Generalized synchronization of complex dynamical networks via impulsive control

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.

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.

Adaptive Pinning Synchronization of Fractional Complex Networks with Impulses and Reaction–Diffusion Terms

In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the Lyapunov method and linear matrix inequality techniques, some sufficient criteria are obtained, ensuring adaptive pinning synchronization of the network under a designed adaptive control strategy. In addition, a pinning scheme is proposed, which shows that the nodes with delayed communication are good candidates for applying controllers. Finally, a numerical example is given to verify the validity of the main results.

Driving-based generalized synchronization in two-layer networks via pinning control

Synchronization of complex networks has been extensively investigated in various fields. In the real world, one network is usually affected by another one but coexists in harmony with it, which can be regarded as another kind of synchronization--generalized synchronization (GS). In this paper, the GS in two-layer complex networks with unidirectional inter-layer coupling via pinning control is investigated based on the auxiliary-system approach. Specifically, for two-layer networks under study, one is considered as the drive network and the other is the response one. According to the auxiliary-system approach, output from the drive layer is designed as input for the response one, and an identical duplication of the response layer is constructed, which is driven by the same driving signals. A sufficient condition for achieving GS via pinning control is presented. Numerical simulations are further provided to illustrate the correctness of the theoretical results. It is also revealed that the least number of pinned nodes needed for achieving GS decreases with the increasing density of the response layer. In addition, it is found that when the intra-layer coupling strength of the response network is large, nodes with larger degrees should be selected to pin first for the purpose of achieving GS. However, when the coupling strength is small, it is more preferable to pin nodes with smaller degrees. This work provides engineers with a convenient approach to realize harmonious coexistence of various complex systems, which can further facilitate the selection of pinned systems and reduce control cost.

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.

Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks

We study the effect of noise on the outer synchronization between two unidirectionally coupled complex networks and find analytically that outer synchronization could be achieved via white-noise-based coupling. It is also demonstrated that, if two networks have both conventional linear coupling and white-noise-based coupling, the critical deterministic coupling strength between two complex networks for synchronization transition decreases with an increase in the intensity of noise. We provide numerical results to illustrate the feasibility and effectiveness of the theoretical results.

Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays

This paper aims to investigate the synchronization problem of coupled dynamical networks with nonidentical Duffing-type oscillators without or with coupling delays. Different from cluster synchronization of nonidentical dynamical networks in the previous literature, this paper focuses on the problem of complete synchronization, which is more challenging than cluster synchronization. By applying an impulsive controller, some sufficient criteria are obtained for complete synchronization of the coupled dynamical networks of nonidentical oscillators. Furthermore, numerical simulations are given to verify the theoretical results.

Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms via periodically intermittent control

We explore the issue of integrating leakage delay, stochastic noise perturbation, and reaction-diffusion effects into the study of synchronization for neural networks with time-varying delays. By using Lyapunov stability theory and stochastic analysis approaches, a periodically intermittent controller is proposed to guarantee the exponential synchronization of proposed coupled neural networks based on p-norm. Some existing results are improved and extended. The usefulness and superiority of our theoretical results are illustrated by a numerical example.

A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization

A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.