaptive coupling strategy to induce hysteresis/explosive synchronization in complex networks of phase oscillators (Sakaguchi-Kuramoto model). The coupling strategy ensures explosive synchronization with significant explosive width enhancement. Results show the robustness of the strategy, and the strategy can diminish (by inducing enhanced hysteresis loop) the contrarian impact of phase frustration in the network, irrespective of the network structure or frequency distributions. Additionally, we design a set of frequency for the oscillators, which eventually ensure complete in-phase synchronization behavior among these oscillators (with enhanced explosive width) in the case of adaptive-coupling scheme. Based on a mean-field analysis, we develop a semi-analytical formalism, which can accurately predict the backward transition of the synchronization order parameter.

Inhibition-induced explosive synchronization in multiplex networks

To date, explosive synchronization (ES) in a network is shown to be originated from considering either degree-frequency correlation, frequency-coupling strength correlation, inertia, or adaptively controlled phase oscillators. Here we show that ES is a generic phenomenon and can occur in any network by multiplexing it with an appropriate layer without even considering any prerequisite for the emergence of ES. We devise a technique which leads to the occurrence of ES with hysteresis loop in a network upon its multiplexing with a negatively coupled (or inhibitory) layer. The impact of various structural properties of positively coupled (or excitatory) and inhibitory layers along with the strength of multiplexing in gaining control over the induced ES transition is discussed. Analytical prediction for the spread of phase distribution of each layer is provided, which is in good agreement with the numerical assessment. This investigation is a step forward in highlighting the importance of multiplex framework not only in bringing phenomena which are not possible in an isolated network but also in providing more structural control over the induced phenomena.

Self-similarity in explosive synchronization of complex networks

We report that explosive synchronization of networked oscillators (a process through which the transition to coherence occurs without intermediate stages but is rather characterized by a sudden and abrupt jump from the network's asynchronous to synchronous motion) is related to self-similarity of synchronous clusters of different size. Self-similarity is revealed by destructing the network synchronous state during the backward transition and observed with the decrease of the coupling strength between the nodes of the network. As illustrative examples, networks of Kuramoto oscillators with different topologies of links have been considered. For each one of such topologies, the destruction of the synchronous state goes step by step with self-similar configurations of interacting oscillators. At the critical point, the invariance of the phase distribution in the synchronized cluster with respect to the cluster size is reported.

Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

The Ott-Antonsen (OA) ansatz [Ott and Antonsen, Chaos 18, 037113 (2008); Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this, we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore, we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step, we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.