Adaptive Modified Function Projective Synchronization of Uncertain Complex Dynamical Networks with Multiple Time-Delay Couplings and Disturbances

This paper studies the modified function projective synchronization of uncertain complex dynamic network model with multiple time-delay couplings and external disturbances. Based on Lyapunov stability theory, the positive definite function is designed and the sufficient conditions of synchronization are given. Both the uncertain parameters and the unknown bounded disturbances are estimated in accordance with the adaptive laws. With the adaptive feedback controller, the complex dynamic network can synchronize with reference node by a scaling function matrix. The reference node can be periodic orbit, equilibrium point, or a chaotic attractor. Finally, two numerical simulations are offered to illustrate the effectiveness of the proposed method.


Introduction
Complex networks widely exist in various fields of science and engineering, ranging from biology, physics, and chemistry to social networks and technological applications.In recent years, complex dynamics networks have been an active research topic and have been developed and systematically studied [1][2][3][4] to extend nonlinear system [5].Synchronization is a fundamental phenomenon in nature which enables the networks to achieve coherent behavior due to interaction.The research of complex network synchronization is one of the most important research directions on complex networks [6][7][8].Up to now, many types of synchronization phenomena on complex networks have been reported, such as complete synchronization [9,10], projective synchronization [11,12], lag synchronization [13], lag projective synchronization [14], bounded synchronization [3], projective cluster synchronization [15], and global synchronization [16].
Modified function projective synchronization (MFPS) is a more general definition of synchronization method which indicates that the drive and response system could be synchronized up to a scaling function matrix.The definition of MFPS contains complete synchronization, projective synchronization, and function projective synchronization.The unpredictability of the scaling function factors in MFPS can additionally enhance the security of communication.MFPS has drawn considerable attention with a lot of research results.Reference [17] studied the MFPS of uncertain chaotic (hyperchaotic) systems based on a novel observer-based finite-time control method.Reference [18] investigated the MFPS of Liu chaotic system and its application to secure communication.References [19][20][21] extended the MFPS to complex domain and studied the MFPS of complex chaotic system with parameter perturbations and external perturbations.
However, the existing researches related to MFPS are mainly concerned with two chaotic systems and there are few theoretical achievements related to MFPS in a general complex network.Recently, some research results on function projective synchronization of complex dynamics networks have been achieved.Based on the adaptive open-plus-closedloop method, [22] investigated function projective synchronization of complex dynamical networks with or without external disturbances using error feedback control scheme.Based on Barbalat's lemma, [23] designed some sufficient synchronization criteria by applying the nonlinear feedback control to realize the generalized function projective synchronization between two different complex networks.Although [22,23] studied the function projective synchronization of two complex networks, the network models are conservative for they did not take the influence of time-delay coupling into consideration.It is well known that, due to the finite speed of information transmission and processing speed among the units, the connection delays in realistic modeling of many large networks must be taken into account.What is more, there always exist some unknown factors in most real systems, covering the uncertain parameters and external disturbances which can result in network instability or poor performance.Therefore, it is significant to study the effects of time-delay coupling, uncertain parameters, and external disturbances in synchronization of complex dynamics networks.Reference [24] proposed an adaptive controller to investigate the problem of function projective synchronization in complex dynamical networks with constant time-delay coupling, uncertain parameters, and disturbance.Reference [25] investigated the modified function projective lag synchronization of dynamical complex networks with disturbance, unknown parameters, and coupling delay based on error feedback control scheme.However, the absolute constant coupling delay may be scarce in the practical networks.In [26], Du et al. achieved the function projective synchronization for general complex dynamical networks with constant or time-varying time-delay coupling by a hybrid feedback control method, but the model uncertain and external disturbances were not taken into account.
Multiple time-delay coupling complex networks indicate that there are more than one coupling delay between two nodes of complex networks.The multiple time-delay coupling complex networks widely exist in the real world such as relationship network, communication network, and transportation network.The multiple time-delay coupling complex networks can be divided into some subnetworks by different time delays [27].The relationship network is given as an example in Figure 1.In accordance with different coupling delays, the relationship network is divided into three subnetworks.The corresponding topological structures are displayed in Figure 1.The single time-delay coupling network as a special case of multiple time-delay couplings complex networks has been studied widely [28,29].However, the synchronization research of complex networks with multiple time-delay couplings is more realistic and representative, which still receives little attention.
Compared with previous work, there are three advantages which can make our research attractive and interesting.Firstly, our paper considers the multiple time-delay coupling complex networks which widely exist in the real world.The single time-delay coupling network is just a special case of multiple time-delay couplings complex networks.Secondly, MFPS is a more general synchronization method which contains complete synchronization, projective synchronization, and function projective synchronization.The unpredictability of the scaling function factors in MFPS can additionally improve the reliability of secure communication.Thirdly, our paper considers the MFPS between a complex network and a reference node.The reference node can be periodic orbit, equilibrium point, or a chaotic attractor.When the reference node is periodic orbit or equilibrium point, the idea mentioned in this paper can control complex network to a stable state.When the reference node is a chaotic attractor, the idea mentioned in this paper can synchronize complex network with a chaotic state, which can be applied in engineering fields such as secure communication and information processing.
The rest of this paper is organized as follows: the network model and some preliminaries are given in Section 2; the MFPS for uncertain complex dynamic networks with multiple time-delay couplings and disturbances is discussed in Section 3; numerical simulations are offered in Section 4; finally, the conclusive remarks are given in Section 5.

