Fixed-Time Synchronization for Hybrid Coupled Dynamical Networks with Multilinks and Time-Varying Delays

This paper concerns the problem of fixed/finite-time synchronization of hybrid coupled dynamical networks. The considered dynamical networks withmultilinks contain only one transmittal time-varying delay for each subnetwork, whichmakes us get hold of more interesting and practical points. Two kinds of delay-dependent feedback controllers with multilinks as well as appropriate Lyapunov functions are defined to achieve the goal of fixed-time synchronization and finite-time synchronization for the networks. Some novel and effective criteria of hybrid coupled networks are derived based on fixed-time and finite-time stability analysis. Finally, two numerical simulation examples are given to show the effectiveness of the results proposed in our paper.


Introduction
The phenomenon of synchronization exists everywhere in the realistic living world.People's applause begins with random rhythm, but it turns to the same rhythm at the end of a splendid drama.Every router of Internet periodically releases routing messages, respectively.However, the researchers have found that different router would eventually send routing messages in synchronous way, which led to the network traffic jam.Synchronization is common in every field and plays an important role.In fact, in physics, mathematics, and other fields, synchronous phenomenon of the coupled dynamical system has been studied widely in the past decade.
We are involved in a variety of complex dynamical networks, such as communication networks, Internet, transportation networks, and electricity networks.And synchronous phenomenon of dynamical networks is common.Sometimes synchronization is good for people's lives; sometimes synchronization may be harmful to people's lives and may even cause huge losses or serious consequences.Therefore, researches on synchronization of complex networks are of great value and significance.In a recent decade, researches on synchronization control of complex dynamical networks attract many researchers' attention [1,2].Meanwhile, lots of relevant achievements have been obtained.Researches on synchronization mainly concentrate on the following: complete synchronization [3,4], projective synchronization [5][6][7], delay synchronization [8][9][10], finite-time synchronization [11][12][13], and so forth.Synchronization methods mainly contain the following: linear feedback method [14,15], nonlinear feedback method [16], adaptive method [17], and so forth.According to the continuity of control law, synchronization techniques can be divided into continuous and noncontinuous synchronization control methods [18][19][20].Compared with continuous control law, noncontinuous control law can reduce some control cost.
Also, synchronization techniques can be divided into finite-time synchronization and infinite-time synchronization.Compared with the asymptotic stability and exponential stability, finite-time stability has relatively lower time complexity.Therefore, in the real world, obtaining synchronization in a setting time has more practical value.In secure 2 Mathematical Problems in Engineering communication, sending and receiving the information in a setting time can not only ensure the security of communication but also improve the efficiency and safety of communications [21].However, achieving synchronization in finite time depends on the initial synchronization error value of the drive-response systems.The finite time for the unknown initial conditions of drive-response systems (the initial conditions of most of practical systems are hard to be achieved) is not fixed.Therefore, Polyakov defined the fixedtime stability theory.The time to obtain fixed synchronization is irrelevant to the initial synchronization error conditions of drive-response systems and only affected and controlled by the controller [22].
In this paper, we study the hybrid coupled dynamical networks containing coupling delay and internal delay that are indeed common in real world, which makes the research work more meaningful.In previous studies, coupling delay is defined as (  ( − ()) −   ( − ())) [23][24][25].In fact, as for the synchronization study of a pair of given driveresponse systems, the time delay influences the variables that are transmitted from one system to another system being studied.Therefore, considering the coupling term of time delay as (  ( − ()) −   ()) is more reasonable [26].
The concept of multilinks means that the edge number linking two nodes in complex networks is greater than one.Multilinked complex networks are ubiquitous in the real world, such as communication networks and transportation networks.Communication links of communication networks can be mail, telephone, QQ, letters, and so forth [27][28][29].Compared with researches of single link networks [30][31][32][33], those of networks with multilinks are more practical and significant.There have been a lot of research achievements in this field so far [34][35][36].However, to my best knowledge, few put their attention to the hybrid coupled networks with both one single time-varying delay coupling and multilinks.Few people study fixed-time synchronization and finite-time synchronization with the conditions of dynamical networks mentioned above.It is interesting to complete such study even to fill the gaps in this field.
Motivated by the foregoing analysis, this paper studies fixed-time synchronization and finite-time synchronization for hybrid coupled dynamical networks with multilinks and one single time-varying delay coupling.Furthermore, we propose new model of multilinked complex dynamical networks with one single time-varying delay coupling term and design two suitable feedback controllers to ensure the synchronization.With the constructed Lyapunov functions, several novel criteria are derived to achieve fixed-time/finitetime synchronization of the complex dynamical networks described in this paper.Finally, we give two numerical simulation examples to verify the effectiveness and correctness of our proposed theoretical results.
The rest of this paper is organized as follows.In Section 2, some preliminaries are introduced, such as notations, some assumptions, and lemmas.In Sections 3 and 4, the main results proposed in this paper are introduced.In Section 5, two numerical simulation examples are provided to verify the effectiveness and correctness of our obtained theoretical results.Finally, we draw the conclusion.

