Nonlinear Measure about l 2 -norm with Application in Synchronization Analysis of Complex Networks via the General Intermittent Control

—Based on the nonlinear measure about l 2 -norm, some simple yet generic criteria are derived by using the general intermittent control, which ensure the exponential stability of dynamic systems. The numerical simulations whose theoretical results are applied to robust synchronization of complex networks demonstrate the effectiveness and feasibility of the proposed technique.

INTRODUCTION Since its origins in the work of Fujisaka and Yamada [1][2][3] ,Afraimovich,Verichev, and Rabinovich [4] , and Pecora and Carrol [5] , the study of synchronization of chaotic systems [6][7][8][9][10][11][12][13] is of great practical significance and has received great interest in recent years.In the above literature, the approach applied to stability analysis is basically the Lyapunov's method. As we all known, the construction of a proper Lyapunov function usually becomes very skillful, and the Lyapunov's method does not specifically describe the convergence rate near the equilibrium point of the system. Hence, there is little compatibility among all of the stability criteria obtained so far.
The concept named the nonlinear measur about l 2norm [14][15][16][17][18][19] has been applied to the investigation of the existence, uniqueness or stability analysis of the equilibrium. Intermittent control [20][21][22][23][24] has been used for a variety of purposes in engineering fields such as manufacturing, transportation, air-quality control and communication. A wide variety of synchronization or stabilization using the periodically intermittent control method has been studied (See [21][22][23][24][25][26][27][28]). Compared with continuous control methods [7][8][9][10][11][12][13][14][15] , intermittent control is more efficient when the system output is measured intermittently rather than continuously. All of intermittent control and impulsive control are belong to switch control. But the intermittent control is different from the impulsive control, because impulsive control is activated only at some isolated moments, namely it is of zero duation, while intermittent control has a nonzero control width. Therefore, it is essential and important to investigate the exponential synchronization of networks with mixed delays by periodically intermittent control.
A special case of such a control law is of the form where k denotes the control strength, 0 δ > denotes the switching width, and T denotes the control period.
The general intermittent controller where ( ) h n is a strictly monotone increasing function on n has been studied (See [29]). In this paper, based on nonlinear measur about l 2 -norm and Gronwall inequality, the general intermittent controller ( ( ) ( )), ( ( ) ( ) ), ( ) 0, ( ( ) ( 1) ), is designed, where ( ) h n is a strictly monotone function on n , then the sufficient yet generic criteria for synchronization of complex networks with and without delayed item are obtained.

II. PRELIMINARIES
Let X be a Banach space endowed with the l 2 - , where , is inner product, and Ω be a open subset of X . We consider the following system: where , F G are nonlinear operators defined on Ω , and ( ), x t ( ) x t τ − ∈Ω , and τ is a time-delayed positive constant, and (0) Definition 1 [ Definition 2 [15,16,18,19] Suppose that Ω is an open subset of n R , and : is called the nonlinear measure of G on Ω with the l 2norm .
Lemma 1 [29] Suppose that Ω is an open subset of n R , and : here, the operator F rI + mapping every point x ∈ Ω denotes ( ) . F x rx + Lemma 2 [29] If the operator G in the system (3) for any , x y ∈ Ω , wher l is a positive constant. The , and is zero, otherwise, and the constraint 1, 1, , A complex network is said to achieve asymptotical synchronization if is a solution of a real target node, satisfying For our synchronization scheme, let us define error vector and control input ( ) i u t as follows, respectively: In this work, the goal is to design suitable function ( ) h n and parameters δ , T and k satisfying the condition (7). The error system follows from the expression When ( ) h n is a strictly monotone increasing function on n with (0) 0, lim ( ) ,   (6) is achieved if the parameters δ , T and , k η satisfy for any ( )

e t hnT n h n T t h n T e t e t n h n T t h n T
( ) (0) exp{ (( ) ) }, is obtained under the condition (9). So the synchronization of the network (6)    Example 1 Consider a delayed Hopfield neural network [31][32] with two neurons: where 1 2 1 ( ) ( ( ), ( )) , ( ( )) (tanh( ( )), , It should be noted that the network is actually a chaotic delayed Hopfield neural network.
We reach the value  which are strictly monotone increasing or decreasing function on n , respectively, then they can be clearly seen that the synchronization of network (6) is realized in Fig.5、Fig.6 ( A≠0), where Fig.7、Fig.8 (A=0), respectively. IV. CONCLUSION Approaches for synchronization of complex networks via general intermittent which use the nonlinear operator named the measure about l 2 -norm have been presented in this paper. Strong properties of global and exponential synchronization have been achieved in a finite number of steps. The techniques have been successfully applied to Chaotic delayed Hopfield neural networks and hyperchaotic Chen system. Numerical simulations have verified the effectiveness of the method.