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In this note, we study the prescribed-time (PT) synchronization of multiweighted and directed complex networks (MWDCNs) via pinning control. Unlike finite-time and fixed-time synchronization, the time for synchronization can be preset as needed, which is independent of initial values and parameters like coupling strength. First and foremost, we reveal the essence of PT stability by improper integral, L'Hospital rule and Taylor expansion theory. Many controllers established previously for PT stability can be included in our new model. Then, we apply this new result on MWDCNs as an application. The synchronization error at the prescribed time is discussed carefully, so, PT synchronization can be reached. The network topology can be directed and disconnected, which means that the outer coupling matrices (OCMs) can be asymmetric and not connected. The relationships between nodes are allowed to be cooperative or competitive, so elements in OCMs and inner coupling matrices (ICMs) can be positive or negative. We use the rearranging variables' order technique to combine ICMs and OCMs together to get the sum matrices, which can make a bridge between multiweighted and single-weighted networks. Finally, simulations are presented to illustrate the effectiveness of our theory., Comment: 18 pages, 3 figures

In this paper, we discuss distributive synchronization of complex networks in finite time, with a single nonlinear pinning controller. The results apply to heterogeneous dynamic networks, too. Different from many models, which assume the coupling matrix being symmetric (or the connecting graph is undirected), here, the coupling matrix is asymmetric (or the connecting graph is directed

The study of complex networks with multi-weights has been a hot topic recently. For a network with a single weight, previous studies have shown that they can promote synchronization. But for complex networks with multi-weights, there are no rigorous analysis to show that synchronization can be reached faster. In this paper, the complex network is allowed to be directed, which will make the synchronization analysis difficult for multiple couplings. In virtue of the normalized left eigenvectors (NLEVec) corresponding to the zero eigenvalue of coupling matrices, we prove that if the Chebyshev distance between NLEVec is less than some value, which is defined as the allowable deviation bound, then the synchronization and control will be realized with sufficiently large coupling strengths, i.e., all coupling matrices do accelerate synchronization. Moreover, adaptive rules are also designed for the coupling strength., Comment: 6 pages

In this paper, we propose several models, which can realize synchronization of complex networks in finite time effectively. The results apply to heterogeneous dynamic networks, too. The mechanism of finite time convergence is revealed. Different from many models, which assume the coupling matrix being symmetric (or the connecting graph is undirected), here, the coupling matrix is asymmetric (or the connecting graph is directed).

In this paper, we discuss distributed adaptive algorithms for synchronization of complex networks, consensus of multi-agents with or without pinning controller. The dynamics of individual node is governed by generalized QUAD condition. We design new algorithms, which can keep the left eigenvector of the adaptive coupling matrix corresponding to the zero eigenvalue invariant. Based on this invariance, various distributive adaptive synchronization, consensus, anti-synchronization models are given.

Octonion-valued neural networks (OVNNs) are a type of neural networks for which the states and weights are octonions. In this paper, the global $\mu$-stability and finite-time stability problems for octonion-valued neural networks are considered under unbounded and asynchronous time-varying delays. To avoid the non-communicative and non-associative multiplication feature of the octonions, we firstly decompose the OVNNs into eight real-valued neural networks (RVNNs) equivalently. Through the use of generalized norm and the Cauchy convergence principle, we obtain the sufficient criteria which assure the existence, uniqueness of the equilibrium point and global $\mu$-stability of OVNNs. By adding controllers, the criteria to ensure the finite-time stability for OVNNs are presented by dividing the analysis of finite-time stability process into two phases. Furthermore, we also prove the adaptive finite time stability theory of above networks. At last, the simulation results of specified examples is given to substantiate the effectiveness and correctness of the theoretical results.

In this brief, we set up the finite time stability (FnTSta) theory for dynamical systems with bounded time-varying delays via aperiodically intermittent control (AIC) and quantized control (QC). A more general QC is designed in this brief. The bound of time-varying delay is required to be less than the infimum in AIC. Two-phases-method (2PM) is applied to solve the FnTSta for delayed system, i.e., it divides the whole proof process into two phases, one phase is that the process for the norm of system error evolving from initial values to $1$, and the other is the process for the norm of system error evolving from $1$ to $0$. By proving that these two phases both use FnT to realize, the whole FnTSta for system is proved. We also design the adaptive rules and prove its validity rigorously. Furthermore, the obtained theories are used to discuss the finite time synchronization (FnTSyn) of neural networks as an application. Finally, some simulations are given to illustrate the effectiveness of our theoretical results.

