The development of generalized synchronization on complex networks

Asymptotically local synchronization in interdependent networks with unidirectional interlinks

Synchronization in complex networks has been investigated for decades. Due to the particularity of the interlinks between networks, the synchronization in interdependent networks has received increasing interest. Since the interlinks are not always symmetric in interdependent networks, we focus on the synchronization in unidirectional interdependent networks to study the control scheme. The mathematical model is put forward and some factors are taken into consideration, such as different coupling functions and strengths. Firstly, the feasibility of the control scheme is proved theoretically by using Lyapunov stability theory and verified by simulations. Then, we find that the synchronization could be maintained in one sub-network by utilizing our control scheme while the nodes in the other sub-network are in chaos. The result indicates that the influence of interlinks can be decreased and the proposed scheme can guarantee the synchronization in one sub-network at least. Moreover, we also discuss the robust of our control scheme against the cascading failure. The scheme is verified by simulations to be effective while the disturbances occur.

Generalized synchronization on the onset of auxiliary system approach

Generalized synchronization is an emergent functional relationship between the states of the interacting dynamical systems. To analyze the stability of a generalized synchronization state, the auxiliary system technique is a seminal approach that is broadly used nowadays. However, a few controversies have recently arisen concerning the applicability of this method. In this study, we systematically analyze the applicability of the auxiliary system approach for various coupling configurations. We analytically derive the auxiliary system approach for a drive-response coupling configuration from the definition of the generalized synchronization state. Numerically, we show that this technique is not always applicable for two bidirectionally coupled systems. Finally, we analytically derive the inapplicability of this approach for the network of coupled oscillators and also numerically verify it with an appropriate example.

Intermittent route to generalized synchronization in bidirectionally coupled chaotic oscillators

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via nonlocal electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.

Generalized synchronization in a conservative and nearly conservative systems of star network

We report the coexistence of synchronized and unsynchronized states in a mutually coupled star network of nearly conservative non-identical oscillators. Generalized synchronization is observed between the central oscillator with the peripherals, and phase synchronization is found among the peripherals in weakly dissipative systems. However, the basin size of the synchronization region decreases as dissipation strength is increased. We have demonstrated these phenomena with the help of Duffing and Lorenz84 oscillators with conservative, nearly conservative, and dissipative properties. The observed results are robust against the network size.

Consistency between functional and structural networks of coupled nonlinear oscillators

In data-based reconstruction of complex networks, dynamical information can be measured and exploited to generate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To address these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching maximum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consistency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world examples are given: corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.

Inapplicability of an auxiliary-system approach to chaotic oscillators with mutual-type coupling and complex networks

The auxiliary system approach being de facto the standard for the study of generalized synchronization in the unidirectionally coupled chaotic oscillators is also widely used to examine the mutually coupled systems and networks of nonlinear elements with the complex topology of links between nodes. In this Brief Report we illustrate by two simple counterexamples that the auxiliary-system approach gives incorrect results for the mutually coupled oscillators and therefore to study the generalized synchronization this approach may be used only for the drive-response configuration of nonlinear oscillators and networks.

Collective almost synchronisation in complex networks

This work introduces the phenomenon of Collective Almost Synchronisation (CAS), which describes a universal way of how patterns can appear in complex networks for small coupling strengths. The CAS phenomenon appears due to the existence of an approximately constant local mean field and is characterised by having nodes with trajectories evolving around periodic stable orbits. Common notion based on statistical knowledge would lead one to interpret the appearance of a local constant mean field as a consequence of the fact that the behaviour of each node is not correlated to the behaviours of the others. Contrary to this common notion, we show that various well known weaker forms of synchronisation (almost, time-lag, phase synchronisation, and generalised synchronisation) appear as a result of the onset of an almost constant local mean field. If the memory is formed in a brain by minimising the coupling strength among neurons and maximising the number of possible patterns, then the CAS phenomenon is a plausible explanation for it.

Generalized synchronization of complex dynamical networks via impulsive control

This paper investigates the generalized synchronization (GS) of two typical complex dynamical networks, small-world networks and scale-free networks, in terms of impulsive control strategy. By applying the auxiliary-system approach to networks, we demonstrate theoretically that for any given coupling strength, GS can take place in complex dynamical networks consisting of nonidentical systems. Particularly, for Barabasi-Albert scale-free networks, we look into the relations between GS error and topological parameter m, which denotes the number of edges linking to a new node at each time step, and find out that GS speeds up with increasing m. And for Newman-Watts small-world networks, the time needed to achieve GS decreases as the probability of adding random edges increases. We further reveal how node dynamics affects GS speed on both small-world and scale-free networks. Finally, we analyze how the development of GS depends on impulsive control gains. Some abnormal but interesting phenomena regarding the GS process are also found in simulations.