Adaptive Synchronization of Complex Dynamical Networks Governed by Local Lipschitz Nonlinearlity on Switching Topology

1 College of Science, North China University of Technology, Beijing 100144, China 2Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China 3 College of Computer and Information Engineering, Beijing Technology and Business University, Beijing 100048, China 4Department of Industrial Engineering and Management, College of Engineering, Peking University, Beijing 100871, China 5 Department of Control Science and Engineering, Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Huazhong University of Science and Technology, Wuhan 430074, China

For a network, the synchronization implies that all nodes will converge to the same state, which can be a homogeneous equilibrium point or a periodic orbit.To enhance the synchronization of such network, a lot of research methods are developed, and one of the most significant methods is to design effective adaptive strategies for the relevant parameters, such as the coupling strengths and the feedback gains [1][2][3][4][8][9][10].In references [4,10,11], there must have information channel by using a special indicator function; however, in reality, the information channel between any two nodes of a network may be lost or changed.Driven by it, we will investigate the synchronization of the complex dynamical networks with switching topology, which can lead to some information channels that occurred as well as another information channels that disappeared, by introducing adaptive strategies to the coupling strengths and feedback gains.Different from [8], the coupling strengths   ()   () are dynamic and variable, which can change according to the switching signal, the neighbor rule, and the switched coupling configuration  ()  .Yet, "fast switching" [5] is hardly to be realized and bears new features and difficulties.Here, we attack the problem by invoking the theory of differential inclusion and nonsmooth analysis [12][13][14][15][16][17][18].
So far, most attentions have been focused on nonlinear dynamics satisfying the globally Lipschitz condition.However, many known systems only satisfy the local Lipschitz condition, such as Lorenz system, Chen system, and FitzHugh-Rinzel system [9][10][11].In this paper, we will also examine nonlinear dynamics of such complex network satisfying the local Lipschitz condition.
The main contribution of the current work lies in characterizing synchronization of complex dynamical networks.First, the nonlinear dynamics of all nodes and the Journal of Applied Mathematics synchronous goal satisfies the local Lipschitz condition.Second, the adaptive strategies are introduced to the coupling strengths and the feedback gains.Third, the topology of complex dynamical networks can switch even though the information is discontinue, we can also solves the synchronization based on differential inclusion and nonsmooth analysis.In this paper, we will prove that all nodes of the complex dynamical network which is steered by the adaptive strategies can converge to the synchronous goal, even though only one node is informed by the synchronous goal if the neighboring graph remains connected.
This paper is organized as follows.Section 2 describes the model with the nonlinear dynamics satisfying the local Lipschitz condition, and some preliminaries about the nonsmooth analysis and the local Lipschitz condition are given.The main results are shown in Section 3, while Section 4 presents some simulations to illustrate our theoretical results.The conclusion is given in Section 5.

Preliminaries and Model Formulation
where ℎ  = 1, if the node  is controlled; otherwise, ℎ  = 0;   () is the feedback gain.() is a desired synchronous state for network (1) with The adaptive strategies on the coupling strengths and the feedback gains are designed as where   (0) ≥ 0,   (0) ≥ 0,   > 0, and   > 0 are the adaptive parameters of the coupling strength and the feedback gains, respectively.
In the th (for all  ∈ ) time period, the weighted coupling configuration matrix of network ( 1) is defined as with

Mathematical Preliminaries.
In this section, we introduce some useful concepts, assumptions, and lemmas.
Definition 2 (see [16], generalized directional derivative).The generalized directional derivative of  at  in the direction V, denoted by  ∘ (; V), is defined as where  is a vector in the Banach space  and  is a positive scalar.
It is worth noting that if the condition (0) = 0 in (i) of Lemma 8 cannot hold; then we can have the following lemma.
Theorem 10.Supposing that Assumptions 1-3 hold, then the combined trajectories of all nodes and the parameters (, , c) in (1) and (2) are geared to a compact hyper-ellipsoid if the initial value is selected from Proof.Taking the derivative of () in [ 0 ,  1 ), we can get Under Assumption 3, Lemma 5 implies that  −  < 0 in [ 0 ,  1 ).Since  is a sufficiently large positive constant, then which implies Similarly, Thus, we have Therefore, we can conclude that the combined trajectories of all nodes with the parameters (, , c) are geared to a compact hyper-ellipsoid This completes the proof.
Theorem 11.If network (1) steered by adaptive laws (3) and the initial value of (, , c) is defined as (18), under Assumptions 1-3, then all nodes will converge to the synchronous state even when only one node is controlled by (2).
where  > 0 is a sufficiently large constant.
By Definition 1, we can obtain a.e.

∈ 𝐾 [𝑘
Because the topology of network ( 1) is switching, we have Also by Definition 3, we know that () is regular, then since  is a sufficiently large positive constant.
From Lemma 9, we can get that all the nodes of network (1) can converge to the synchronous state ().This completes the proof.
Remark 12.Note also that Theorem 11 is still established when   is asymmetric.
Similar to the proof of Theorem 11, we also have since ( +   )/2 is symmetric.

Simulations
In this section, number simulations are given to illustrate our theoretical results.All nodes of network (1) and the synchronous goal share the same nonlinear dynamics described as the Lorenz system as follows: as shown in Figure 1.Then the network can be shown as follows: [ Figure 2 shows the topology of network (1) with subgraphs (a), (b), and (c) during the time intervals [0, 4], [4,12], [12,50], respectively.The initial values of the 10 nodes are chosen randomly, and the initial value of the synchronous goal is  0 (0) = [10, 10, 10]  .There is only one node informed by the synchronous goal, and the 10th node is given, that is, ℎ  ( = 1, 2, . . ., 9) = 0 and ℎ 10 = 1.The initial values of the adaptive parameters are   (0) = 0 with the weight of   (0) = 0 for all  and , while   (0) = 0 with the weight of   (0) = 0 for all .
Figure 3 describes the convergence of the state errors on the x-axis, y-axis, and z-axis, respectively.From this figure, we can see that all nodes of network (1) can synchronize to the synchronous state when the neighboring graph G remains connected with switching topology even though only one node is informed by the synchronous state.Figure 4 shows the change trends of the adaptive coupling strengths and the adaptive feedback gains, respectively, and all these parameters converge to the constants.

Conclusion
In this paper, we have investigated the synchronization of complex dynamical networks with switching topology via differential inclusion method.Different from the most previous work, all nodes and the synchronous state in this paper share the same intrinsic nonlinear dynamics governed by the local Lispchitz condition.By adding decentralized adaptive strategies to the coupling strengths and the feedback gains, all nodes can converge to the synchronous state even when only one node is pinning controlled by the synchronous state if the neighboring graph G of the switching topology remains connected.

Figure 1 :
Figure 1: Trajectories of the synchronous state.

Figure 4 :
Figure 4: Convergence of the adaptive coupling strengths and the feedback gain.
Assumption 2. The synchronous state  0 is bounded; that is, there exists a compact set S = S( 0 (0)) ∈   such that the trajectory of (3) starting from  0 (0) is always in the compact set S. Note that the nonlinear dynamics of network (1) only satisfies the local Lipschitz condition.If the Jacobian matrix of  is continuous, then  is at least local Lipschitz.Many famous systems may not be governed by global Lipschitz nonlinearity but by local Lipschitz nonlinearity, such as Lorenz system and Chen system; therefore, it is worthy of discussing the local Lipschitz nonlinearity dynamics.