The finite time stability (FnTSta) theory of delayed systems has not been set up until now. In this paper, we propose a two-phases-method (2PM), to achieve this object. In the first phase, we prove that the time for norm of system error evolving from initial values to 1 is finite; then in the second phase, we prove that the time for norm of the error evolving from 1 to 0 is also finite, thus FnTSta is obtained. Considering the cost and complexity of controller, we use only two simple terms to realize this aim. For the proposed 2PM, time delays can be the same or asynchronous, bounded or unbounded, etc; the norm can be $2$-norm, $1$-norm, $\infty$-norm, which show that 2PM is powerful and has a wide scope of applications. Furthermore, we also prove the adaptive finite time stability (AFnTSta) theory of delayed systems using 2PM. As an application of the obtained FnTSta theory, we consider the finite time outer synchronization (FnTOSyn) for complex networks with asynchronous unbounded time delays, corresponding criteria are also obtained. Finally, two numerical examples are given to demonstrate the validity of our theories.

In this paper, we studied the finite time anti-synchronization of master-slave coupled complex-valued neural networks (CVNNs) with bounded asynchronous time-varying delays. With the decomposing technique and the generalized $\{\xi,\infty\}$-norm, several criteria for ensuring the finite-time anti-synchronization are obtained. The whole anti-synchronization process can be divided into two parts: first, the norm of each error state component will change from initial values to $1$ in finite time, then from $1$ to $0$ in fixed time. Therefore, the whole time is finite. Finally, one typical numerical example is presented to demonstrate the correctness of our obtained results., Comment: 7 figures

In this paper, the complete synchronization problem of linearly coupled neural networks with reaction-diffusion terms and time-varying delays via aperiodically intermittent pinning control is investigated. The coupling matrix for the network can be asymmetric. Compared with state coupling in the synchronization literature, we design a novel distributed coupling protocol by using the reaction-diffusion coupling-spatial coupling, which can accelerate the synchronization process. This can be regarded as the main difference between this paper and previous works. Using the Lyapunov function and theories in the aperiodically intermittent control, we present some criteria for the complete synchronization with a static coupling strength. In this case, there is no constraint on the bound of time-varying delays, so it can be larger than the length of control span. On the other hand, for the network with an adaptive coupling strength, we propose a simple adaptive rule for the coupling strength and prove its effectiveness rigorously. In this case, the bound of time-varying delay is required to be less than the infimum of the control time span. Finally, numerical simulations are given to verify the theoretical results., Comment: 33 pages, 8 figures

In this paper, the fixed-time cluster synchronization problem for complex networks via pinning control is discussed. Fixed-time synchronization has been a hot topic in recent years, which means that the network can achieve synchronization in finite-time and the settling time is bounded by a constant for any initial values. To realize the fixed-time cluster synchronization, a simple distributed protocol by pinning control technique is designed, whose validity is rigorously proved, and some sufficient criteria for fixed-time cluster synchronization are also obtained. Especially, when the cluster number is one, the cluster synchronization becomes the complete synchronization problem; when the intrinsic dynamics for each node is missed, the fixed-time cluster synchronization becomes the fixed-time cluster (or complete) consensus problem; when the network has only one node, the coupling term between nodes will disappear, and the synchronization problem becomes the simplest master-slave case, which also includes the stability problem for nonlinear systems like neural networks. All these cases are also discussed. Finally, numerical simulations are presented to demonstrate the correctness of obtained theoretical results.

In this letter, by regarding finite-time stability as an inverse problem, we reveal the essence of finite-time stability and fixed-time stability. Some necessary and sufficient conditions are given. As application, we give a new approach for finite-time and fixed-time synchronization and consensus. Many existing results can be derived by the general approach.

In this paper, we study the relationship between the network-based inference method and global ranking method in personal recommendation. By some theoretical analysis, we prove that the recommendation result under the global ranking method is the limit of applying network-based inference method with infinity times., Comment: 13 pages, 3 figures

Stability of the zero solution plays an important role in the investigation of positive systems. In this note, we revisit the $\mu$-stability of positive nonlinear systems with unbounded time-varying delays. The system is modelled by continuous-time differential equations. Under some assumptions on the nonlinear functions like homogeneous, cooperative, nondecreasing, we propose a novel transform, by which the nonlinear system reduces to a new system. Thus, we analyze its dynamics, which can simplify the nonlinear homogenous functions with respect to (w.r.t.) arbitrary dilation map to those w.r.t. the standard dilation map. We finally get some criteria for the global $\mu$-stability. A numerical example is given to demonstrate the validity of obtained results.

In this paper, we investigate the global exponential stability for complex-valued recurrent neural networks with asynchronous time delays by decomposing complex-valued networks to real and imaginary parts and construct an equivalent real-valued system. The network model is described by a continuous-time equation. There are two main differences of this paper with previous works: (1), time delays can be asynchronous, i.e., delays between different nodes are different, which makes our model more general; (2), we prove the exponential convergence directly, while the existence and uniqueness of the equilibrium point is just a direct consequence of the exponential convergence. By using three generalized norms, we present some sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for delayed complex-valued neural networks. These conditions in our results are less restrictive because of our consideration of the excitatory and inhibitory effects between neurons, so previous works of other researchers can be extended. Finally, some numerical simulations are given to demonstrate the correctness of our obtained results.