A BRIEF SURVEY ON FINITE TIME AND FIXED TIME SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS AND ITS APPLICATIONS

Complex networks have, in recent years, brought many innovative impacts to large-scale systems. However, great challenges also come forth due to distinct complex situations and imperative requirements in human life nowadays. This paper attempts to present an overview of recent progress of synchronization of complex dynamical networks and its applications. We focus on Finite-time and Fixed-time synchronization of complex dynamical networks with nonidentical discontinuous nodes, time delay, Class of Output-Coupling via continuous control and Markovian jump complex networks. Then, were view several applications of synchronization in complex networks, especially in neuroscience and power grids. The present limitations are summarized and future trends are explored and tentatively highlighted.


INTRODUCTION
Complex networks have been extensively studied in various fields in recent decades, including biology, chemistry, physics, and mathematics. A complex dynamic network is typically a broad collection of interconnected nodes, in which each node is a nonlinear dynamic system. Synchronization is unique in nature and can play an extremely important role in many science fields including genetics, climatic science, sociology and ecology. Consequently, study of synchronization of complex networks is essential in both theory and practice.
To analyze a complex network's synchronous behaviours, first, some useful modelling techniques need to be explored. Algebraic graph theory is commonly accepted as one of the most convenient approaches and was widely used to provide a universal analysis model for node-to-node interactions within a network. In particular, an undirected or a directed graph may define the communication topology of a complex network, where a vertex signifies an individual node within a network and an edge denotes a communication connection between nodes. Therefore, an individual system's intrinsic dynamics reflect its evolutionary law and strongly influence the dynamic processes of the entire network. It should also be taken into account in the modelling process. The developed model for a complex dynamic network will therefore concern both the nonlinear dynamic properties of nodes and the exchange of information between nodes.
In recent years, researchers have paid growing attention to the synchronization issue of complex dynamic networks in order to get a deep understanding of the synchronization process 5 A BRIEF SURVEY ON FINITE TIME AND FIXED TIME SCDN AND ITS APPLICATIONS

PRELIMINARIES
In this section we remember some concepts and synchronization properties of complex dynamic networks, which will be used throughout the paper. (i) A digraph is strongly connected if there is a directed path between any two vertices.
(ii) A digraph is quasi-strongly connected if at least one root exists. 7 A BRIEF SURVEY ON FINITE TIME AND FIXED TIME SCDN AND ITS APPLICATIONS Remark 4: From Definition 10, one can observe that the Laplacian matrix of a graph is an M-matrix. Another interesting finding is that the Laplacian matrix is irreducible if and only if the corresponding digraph is strongly connected. If a digraph is strongly connected, then its Laplacian matrix has a simple zero eigenvalue and a positive left eigenvector associated with the zero eigenvalue. It should be noted that the above statement is a sufficient condition, rather than a necessary condition. Moreover, the zero eigenvalue of the Laplacian matrix is simple if and only if the associated graph has a directed spanning tree (or quasi-strongly connected).

Assumption 1:
Areal-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, ( ) ( ) for all real x1 and x2  R.
  are two constants. Then, for any given 0 t ,

( )
Vt satisfies the following differential inequality:  Consider a complex N-non-identical node model with diffusively linear couplings in which each node is a dynamic n-dimensional structure, i.e.

Synchronization of Complex Networks with nonidentical discontinuous nodes
represents the state vector of the i th dynamical node, the dynamics of the uncoupled i th node is ( ) Because although the sign function in (4) meets the basic requirements, the Filippov solutions of (2) and (3) are available and can be described as Subtracting (5) from (6) produces the following error dynamical system: The other parameters are defined in Corollary 1, in Theorem 1.
Then, in a finite time defined in Theorem 1, the complex network (3) is synchronized to (2) If all of the system nodes (3) are similar to the isolate system (2),  the error mechanisms (7) and (8) then turn out to be as follows: By contrasting the program (17) with (8), the following corollary can be easily obtained by taking where in Theorem 1, the other parameters are defined, then, in a finite 1 t time, the complex network (3) with identical nodes is synchronized to (2) where 1 t is the same as above.
Theorems 1 and 2 are formulated in terms of LMIs, while the algebraic inequalities are used to describe Corollaries 1 and 2. While results with algebraic inequalities are more conservative than those with LMIs, computing using algebraic inequalities is simpler than using LMIs particularly for complex networks, as complex networks typically have a large number of nodes. The error mechanisms (7) and (8) derive specific synchronization parameters respectively. All the synchronization requirements in Theorems 1, 2, and Corollaries 1, 2 are valid for regulation of 13 A BRIEF SURVEY ON FINITE TIME AND FIXED TIME SCDN AND ITS APPLICATIONS finite-time synchronization of complex networks with non-identical discontinuous nodes.
Although operators should choose less restrictive parameters for synchronization according to the realistic situation in actual applications.

