Finite-Time Synchronization and Synchronization Dynamics Analysis for Two Classes of Markovian Switching Multiweighted Complex Networks from Synchronization Control Rule Viewpoint

This paper is mainly concerned with how nonlinear coupled one impacts synchronization dynamics of a class of nonlinear coupled Markovian switching multiweighted complex networks (NCMSMWCNs). Firstly, sufficient conditions of finite-time synchronization for a class of NCMSMWCNs and a class of linear coupled Markovian switching multiweighted complex networks (LCMSMWCNs) are investigated. Secondly, based on the derived results, how nonlinear coupled one affects synchronization dynamics of the NCMSMWCNs is analyzed from synchronization control rule. Thirdly, in order to further explore how nonlinear coupled one affects synchronization dynamics of theNCMSMWCNs, synchronization dynamics relationship of theNCMSMWCNs and the LCMSMWCNs is built. Furthermore, this relationship can also show how linear coupled one affects synchronization dynamics of the LCMSMWCNs. At last, numerical examples are provided to demonstrate the effectiveness of the obtained theory.


Introduction
In recent years, synchronization of nonlinear coupled complex networks has gained considerable attention because synchronization is one of the most important collective behaviors of complex networks [1][2][3][4][5][6][7][8].Furthermore, nonlinear coupled one was important factor impacting synchronization dynamics of nonlinear coupled complex networks.In [1], Zhang et al. considered outer synchronization for a class of nonlinear coupled drive-response networks.In [2], exponential quasi-synchronization for a class of nonlinear coupled drive-response memristive neural networks was considered.In [3], a class of nonlinear inner coupling drive-response complex networks was proposed and sufficient conditions of its generalized matrix projective outer synchronization were given.By analyzing [1][2][3][4][5][6][7][8], it is worth noting that, except [4], there was no topic on how nonlinear coupled one affects synchronization dynamics of nonlinear coupled complex networks.The literature [4] adopted simulation method to discuss how nonlinear coupled one impacts synchronization dynamics of the addressed nonlinear coupled complex networks.It is pity that there was no theoretical analysis in [4].The literatures [1][2][3][5][6][7][8] mainly emphasized how to obtain sufficient conditions of the considered nonlinear coupled complex networks.Therefore, it is necessary and significant to further develop the above issue from theoretical aspect.Until now, to the best of our knowledge, there are a few literatures to mention the issue.In addition, linear coupled one can also affect synchronization dynamics of linear coupled complex networks [7].Notice that although a lot of results for synchronization problems of linear coupled complex networks have been obtained [8][9][10], how linear coupled one impacts synchronization dynamics of linear coupled complex networks still needs to be further researched.For instance, in [8], Ali et al. studied extended dissipative synchronization for a class of linear coupled complex networks with additive time-varying delay and discrete-time information.The literature [9] was concerned with asymptotical synchronization of a class of linear coupled complex networks with actuator faults and unknown coupling weights via adaptive control schemes.From [8][9][10], it is seen that although some feasible synchronization theories were derived, the relationship of linear coupled one and synchronization dynamics was still not noticed.
As a matter of fact, because the connections of many real networks have different weights, such as email networks and public traffic networks, these networks can be modeled by multiweighted complex networks [11][12][13][14][15][16][17][18][19].Therefore, recently, some works for synchronization problems of multiweighted complex networks began to develop [11][12][13][14][15][16].For example, in [11], the authors studied synchronization of complex networks with multiweights and its application in public traffic network.Qiu et al. [12] investigated synchronization and  ∞ synchronization of multiweighted complex delayed dynamical networks with fixed and switching