Adaptive Finite-Time Mixed Interlayer Synchronization of Two-Layer Complex Networks with Time-Varying Coupling Delay

This paper is concerned with two-layer complex networks with unidirectional interlayer couplings, where the drive and response layer have time-varying coupling delay and different topological structures. An adaptive control scheme is proposed to investigate finite-time mixed interlayer synchronization (FMIS) of two-layer networks. Based on the Lyapunov stability theory, a criterion for realizing FMIS is derived. In addition, several sufficient conditions for realizingmixed interlayer synchronization are given. Finally, some numerical simulations are presented to verify the correctness and effectiveness of theoretical results.Meanwhile, the proposed adaptive control strategy is demonstrated to be nonfragile with the noise perturbation.


Introduction
As an important and typical dynamic behavior of complex networks, synchronization has been extensively investigated in many fields, such as physics, mathematics, information science, biology, and sociology.In the literatures, many works have primarily focused on synchronization within a network having no connection with other networks.However, in realworld situations, many systems are often composed of some interacting networks.For example, a transportation system consists of road network, railway network, and air network.A communication system is composed of some subnetworks depending on phone, email, QQ, Wechat, etc.Thus, multiplex networks, proposed by Mucha et al. [1], would be more appropriate for describing systems in the real-world than traditional (single-layer) complex networks.In the past few years, many efforts have been made to investigate various problems of multiplex networks, such as topological structure, dynamic behavior [2], synchronizability [3,4], spectral property [5], diffusion process [6], and synchronization [7][8][9][10].
Interlayer synchronization is also called counterpart synchronization [11] which describes how the nodes in one network behave coherently with the corresponding ones in other connected networks.This concept can be regarded as the development of synchronization for coupled driveresponse systems [12].In a drive-response chaotic system, the response network is driven by signals from the drive network, but the latter is not influenced by the former.These two coupled single-layer networks, whose topological structures may be different, can be viewed as a two-layer network.For example, a two-layer network with unidirectional interlayer couplings is shown in Figure 1, where the layer  and the layer  represent the drive layer and the response layer, respectively.Synchronization between the layer  and the layer  exists extensively in the real world.This kind of interlayer synchronization can be also understood as outer synchronization [13] between two coupled networks.In the past decade, it has gained considerable attention-see [11,[13][14][15][16][17][18][19][20] and the references therein.Some control schemes have been proposed to realize all kinds of outer synchronization such as complete outer synchronization [13][14][15][16][17], inverse outer synchronization [18], generalized outer synchronization [19], and finite-time outer synchronization [21].As a special case of generalized outer synchronization, mixed outer synchronization (MOS) was first proposed in [22].Wang et al. [22] studied MOS between two complex networks with the same topological structure and time-varying coupling delay by designing robust linear feedback controllers.Later, Zheng and Shao [23] reported MOS between two complex networks with the same topological structure and output couplings via impulsive hybrid control.Sheng et al. [24] considered MOS between two complex networks with time-varying delay coupling and nondelay coupling by using pinning feedback control and impulsive control.
Finite-time mixed outer synchronization (FMOS) [25] between two complex networks is a recently developed MOS.Essentially, FMOS can be regarded as finite-time mixed interlayer synchronization (FMIS).In the state of FMIS, different state variables of the corresponding nodes can attain finite-time synchronization, finite-time antisynchronization, and even finite-time amplitude death simultaneously.FMIS is a kind of finite-time synchronization, in which the synchronization error remains within a prescribed range in a fixed time interval for a given range of initial error.Notice that finite-time synchronization is defined in a fixed finitetime interval.Hence, it has attracted increasing attention [26][27][28][29][30][31].In [25], He et al. discussed FMOS between two complex networks with time-varying coupling delay and the same topological structures by designing a simple and robust linear state feedback controller.However, for most twolayer networks in real-world situations, the nodes are not always identical and topological structures of two layers are not always the same.In this paper, we consider FMIS of two-layer networks with time-varying coupling delay and different topological structures by designing an adaptive control scheme.Based on the Lyapunov stability theory, a sufficient condition for realizing FMIS of two-layer networks is derived.In addition, a criterion for realizing MIS is obtained.Finally, some numerical simulations are provided to verify the effectiveness of our theoretical results.Moreover, FMIS is analyzed when the two-layer networks are disturbed by the additive noise.
The rest of this paper is organized as follows.In Section 2, we formulate the two-layer network model and introduce some preliminaries.In Section 3, some sufficient conditions for realizing FMIS or MIS of two-layer networks are given under the proposed adaptive controllers.In Section 4, some numerical simulations are presented.Conclusions are finally drawn in Section 5.