MFPS Synchronization Scheme
Theorem 5.For a given synchronization scaling function matrix () and any initial conditions   (0) and (0), if Assumptions 2 and 3 are satisfied, the uncertain complex dynamic network model (1) and the isolated node () will realize MFPS with the following adaptive control laws: where   > 0,  = 1, 2, 3, are three positive constants; θ is the estimated parameter for   ; d is the estimated parameter for   ; q is adaptive feedback control gains; sign(⋅) is the sign function.
Remark 9.In a lot of literature about complex network synchronization, the controller always includes the time delay ().However, it is difficult to measure the delay and implement the delay term, especially the time-varying delay term in the real control system.In our work, the controller does not include (), so the proposed method is more general and realistic.

Illustrative Examples
In this section, we will study the MFPS of the uncertain complex dynamical networks with multiple time-delay couplings and disturbances.We take the Lü chaotic system as reference node to verify the effectiveness of the proposed method.The L ü system is described as follows: where  1 ,  2 ,  3 are the state variables and , ,  are real constants.When  = 36,  = 20,  = 3, the system is chaotic attractor, which is shown in Figure 2.
Example 2. Consider a coupled complex dynamical network consisting of four nodes with two time-delay couplings, that is,  = 4,  = 3.The network model can be written as where  = 1, 2, 3, 4 and  1 () = 0.6 s is a constant time delay. 2 () =   /(2 +   ) is a time-varying delay and then τ 2 () = 2  /(2 +   ) 2 ∈ (0, 1/2].The topological structure matrices  0 ,  1 ,  2 are as follows:      and the numerical results are given to show that the theoretical results are also effective to the complex dynamical networks with multiple time-varying delay couplings.

Conclusion
This paper studied the MFPS of uncertain complex dynamic network with multiple time-delay couplings and external disturbances.The complex networks can be divided into some subnetworks by different coupling delays.Through the adaptive feedback controller, the complex network can synchronize with reference node according to the expected scaling function matrix.The uncertain parameters, feedback gains, and bounds of the external disturbances are all estimated by the adaptive laws.The controller does not include the delay term, so the proposed method is more general and realistic.The corresponding theoretical proofs and numerical simulations are given to demonstrate the validity and feasibility of the proposed control technique.The idea may be applied in engineering fields such as secure communication and information processing.How to realize MFPS of uncertain complex dynamical networks with multiple time-delay couplings in actual practice is our next research topic.

Figure 1 :
Figure 1: Relationship network and its division.According to different coupling delays, the relationship network is split into three subnetworks.

Figure 2 :
Figure 2: 2D and 3D projections of chaotic attractor of the Lü system.

( 25 )
Similar to Example 1, the simulation results are given in Figures7-10 ,