topologies.Furthermore, although the literatures [17][18][19] gave several classes of multiweighted complex networks, they addressed some passivity problems, instead of synchronization issue.From [11][12][13][14][15][16][17][18][19], it is not difficult to find that until now there are still a few literatures to discuss synchronization of multiweighted complex networks.Besides these, it should be noted that, in many engineering areas, there sometimes exist abrupt variations which are often caused by random failures or repairs of the components in systems.In this case, Markovian switching can model this phenomenon [20].This caused that Markovian switching systems have gained widespread attention [21][22][23][24][25][26][27].Simultaneously, due to processing speeds, finite information transmission, and random perturbations which are often from environment elements, it is inevitable that time-delays and stochastic noises often happen [28].Thus, many useful results for synchronization of Markovian switching systems with time-delays and stochastic noises were derived [21,29,30].
In fact, from the viewpoint of system dynamics, if a system can achieve stability state from initial state, there should exist convergence time  * .That is to say, if a system satisfies stability conditions, it can get to stability state within finitetime  * .How to get  * ?In this situation, finite-time problems of many systems have been attracting increasing interest [31][32][33][34][35][36][37][38][39][40][41][42].For example, Liu et al. [31] investigated finite/fixedtime synchronization of complex networks with stochastic disturbances.The literature [32] was concerned with finitetime consensus of multiple nonholonomic chained-form systems based on recursive distributed observer.Liu et al. [33] studied finite-time synchronization switched coupled neural networks with discontinuous or continuous activations.In [34], finite-time robust passive control for a class of switched reaction-diffusion stochastic complex dynamical networks with coupling delays and impulsive control was considered.According to the obtained results of [31][32][33][34][35][36][37][38][39][40][41][42], it is observed that in some effective finite-time  * estimation approaches for synchronization, passivity, and consensus, the proposed systems were derived.Unfortunately, to our knowledge, few researchers focused attention on finite-time synchronization of linear or nonlinear coupled Markovian switching multiweighted complex networks.
Inspired by the above discussion, this paper investigates sufficient condition of finite-time synchronization for a class of NCMSMWCNs.Based on the derived results and Lyapunov stability theory, how nonlinear coupled one affects synchronization dynamics of the NCMSMWCNs is analyzed from synchronization control rule viewpoint.Besides this, sufficient condition of finite-time synchronization for a class of LCMSMWCNs is proposed.Moreover, by comparing synchronization dynamics of the NCMSMWCNs and the LCMSMWCNs, how nonlinear coupled one impacts synchronization dynamics of the NCMSMWCNs is further explored.Numerical simulation examples illustrate the effectiveness of the derived results.The novelties and contributions of the paper are highlighted as follows: (1) Based on the derived sufficient condition of finitetime synchronization of the NCMSMWCNs, how nonlinear coupled one impacts synchronization dynamics of the NCMSMWCNs is discussed.
(2) By building sufficient condition relationship of finitetime synchronization for two classes of the considered complex networks, synchronization control rule relationship of the NCMSMWCNs and the LCMSMWCNs is presented.
(3) How nonlinear coupled one and linear coupled one affect synchronization dynamics of the NCMSMWCNs and the LCMSMWCNs is analyzed from synchronization control rule relationship aspect.
The rest of this paper is organized as follows.Section 2 gives model and preliminaries.In Section 3, sufficient conditions and sufficient condition relationship of finite-time synchronization of the NCMSMWCNs and the LCMSMWCNs are derived.Furthermore, synchronization dynamics of the addressed systems are analyzed.Sections 4 and 5 provide simulation results and the conclusions, respectively.At last, acknowledgments and conflicts of interest are given.