Synchronization for a Class of Output-Coupling Networks via Continuous control
Most studies have concentrated on state-coupling complex networks in recent years, while output-coupling complex networks attract comparatively less consideration, let alone research on fixed-time and finite-time synchronization issues.
As we all know, coupling power between the nodes of complex networks plays a very significant role in the issue of complex network synchronization. In general, reinforcing the coupling effect to understand the synchronization for CNs is a simple and prime concept. In this condition, however, the coupling force is also required to be as strong as possible.
It is therefore necessary and desirable to find a suitable coupling strength which is appropriate. Use of adaptive technique [21] is a simple and successful way of achieving this goal.
The authors of [22] studied the problem of synchronization with delays of coupled connected NN.
The authors investigated in [23] that a single controller used pinning control for complex networks, but the coupling strength was expected to be very high, which was very rigorous. The problem of adaptive coupling power for fixed-time synchronization of complex networks with output nodes is less discussed in [24].
Zhiwei Li discussed the above problem in detail in his paper entitled "Fixed-Time and

Finite-Time Synchronization for a Class of Output-Coupling Complex Networks by Continuous
Control." He was investigating the problem of finite-time and fixed-time synchronization with output feedback nodes for a class of general output-coupling CNs by using the Lyapunov stability principle, LMI and adaptive methodology, many ample conditions ensuring fixed-time and finite-time synchronization are extracted.
Find output coupling complex networks with output nodes as observes:  (20) can therefore be used to define the complex networks, guided and undirected weighted.
If output matrix C is matrix of identity n I , (1) then degrade to This has been examined in detail by [25][26][27]. Hence, system (20) is wider than system (21).
The author developed appropriate controllers ( ) i utin such a way that complex network states (20) synchronize in a finite time and fixed time into the state of the following target system; The following dynamical error systems are obtained by subtracting (22) from (20): are the i th node 'status error and output error, respectively. We definitely have

Finite-Time Synchronization
The author developed suitable controllers ( ) The following theorem is given by constructing the Lyapunov function ( ) ( ) ( ) give the finite-time synchronization for complex networks (22) to the homogenous trajectory (21).

Theorem 3:
Suppose that 1 holds that. If any positive constants exist then, the controlled network

Fixed-Time Synchronization
The author has developed suitable controllers ( ) i utto synchronize complex output-coupling networks (1) into (3) within a specified time-limited settlement period as follows:  From Theorem 4 results it can be found that the settling time is independent of the initial state value xi (0) and x0 (0). In addition, the settling time can be determined by the node dimension n, the design parameters and group order N.

Adaptive Adjustment of the Coupling Strength
Using adaptive technique, the adaptive coupling intensity problem for fixed-time synchronisation of complex networks with output nodes is investigated. The regulated complex network with output nodes is followed by the adaptive coupling rule to: where  is a small positive constant.
The following theorem is presented to give the homogenous trajectory a fixed-time synchronization for adaptively controlled complex networks by creating the Lyapunov function , ,,

Synchronization of complex networks with time delay via impulsive control with finite-time
To evaluate finite-time impulsive synchronization with time delay process of general complex dynamic networks as follows: The above complex dynamic network (28) can be rewritten in the following form of impulsive differential equation, without loss of generality: ii P e u t e t k sign e k q e s Pe s ds i P e u t e Where 0 i   are constants to be determined, k > 0 is a tunable constant and real number.
Pe q e s Pe s ds The theorem above was implemented by constructing the following LKF

Synchronization of coupled networks via discontinuous controllers with finite time
Some of the earlier research on the synchronization of coupled neural networks have implemented linear feedback controller and can only achieve exponential or asymptotic convergence. Nevertheless, in practical application the finite-time convergence is more significant. The finite-time synchronization advantage requires robustness and higher convergence rate against uncertainties. As is well known, certain discontinuous dynamic systems are more likely to achieve convergence of the finite-time. Discontinuous controllers are often deliberately configured to monitor and stabilize finite-time. Jun shen explores the finite-synchronization by discontinuous controllers of an array of coupled neural networks. Some appropriate parameters for finite-time synchronization are obtained based on the Lyapunov method. In addition, author suggested strategies for shifted control and adaptive tuning parameter to reduce the settling time. Additionally, the pinning control scheme is also configured for finite-time synchronization via a single controller. With the hypothesis that the topology of the coupling network includes a directed spanning tree and that each of the strongly connected components is detail-, it has been shown that finitesynchronization can be accomplished by pinning. With the hypothesis that the topology of the coupling network includes a directed spanning tree and that each of the strongly connected components is detail-, it has been shown that finite-synchronization can be accomplished by pinning.
Find the following configuration for a neural network: Under the above control scheme (37), systems (35) and (36) can be represented as follows:

Finite-time synchronization of Markovian jump complex networks
Over the past decade [30][31][32][33][34][35][36][37], Markovian Jump Complex Networks (MJCNs) have provided a wide range of coverage. This is partly due to Markovian jump being a suitable mathematical pattern for representing a class of complex networks subject to spontaneous abrupt structural variations [31][32][33][34][35]. Additionally, MJCNs may be viewed as a special class of stochastic network systems. This class of network systems has finite modes that move at various times from one to the next [38]. In addition, such a turn (or jump) can be controlled by a Markovian chain [37]. MJCNs occur in a variety of fields, such as NNs [35], genetic regulatory networks [36] and hopfield networks [33].The study of the dynamic and topological structure of MJCNs is therefore of fundamental importance for understanding the real-world functions.
Synchronization has been intensively studied in recent years for MJCNs, with or without delays, as one of the most important dynamic behaviours [31][32][33][34][35][36][39][40][41][42]. For example, synchronization issues with mode-dependent mixed time delays in [39] were resolved for the discrete MJCNs. Exponential synchronization for MJCNs with mixed time delays has been studied in [32]. And synchronization of hybrid coupled MJCNs with mode-dependent mixed delays has studied in [41,42]. In practice engineering, however, synchronization is often needed

Finite-time Synchronization of MJCNs with partially unknown transition rates
Consider a complex Markovian jump network consisting of N identical nodes with diffusive couplings, in which each node is a dynamic system of n dimensions, i.e.

Finite-time Synchronization of MJCNs with partially unknown transition rates and time delays
Time delays are well known to be inevitable when designing complex network models.
Hence, considering the dynamics for the complex networks with time delays is very important.. The Network Isolated Node (50) is given by Where the basic device solution is (46).
The following dynamical error system is generated by subtracting (49) from (48):

APPLICATIONS OF SYNCHRONIZATION OF COMPLEX NETWORKS
Complex network synchronization or controllability has been widely applied in many fields such as power grids and neuroscience and other fields which are briefly evaluated as follows:   [19,20].

CONCLUSIONS
A brief review is presented in this paper on recent developments in finite-time and fixed-time synchronization of complex dynamic networks with non-identical discontinuous nodes, time delay, output-coupling class by continuous control and complex Markovian jump networks.
Then, several synchronization applications in complex networks, especially in neuroscience and power grids, were viewed. This paper's main aim is to make some new innovations and serve as good advice for people working in the area. If any of the latest published studies on the subject are missed, we give the writers and readers our apologies. There are some issues that should be addressed in future research given diverse findings. Some of them we high-light as follows: (i) One can investigate the effect of communication network topologies and network-induced constraints on pinning controllability efficiency, pinning observability and pinning synchronization for a complex dynamic network with general topology. Since the shared communication network is vulnerable to malicious attacks and exploits, the issue of cyber security has received increasing interest in research and needs to be investigated in depth, especially for application-level multi-agent systems. In addition, problems related to stochastic complex network and randomly occurring pinning control strategies are interesting topics for study. It is also interesting and important to consider a complex dynamic network with non-identical nodes and the case that the pinning cost and the quantity of pinned nodes are finite for a large-scale, distributed, directed network. (ii) It is still important to further examine the design of controllers with delays to achieve finite-time synchronization of coupled neural networks.
(iii) Another interesting but challenging problem is the analysis of finite-time synchronization of coupled neural networks with discontinuous activation functions through discontinuous controllers.
(iv) Power grids are becoming more distributed, smarter and more versatile. Nowadays, as small distributed power generators and decentralized energy storage systems need to be connected to the power network, smart grids are being introduced to supply electricity from producers to customers in order to conserve resources, thus lowering costs and increasing efficiency and transparency.
(v) Due to the advent of micro grids in power grids nowadays, it is becoming more important to use a droop controller to prevent the propagation of currents between converters without any essential communication). It is therefore imperative that complex network theory and control theory be used to enhance the controllability of power grids by the use of hierarchical control. (ix) Intelligent methods for modelling dynamics of complex networks, such as neural networks and fuzzy structures can be adopted. In addition, single objective or multi-objective evolutionary algorithms and constraint evolutionary algorithms are promising to serve as a candidate for managing complicated problems of optimization in complex networks such as problems of controllability.
(x) It should be noted that comprehensive studies incorporating some of the above topics are not yet appropriate, especially for the controllability of interdependent, complex networks and the robustness of complex network control (Bakule, 2014). Considering controllability, robustness, multiple layers, particularly applying the results in robotic systems, neuroscience and power grids would be challenging and promising at the same time. Experimental findings, including a decrease in histogram variance, low PSNR, entropy closeness to 8 and a small association between plain images and ciphered images, indicate successful implementation of the theoretical results obtained in (Tengda Wei et al, 2017). Future work will concentrate on implementing permutation operation in encryption scheme, as NPCR (number of pixel change rate) and UACI (unified average change intensity) cannot achieve desired performance and hopefully the permutation operation will solve the problems.

ACKNOWLEDGMENT
Authors would like to thank the anonymous referees for their valuable comments and careful reading to the improvement of the manuscript.