Synchronization in duplex networks of coupled Rössler oscillators with different inner-coupling matrices

Abstract

The traditional and extended master stability framework cannot work for multiplex networks with different inner-coupling functions on different layers. In this paper, we thoroughly investigate the influence of inner-coupling functions on the synchronized regions with respect to intra- and inter-layer coupling strength in two-layer multiplex networks composed of linearly coupled Rössler oscillators with identical or nonidentical intra-layer topological structures. We focus on three typical inner-coupling matrices, representing the classical bounded, unbounded and empty synchronized regions. We numerically investigate intra-layer synchronization (synchronization of nodes within each layer), inter-layer synchronization (synchronization of node pairs across layers) and global synchronization (synchronization of all nodes) in duplex networks with identical or nonidentical intra-layer structures, for all the combinations of the three typical inner-coupling matrices. Interestingly, we find that whatever the inner-coupling matrices are, the intra-layer synchronized regions of the two layers will become the same type in most cases, which should be due to the interaction between layers. Particularly, the inner-coupling matrix that leads to an empty synchronized region in an isolated network does not necessarily result in empty synchronized regions within or across layers, though it will lead to empty global synchronized regions while working as the inter-layer inner-coupling function. Numerical results also imply that structural difference of the two intra-layer networks will weaken inter-layer as well as intra-layer synchronizability.

Introduction

Many real-world systems can be described as a network of connected elements [1], which has stimulated tremendous investigations into consensus and synchronization [2], [3], [4], [5], [6], [7], control [8], [9], [10], [11], [12], bifurcation [13], ranking [14], structure identification [15], [16], among many others. Yet, many phenomena do not involve a single network in isolation but rely on a group of connected networks or multilayer networks. In fact, the model of multilayer networks [17], [18], [19], [20], [21] widely exists and has been applied to numerous research and practical fields. For instance, a person has several social network accounts [22], [23], various kinds of diseases are propagated in the crowd [24], [25], and the airport in a metropolis is always a traffic hub for different transportation systems [26], [27]. We can easily find that, an individual will play different roles in different network layers. Later on, researchers used multiplex networks to characterize the above scenarios. Actually, a multiplex network is a multilayer network in which there exists a ‘one-to-one’ correspondence between those nodes in different layers, where the topological structures and the coupling functions between nodes within each layer might be different, and the number of nodes in one layer is the same as those in other layers. Especially, when there are only two layers, we call it a duplex network.

Synchronization is a universal phenomenon in nature, such as fish gathering, ant clustering, and applause synchronization. Generally speaking, synchronization is about a group of nodes arriving at the same behavior. During the last two decades, synchronization in single-layer networks has been widely investigated [28], [29], [30], [31], [32]. With the development of theoretical and computation techniques, more and more interests have been placed on synchronization analysis of multiplex networks, since the multiplex network model is more proper for describing real complex systems than isolated networks. In multiplex networks, there are three typical kinds of synchronization, that is, intra-layer synchronization, inter-layer synchronization, and global synchronization. As shown in Fig. 1, the left panel represents intra-layer synchronization in a duplex network with nonidentical intra-layer structures, where nodes within each layer reach synchronization, and the right panel represents inter-layer synchronization with each node in one layer being synchronized with a counterpart in the other layer. In this paper, the three typical kinds of synchronization mentioned above are investigated in duplex network. Specifically, synchronized regions of these three kinds of synchronization will be taken into consideration, in the spanning parameter space of the intra- and inter-layer coupling strength.

In 2016, Sevillaescoboza et al. [33] analytically derived the necessary conditions for the existence and stability of inter-layer synchronization. Gambuzza et al. [34] studied synchronization of N oscillators indirectly coupled through a network medium, where the model used in their work is a duplex network and one of its two layers has no intra-layer couplings. Nicosia et al. [35] investigated the collective phenomena emerging from the interactions between different dynamical processes in a duplex network that can model the neural activity and energy transport in human brain. Gómez et al. [36] used supra-Laplacian matrix to analyze the physics of diffusion processes on top of multiplex networks. Granell et al. [37] built a duplex network with an information layer and an epidemics layer, and then discussed the interaction between information dissemination and disease propagation.

It is well-known that, in networks, nodal dynamics, network topologies and inner-coupling functions play crucial roles in determining the networks’ evolutionary mechanisms and functional behaviors. Besides, nodes in multiplex networks are always involved with different features and functions. Naturally, the coupling functions in each layer are usually different [38], [39], [40], [41]. Take the brain as an example [40]. On one hand, the brain functioning is the synchronization of neurons, on the other hand, the brain also needs blood flow to supply necessary oxygen. The ways that neurons transmit electrical signals and blood cells deliver nutrients are obviously different. Thus, a duplex network characterizing this process consists of the neurons layer and the vascular layer, and the coupling functions in each layer are diverse. Furthermore, it is worth mentioning that, the coupling function between nodes across different layers is also particular and different from those within layers [41]. Therefore, duplex network models with different intra- and inter-layer coupling functions are practical and worth further investigation, which have actually attracted wide research passion in the past few years.

During the past decades, the master-stability function (MSF), one of the most important methods to study network synchronization, has achieved many works [42], [43], [44], [45]. In 2009, Huang et al. [42] simulated the master-stability synchronized regions of chaotic systems in single-layer networks when the oscillators are coupled in different ways. They also divided master stability regions into three representative categories, and presented typical inner-coupling matrices of each category. Very recently, Tang et al. [45] extended the traditional master stability function framework to a certain type of multiplex networks and derived three master stability equations that can determine the necessary regions of global synchronization, intra-layer synchronization and inter-layer synchronization. However, it is required that the intra-layer inner-coupling functions of different layers be identical, and the inter-layer and intra-layer supra-Laplacian matrices be commutable. Thus, we wonder what the synchronized regions will be for more general duplex networks, especially those with various inner-coupling functions.

Motivated by above discussions, we will intensively investigate the influence of different inner-coupling functions on synchronization of duplex networks. The rest of this paper is organized as follows: in Section 2, the duplex network model and two order parameters describing synchronization capabilities are introduced. Then, three typical inner-coupling matrices and their corresponding synchronized intervals with respect to coupling strength in single-layer networks of Rössler oscillators are presented. In Section 3, synchronized regions with respect to the parameter space of intra- and inter-layer coupling strength for duplex networks with the same as well as different intra-layer structures are explored, where all the combinations of the three typical inner-coupling matrices are taken into consideration. Finally, conclusions are drawn in Section 4.