Problem Formulation and Preliminaries
Consider two classes of Markovian switching multiweighted complex networks as follows: where f(⋅) : R  → R  is the activity function of ĩth node, zĩ () = (z ĩ1 (), zĩ 2 (), . . ., zĩ stands for the kth outer-coupling weight matrix.If there exists a connection from node  to node  ( ̸ = ), Dk (r()) ̸ = 0, otherwise, Dk (r()) = 0. τ() is coupling time-varying delay, ũĩ (, r() is the controller of networks ( 1) and ( 2), σĩ (⋅, ⋅, ⋅, ⋅) is the noise intensity, and r() is right-continuous Markov chain with known transition rates and is independent of the Brownian motion   ().In order to make notation simplicity, let Dk (r()) = Dr k , dk ĩ j(r()) = dk ,r ĩ j , ũĩ (, r()) = ũĩ (, r), and r() = r.Remark 1. Comparing with models (1) and (2), it is seen that there is no difference except coupled functions.The coupled functions of models (1) and ( 2) are nonlinear function g(z j(), zj ( − τ())) and linear coupled function zj () + zj ( − τ()), respectively.Why are the two similar models considered?The main reason is related to the motivations of this paper.From the analysis of introduction, it is known that one of the motivations is under the same synchronization control rule, and comparing with synchronization dynamics of models (1) and (2), it is derived how nonlinear coupled function and linear coupled function affect synchronization dynamics of the addressed system.The detailed analysis is in Remark 15.Besides this, in models ( 1) and ( 2), diffusive coupling condition ) is removed.Thus, the obtained results are more general.Actually, the scheme has been adopted in [19].It is necessary to emphasize that although some classes of multiweighted complex networks have been proposed [11][12][13][14][15][16][17][18][19], these models are liner coupled multiweighted complex networks.Furthermore, in these models, there is no Markovian switching.Therefore, models (1) and (2) are two classes of new multiweighted complex networks.
In order to obtain the derived and analysis results, the definition of finite-time synchronization of networks (1) and (2), some assumptions, and lemmas are needed.
Assumption 4. The functions f(⋅) and g(⋅, ⋅) satisfy the Lipschitz conditions and g(0, 0) = 0.That means there exist constants L, L1 > 0 and L2 > 0 such that Remark 5.In [1][2][3][4][5][6][7], some synchronization problems for some classes of nonlinear coupled complex networks have been proposed.Furthermore, it is observed that some methods can be used to deal with nonlinear coupled one [1][2][3][4][5][6][7].According to these methods, it is derived that these methods satisfy the following two properties.The first is that these methods are linearization techniques.That is to say, by using these methods, nonlinear coupled function will become linear function.For example, in [5][6][7], nonlinear coupling function of the addressed networks is (()).In order to obtain the derived results, one assumed that (()) satisfied ‖  (
Remark 12. From Lemma 11, it is observed that if  ≥  * , there is () = 0.Because () is Lyapunov function with respect to a system, this causes that if () = 0, the system must achieve stability state within finite-time  * .Finite-time synchronization, which is as a special case of stability, can be processed by the above Lemma 11.According to Lemma 11, if ()/ ≤ −  (), there is Therefore, in order to get  * , it is needed to prove that Lyapunov function () of the system satisfies ()/ ≤ −  ().It should be pointed out that finite-time algorithm of Lemma 11 is a feasible finite-time estimation approach.Besides this, in  ż ĩ() of networks ( 1)-( 2) and  ĩ() of the error systems ( 5)-( 6) there exists the notation .Actually,  of  ż ĩ() and  ĩ() stands for small change in any quantity over a small subsequent time interval , and   () of systems ( 1), ( 2), ( 5), and (6) represents a bounded vector-form Brownian motion process [48].Therefore, systems (1), ( 2), ( 5), and ( 6) are two classes of stochastic differential equations with Markovian switching in nature.Because Definition 3 is built on systems (1) and ( 2), E in Definition 3 represents expectation.For instance, E[] means the expectation of the random variable .

Main Results
In this section, sufficient conditions of finite-time synchronization of networks ( 1) and ( 2) are derived.Based on sufficient condition of network (1), how nonlinear coupled one g(⋅, ⋅) affects synchronization dynamics of network ( 1) is analyzed.Besides this, according to the derived results, synchronization dynamics relationships of the networks ( 1) and ( 2) are built.By using synchronization dynamics relationship and synchronization control rule viewpoint, how nonlinear coupled one g(⋅, ⋅) and linear coupled one zj () + zj ( − τ()) impact synchronization dynamics of networks ( 1) and ( 2) is further explored.

Corollary 20. Under Theorem 16, network (2) with controller (4) must satisfy Theorem 13 if the following inequalities hold:
(2) Proof.The proof is similar to that of Corollary 17.