Preliminaries
Throughout this paper, we always have the functions (⋅) and (⋅) to satisfy the Lipschitz condition, which mean the following assumptions [37].
In order to prove the correctness of the results proposed in this paper, we need the following lemmas.

Finite-Time Synchronization for Hybrid Coupled Networks with Multilinks
In this section, we study finite-time synchronization for the multilinked complex dynamical networks with hybrid coupled terms and time-varying delays.By constructing Lyapunov function and designing appropriate controller, some useful criteria are derived to ensure that the multilinked complex dynamical networks proposed in this paper are synchronized.
Note 1.We propose a new multilinked complex dynamical networks model with hybrid coupled terms and time-varying delays.Bringing in inner coupling matrices &   , our study of the finite/fixed-time synchronization for the hybrid coupled dynamical networks with both one single timevarying delay coupling and multilinks is more practical for the real world.To the best of our knowledge, this study is the first exploration in such a field.
Assume that ([−, 0],   ) is a Banach space which consists of continuous functions mapping the interval [−, 0] into   with the norm ‖‖ = sup −≤≤0 ‖()‖.For the drive system (11) and the response system (13), their initial conditions are, respectively, given by Assume that in each of model equation (11) and model equation ( 13) a unique solution exists in regard to the initial conditions mentioned.

Finite-Time Synchronization between the Drive System and the Response System.
Combining with the given lemmas, assumptions, and conditions, we study finite-time stability of the origin of the given error dynamical system, which denotes the finite-time synchronization for the multilinked complex dynamical networks (11) and (13).
Definition 8 (the finite-time stability).Through controlling of the designed controller, there always exists a positive constant  1 , and the value of  1 depends on the value of the initial state vector then, the error dynamical system ( 15) is said to be finite-time stable, and this state of the error dynamical system is named as the finite-time stability.And  1 is called the settling time.
In order to obtain the target of finite-time synchronization for systems (11) and ( 13), we design the following controller as the control law of complex dynamical networks (13).The control law is given as follows: where  0 > 0, 0 <  < 1, and  = 1, 2, . . ., ; the parameters of   ,   ,    will be determined as follows.The following theorem can be derived for the driveresponse systems of models ( 11) and ( 13) with the given controller (17) then, the drive system (11) and response system (13) can realize the synchronization in finite time.And the settling time  1 can be determined as Proof.In this paper, the Lyapunov function is constructed as follows: Applying Lemma 4 and the error system to the process of the proof, we calculate the derivative of () step by step as follows: The calculating result (20) contains many details.Firstly, in order to calculate conveniently later, we deal with the function ∑  =1 ((sign  (  ())/√)  ()) in (20).Then, we substitute the partial processing result into the previous result (20) Secondly, we substitute result (21) into result (20).We get the following result: () According to Lemma 5, the error dynamical system (15) can be stable in finite time.And the settling time is determined by  1 =  0 +  1− ( 0 )/ 0 (1 − ).
This completes the proof.
Remark 10.Lemma 5 plays the role of an important theoretical basis for the previous proof, which supports the analysis of finite-time synchronization for hybrid coupled complex networks with multilinks and internal/coupling delay terms with single time-varying delays.Lemma 5 can be traced from [40].
According to Lemma 6, the following corollary can be obtained.

And let the parameters
where   > 0,   > 0, and . Then, we can obtain the conclusion that the complex system (11) and complex system (13) will realize synchronization in finite time.Remark 12.According to Theorem 9 and Corollary 11, some sufficient finite-time synchronization criteria of the multilinked hybrid coupled networks are derived for the driveresponse systems of models (11) and (13) with the designed controller in this paper.The parameters   > 0,   > 0, and    > 0 ( = 1, 2, . . ., ) are critical to the finite-time synchronization of networks (11) and (13).Compared with the previous researches, the models of complex dynamical networks proposed in our paper are more general.

