Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. For the multiplex network of coupled Rössler oscillators, here we show how the master stability function, a celebrated framework for analyzing synchronization on a single network, can be extended to certain classes of multiplex networks with different intralayer and interlayer coupling functions. We derive three master stability equations that determine, respectively, the necessary regions of complete synchronization, intralayer synchronization, and interlayer synchronization. We calculate these three regions explicitly for the case of a two-layer network of Rössler oscillators and show that the overlap of the regions determines the type of synchronization achieved. In particular, if the interlayer or intralayer coupling function is such that the interlayer or intralayer synchronization region is empty, complete synchronization cannot be achieved regardless of the coupling strength. Furthermore, for any network structure, the occurrence of intralayer and interlayer synchronization depends mainly on the coupling functions of nodes within a layer and across layers, respectively. Our mathematical analysis requires that the intralayer and interlayer supra-Laplacians commute. But, we show this is only a sufficient, and not necessary, condition and that the results can be applied more generally.

The coexistence of multiple types of interactions within social, technological, and biological networks has motivated the study of the multilayer nature of real-world networks. Meanwhile, identifying network structures from dynamical observations is an essential issue pervading over the current research on complex networks. This paper addresses the problem of structure identification for multilayer networks, which is an important topic but involves a challenging inverse problem. To clearly reveal the formalism, the simplest two-layer network model is considered and a new approach to identifying the structure of one layer is proposed. Specifically, if the interested layer is sparsely connected and the node behaviors of the other layer are observable at a few time points, then a theoretical framework is established based on compressive sensing and regularization. Some numerical examples illustrate the effectiveness of the identification scheme, its requirement of a relatively small number of observations, as well as its robustness against small noise. It is noteworthy that the framework can be straightforwardly extended to multilayer networks, thus applicable to a variety of real-world complex systems.

This article investigates uniformly predefined-time bounded consensus of leader-following multiagent systems (MASs) with unknown system nonlinearity and external disturbance via distributed adaptive fuzzy control. First, uniformly predefined-time-bounded stability is analyzed and a sufficient condition is derived for the system to achieve semiglobally (globally) uniformly predefined-time-bounded consensus. Therein, the settling time is independent of initial conditions and can be defined in advance. Then, for first-order MASs, distributed adaptive fuzzy controllers are designed by combining neighboring consensus errors to drive all following agents to globally track the leader’s state within predefined time. For second-order MASs, by formulating filtered errors, the consensus errors between following agents and the leader are shown to be bounded if the filtered errors are bounded. Furthermore, with the distributed controllers designed based on filtered errors, second-order MASs achieve semiglobally uniformly predefined-time-bounded leader-following consensus. Finally, two numerical examples are simulated, including: 1) a first-order leader-following MAS and 2) a second-order Lagrangian system consisting of single-link manipulators, to demonstrate the performance of the proposed controllers.

This article is concerned with fixed-time synchronization of the nonlinearly coupled neural networks with discontinuous activation and mismatched parameters. First, a novel lemma is proposed to study fixed-time stability, which is less conservative than those in most existing results. Then, based on the new lemma, a discontinuous neural network with mismatched parameters will synchronize to the target state within a settling time via two kinds of unified and simple controllers. The settling time is theoretically estimated, which is independent of the initial values of the considered network. In particular, the estimated settling time is closer to the real synchronization time than those given in the existing literature. Finally, two numerical simulations are presented to illustrate the effectiveness and correctness of our results.

Fixed-time synchronization of complex networks is investigated in this article. First, a completely novel lemma is introduced to prove the fixed-time stability of the equilibrium of a general ordinary differential system, which is less conservative and has a simpler form than those in the existing literature. Then, sufficient conditions are presented to realize synchronization of a complex network (with a target system) within a settling time via three different kinds of simple controllers. In general, controllers designed to achieve fixed-time stability consist of three terms and are discontinuous. However, in our mechanisms, the controllers only contain two terms or even one term and are continuous. Thus, our controllers are simpler and of more practical applicability. Finally, three examples are provided to illustrate the correctness and effectiveness of our results.

For strict-feedback systems with mismatched uncertainties, adaptive fuzzy control techniques are developed to provide global prescribed performance with prescribed-time convergence. First, a class of prescribed-time prescribed performance functions are designed to quantify the performance constraints of the tracking error. Additionally, a novel error transformation function is provided to eliminate the initial value limitations and resolve the singularity issue in previous research. To ensure the convergence of the tracking error into a prescribed bounded region within a prescribed time and satisfactory transient performance, controllers with or without approximating structures are established. Notably, the settling time and initial condition of the prescribed performance function are completely independent of the initial tracking error and system parameters, thereby improving upon existing results. Furthermore, the disadvantage of the semi-global boundedness of tracking error induced by dynamic surface control can be eliminated through the use of a novel Lyapunov-like energy function. Finally, the effectiveness of the proposed strategies is validated through numerical simulations performed on practical examples.