Corollary 21. Under Corollary 20, synchronization control rule for network (2) must make network (1) get synchronization within finite-time 𝑡
Proof.The proof is similar to that of Corollary 18.
Remark 22.Under Corollary 21, there are 0 < L2 r and synchronization dynamics relationship of networks ( 1) and ( 2) is similar to that of Remark 19.From the above, it is seen that under Corollaries 18 and 21, synchronization dynamics relationship of networks ( 1) and ( 2) is decided by the initial state zĩ (0), synchronization state s(), nonlinear coupled function g(⋅, ⋅), linear coupled function z() + z( − τ()), and coupled matrix Dr k.From synchronization dynamics relationship of networks ( 1) and ( 2), it is derived that how nonlinear coupled function g(⋅, ⋅) and linear coupled function z() + z( − τ()) impact synchronization dynamics of networks ( 1) and ( 2) is not only related to coupled function, but also connected with coupled matrix, the initial state, and synchronization state.
Remark 23.Compared with the recent results of synchronization problems for complex networks such as [8,10,16], the effectiveness of this paper can be reflected by synchronization dynamics analysis ideas of the addressed complex networks.From the analysis of Remarks 15-19, it is seen that the effect of nonlinear coupling function for synchronization dynamics is not only related to nonlinearity of nonlinear Complexity coupling function, but also connected with coupled matrix, synchronization state, and the initial state of the considered system.Although some feasible synchronization results of complex networks were derived in [8,10,16], these papers mainly focused on how to obtain sufficient conditions of synchronization problems.Besides this, according to introduction of this paper, it is observed that the literature [4] only used simulation method to analyze how nonlinear coupling function impacts synchronization dynamics of the proposed complex networks.All these show that the work in this paper can extend the existed analysis ideas of synchronization problems for complex networks.
Remark 24.According the above analysis, it is not difficult to find that the results of Remarks 14-19 are closely related to the common synchronization control rule.This causes that although nonlinearity of nonlinear coupled function g(⋅, ⋅) can affect synchronization dynamics of network (1), it is not reflected by synchronization finite-time  * .The reason is that there is no function relationship between the estimation approach of synchronization finite-time  * and nonlinearity of nonlinear coupled function g(⋅, ⋅).This can be seen from condition (3) of Theorem 13.How to solve it?Besides this, the ideas of this paper can be extended to finite/fixedtime pinning synchronization of complex networks [31], nonsmooth finite-time synchronization of switched coupled neural networks [33], finite-time consensus of multiagent systems [35], tracking control uncertain interconnected nonlinear systems [49], distributed formation control of multiple quadrotor aircraft [50] and so on.All these are further works in the future.

Conclusions
In this paper, sufficient conditions of finite-time synchronization for a class of NCMSMWCNs and a class of LCMSMWCNs are studied.Based on the derived results, synchronization dynamics problems of the NCMSMWCNs and synchronization dynamics relationships of the NCMSMWCNs and the LCMSMWCNs are analyzed, respectively.Comparing synchronization dynamics between the NCMSMWCNs and the LCMSMWCNs, how nonlinear coupled one and linear coupled one affect synchronization dynamics of the NCMSMWCNs and the LCMSMWCNs is further explored.Numerical simulation results show the effectiveness of the derived theory.

Figure 1 :
Figure 1: Synchronization trajectories, synchronization total error trajectories of network (1) with controller (4) and the curves of controller (4) for Case I of Example 25.

Figure 2 :
Figure 2: Synchronization trajectories, synchronization total error trajectories of network (1) with controller (4) and the curves of controller (4) for Case II of Example 25.

Figure 3 :
Figure 3: Synchronization trajectories, synchronization total error trajectories of network (1) with controller (4) and the curves of controller (4) for Case III of Example 25.

Figure 4 :
Figure 4: Synchronization trajectories, synchronization total error trajectories of network (1) with controller (4) and the curves of controller (4) for Case IV of Example 25.
The curves of the controller(4)