Model and Preliminaries
Consider a dynamical two-layer complex network (See Figure 1) which is composed of the drive layer  and the response layer .Assume that each layer consists of  nodes.The drive layer  and the response layer  are described as follows: where =   ,  = 1, 2, ⋅ ⋅ ⋅, . = (  ) × ∈ R × denotes the weight configuration matrix of the response layer  and has the same meanings as that of .() ≥ 0 is the time-varying coupling delay.  is a controller for node  to be designed later.Now, we introduce some assumptions, definitions, and lemmas, which are necessary to the proofs of main results.
Assumption 3. The time-varying coupling delay () is a differential function with Notice that this assumption is obviously satisfied when the delay () is a constant.

Main Results
In this section, we will give some sufficient conditions for realizing FMIS or MIS of two-layer networks based on adaptive control.The adaptive controllers are designed as follows: where Here,   ( = 1, 2, . . ., ) is a positive constant.
Integrating from 0 to  ( ∈ [0, ]), it follows that By ( 14), we get where Combining this with (13), we have The proof of Theorem 8 is finished.
Remark 12.For the two-layer network (1), the nodes in each layer may be nonidentical and topological structures of two layers may be different.For each node, the inner connecting function is nonlinear.In addition, two configuration matrices  and  are not assumed to be symmetric or irreducible.Consequently, Theorems 8 and 10 can be widely applied in practice.
Corollary 14. Suppose that Assumptions 1-3 hold.If the configuration matrix of the drive layer is the same as that of the response layer, i.e.,  = , then MIS of two-layer network (1) can be realized with the following simplified adaptive controllers: where ḋ =      ()  () and   is any positive constant.
The coupled two-layer network is described as follows: Let  1 and  2 be two matrices defined as follows: Advances in Mathematical Physics ) . (39) 4.1.FMIS of Two-Layer Networks with respect to ( 1 ,  2 , , ).
In the subsection, we consider FMIS of the two-layer network (36) under two cases:  ̸ =  and  = .
Case 1. () is added to the right hand of two equations in (36).This means that every node in (36) is affected by the same noise.
Case 2. () and () are added to the right hand of two equations in (36), respectively.This indicates that the nodes in the drive layer and the ones in the response layer are affected by different noise.
For these two cases, the steps of above numerical simulation are repeated with the adaptive controllers (37).The synchronization errors for two cases are shown in Figures 3  and 4. Hence, the proposed scheme is demonstrated to be nonfragile with the noise perturbation.
Suppose that  =  =  1 .Figure 9 presents the synchronization error with the simplified adaptive controllers in Corollary 14.Hence, the MIS in two-layer network (36) can be realized.

Conclusions
In this paper, we have investigated finite-time mixed interlayer synchronization (FMIS) of two-layer complex network with unidirectional interlayer couplings.For the two-layer network, the nodes in each layer are nonidentical and two single-layer networks have time-varying coupling delay and different topological structures.We have proposed an adaptive control scheme to realize FMIS of two-layer networks.Based on the Lyapunov stability theory, a sufficient condition for realizing FMIS has been derived.In addition, some criteria for realizing MIS have been obtained.Finally, some numerical simulations have been presented to verify the correctness and effectiveness of theoretical results.Moreover, FMIS has been analyzed when the two-layer networks are disturbed by the additive noise.Two numerical simulations show that the proposed adaptive control strategy is nonfragile with the noise perturbation.

Remark 5 .Definition 6 .
In view of Definition 4, when  =  and  = −, FMIS is reduced to the known notions: finite-time outer synchronization and finite-time outer antisynchronization, respectively.The two-layer network (1) is said to achieve mixed interlayer synchronization (MIS) with respect to the scaling matrix  if there exists a controller   () ( = 1, 2, ⋅ ⋅ ⋅, ) for node  such that lim →∞       ()     = 0.
the state vectors,   and   are the diagonal and nondiagonal matrices representing the linear part of the -th node dynamics,   : R  → R  is a smooth nonlinear