Finite-Time Synchronization between Hybrid Coupled
Dynamical Network and an Isolated Node.Suppose that we need multilinked hybrid coupled complex networks to synchronize to an isolated node in finite time.Then we will design the following complex networks model (25) as follows: where  = 1, 2, . . ., .
Use   () to denote the synchronization errors, where   () =   ()−(),  = 1, 2, . . ., .Combining with the given condition (12),   () can be described as follows: The process of the proof for system (25) and isolated node (26) to obtain finite-time synchronization is the same as that for Theorem 9. Hence, we omit it.
Then, the following corollary can easily be obtained.
The controller designed to obtain the finite-time synchronization is described as follows: where  0 > 0 and 0 <  < 1.
Then, the multilinked hybrid coupled complex networks (25) with control law ( 29) can realize the finite-time synchronization.

Fixed-Time Synchronization for Hybrid Coupled Networks with Multilinks
In this section, combining with the given lemmas, assumptions, conditions, and so on, we study the fixed-time stability of the origin for the given error dynamical system (15), which denotes the fixed-time synchronization for the multilinked complex dynamical networks ( 11) and ( 13).So, we need to define what the fixed-time stability is, and its definition is described as follows.
Note 2. Through analyzing finite-time stability and fixedtime stability, we can easily find that finite-time synchronization depends on the initial conditions of the complex networks; on the contrary, fixed-time synchronization is irrelevant to the initial conditions of the complex networks.Therefore, it is meaningful to study fixed-time synchronization, especially when the initial conditions are hard to be achieved.
then, according to Lemma 7, the drive system (11) and response system (13) can realize synchronization in fixed-time  0 .And the fixed settling time  0 can be determined as The following is the process of the proof.
Applying Lemma 4 and the error system to the process of the proof, we calculate the derivative of () step by step as follows: The front part of the proof here is the same as that of Theorem 9. Hence, we omit the front part of proof and step to the calculating result (34) directly.The calculating result (34) contains many details.Firstly, in order to calculate conveniently later, we deal with the function ∑ N =1 ((sign  (  ())/√)  ()) in (34).Then substitute the partial processing result into the previous result (34) Substituting the calculating result ( 35) into (34), continue to calculate as follows: Taking expressions (32) and Lemma 3 into consideration, we can get the following calculating result step by step: Through comprehensive consideration, the given error dynamical system ( 15) is fixed-time stable.According to Lemma 7, the fixed settling time can be calculated as follows: The proof is completed here.
Remark 17.So far, lots of achievements for finite-time synchronization of complex networks have been obtained, while, to the best of our knowledge, there is little attention drawn to the fixed-time synchronization of the multilinked hybrid coupled complex networks with time-varying delays or few published papers in this field.That is why this study can draw our attention so much.
Proof.The process of the proof for system (25) to obtain fixed-time synchronization is the same as that for Theorem 16.Hence, we omit it.

Numerical Simulations
We provide two simulation examples to illustrate the correctness and effectiveness of the results obtained by our paper in this section.Example 1.This example considers the drive system (41) which is the 3-linked hybrid coupled complex networks and consists of 3 nodes.Each of the nodes is 2-dimensional.
ẋ  () =  (  ()) +  (  ( −  ())) where The time delays are chosen as follows: The configuration matrices are chosen as follows: Mathematical Problems in Engineering The inner coupling matrices are chosen as follows: Correspondingly, the response system is given as follows: ẏ  () =  (  ()) +  (  ( −  ())) The initial conditions for simulation are given as follows: We define the convergence errors as We divide simulation Example 1 into three cases to illustrate our obtained results in more detail and in more normalized manner.Let  = 3.9 and  = 1.
Case 2. We simulate the result of Theorem 9 when inequalities in (18) are satisfied.We choose the values of the parameters as follows:  1 = 12,  1 = 1,   The convergence errors between system (41) and system (46) under controller (17).
Theorem 9, we can calculate that the value of the settling time  1 is 4.4496.The simulation of finite-time synchronization for the drive-response systems ( 41) and (46) in settling time  1 with control inputs ( 17) is showed in Figure 3.

Conclusions
In this paper, we study the finite-time synchronization of new multilinked hybrid coupled complex dynamical networks  with one single time-varying delay coupling for the first time.Furthermore, we study the fixed-time synchronization of new multilinked hybrid coupled complex dynamical networks with one single time-varying delay coupling for the first time.Two suitable feedback controllers are designed in order to obtain the finite/fixed-time synchronization.Several synchronization criteria are proposed through strict calculating and theorem derivation.Finally, we use two appropriate numerical examples to illustrate the correctness and effectiveness of the results proposed in this paper.The problem of fixed-time synchronization and finite-time synchronization for the hybrid coupled dynamical networks whose coupling terms contain only one transmittal time-varying delay comes naturally for multilinked systems which simulate the real world in closer and more realistic way.It is the first time to study this exact problem and the significance of solving the problem that make our work interesting and meaningful.