Modified Function Projective Synchronization of Complex Networks with Multiple Proportional Delays

This paper deals with the modified function projective synchronization problem for general complex networks with multiple proportional delays. With the existence of multiple proportional delays, an effective hybrid feedback control is designed to attain modified function projective synchronization of networks. Numerical example is provided to show the effectiveness of our result.

Modified function projective synchronization(MFPS) has been proposed and extensively investigated in the latest.MFPS means that the drive and response systems could be synchronized up to a desired scaling function matrix [15].It is easy to see that the definition of MFPS encompasses projective synchronization and function projective synchronization.The MFPS of general complex networks can reveal that the nodes of complex networks could be synchronize up to an equilibrium point or periodic orbit with a desired scaling function matrix.Because the unpredictability of the scaling function in MFPS can additionally enhance the security of communication, MFPS has attracted the interest of many researchers in various fields.On the basis of an adaptive fuzzy nonsingular terminal sliding mode control scheme, a general method of MFPS of two different chaotic systems with unknown functions was investigated in [16].The work in [17] gives MFPS of a class of chaotic systems.MFPS of a classic chaotic systems with unknown disturbances was investigated by adaptive integral sliding mode control [18].Ref. [19] investigates the adaptive MFPS of a class of complex four-dimensional chaotic system with one cubic cross-product term in each equation.Ref. [20] investigates the MFPS of two different chaotic systems with parameter perturbations.
A simple general scheme of MFPS in complex dynamical networks (CDNs) is investigated in this paper, considering that external disturbances and unmodeled dynamics are always unavoidably in the practical evolutionary processes of synchronization, MFPS in CDNs with proportional delay and disturbances will be investigated by the proposed scheme.The rest of this paper is organized as follows.Some definitions and a basic lemma are given in Section 2. In Section 3, the synchronization of the complex networks with proportional delays by the pinning control method is discussed by the way of equivalent system.Finally, computer simulation is performed to illustrate the validity of the proposed method in Section 4.

Preliminaries
Consider a generally controlled complex dynamical networks consisting of N identical linearly coupled nodes with multiple proportional delays by the following equations: , , , R is a continuously differentiable vector function determining the dynamic behavior of the nodes, is the coupling configuration matrix representing the topological structure of the network, where 0 ij g > if there is a connection between node i and node j; otherwise , and the diagonal elements of matrix G are defined by , , 1, 2, , ij q i j n =  are proportional delay factors and satisfy Furthermore, the complex network described in (2.1) δ is a finite constant, and ( ) n t ∈ x R can be an equilibrium point, or a periodic orbit, or an orbit of a chaotic attractor, which satisfies ( ) ( ) ( ) Considering the actual evolutionary processes of synchronization, external disturbances and unmodeled dynamics are always unavoidable.MFPS in CDNs with disturbances will be investigated further as follows: where ( ) , , , is the coupling configuration matrix representing the topological structure of the network, and the diagonal elements of matrix G are defined by Equation (2.2).
In the following, some necessary assumptions are given.
Assumption 1.The derivative of scaling function for all t R + ∈ , where * a R + ∈ is the upper limit of the ( ) , 1, 2, , Assumption 2. The norm of the mismatched terms ( ) ( ) where is the upper limit of the norm of ( ) In this paper, , M M denote the upper limit of the norm of ( ) ( ) ( )

MFPS in Complex Networks with Multiple Proportional Delays
In this section, a hybrid feedback control method for realizing modified function projective synchronization in complex dynamical networks with multiple proportional delays is proposed.
Let ( ) ( ) , then a couple of networks (2.1) and (2.4) is equivalently transformed into the following couple of complex networks with constant delay and time varying coefficients and Definition 2. The network (3.2) is said to achieve modified function projective synchronization if there exists a continuously differentiable scaling function matrix where ⋅ stands for the Euclidean vector norm, is a modified function matrix, and each modified function α is a continuously differential function and is bounded as finite constant, and ( ) n t ∈ y R can be an equilibrium point, or a periodic orbit, or an orbit of a chaotic attractor, which satisfies ( ) ( ) ( ) Theorem 1. Suppose Assumptions 1 and 2 hold.For a given synchronization scaling function matrix with disturbance (3.2) can realize modified function projective synchronization via the control law : e , 1, 2, , , where ( ) sgn ⋅ denotes the sign function. Proof.Define , , The time derivative of V(t) along the trajectories of system (3.6) is (3.9) , we obtain where ( ) According to the Lyapunov stability theory, the error system (3.6) is asymptotically stable.This completes the proof.
Corollary 1. Suppose Assumptions 1 hold.For a given synchronization scaling function matrix
By Theorem 1, it is easy to see that a similar proof holds for ( ) ( ) Thus, the proof is omitted here.
Though the proposed error feedback control method is very simple, Choosing the appropriate feedback gains , 1, 2, , , e , 1, 2, , , , 1, 2, , , e e e e (3.18)The time derivative of V(t) along the trajectories of (3.6) is      Because chaos systems and the scaling function are bounded, ( )   where ( )  By Theorem 2, it is easy to see that a similar proof holds for ( ) ( ) . Thus, the proof is omitted here.

Computer Simulation
In this section, the chaotic Lorenz system is taken as nodes of CDNs to verify the effectiveness of the proposed scheme in Corollary 2.
Consider the following single Lorenz system:

G
Complex networks with proportional delays can be described as follows:

Concluding Remarks
In this paper, function projective synchronization schemes for complex networks with proportional delays are given by a error feedback control method.
Numerical simulation is provided to show the effectiveness of our result.

l
3.17) where ( ) sgn ⋅ denotes the sign function.> are arbitrary positive constants.Proof.Construct the Lyapunov function

Figure 2 .
Figure 2. Components of the Lorenz system.

1
In this numerical simulation, we take the initial states as

Figure 3 Figure 4 ,
Figure 4, which displays ( ) 0 t → e with t → ∞ .These results show that function projective synchronization takes place with the desired scaling function in complex networks (4.2).

Figure 3 .
Figure 3. State phases of the Lorenz system.

Figure 4 .
Figure 4.The time evolution of the synchronization errors e.
denotes the state vector of the ith node, represent the Kroncecker product, 1, 2, , ,