#### Abstract

This paper is concerned with the quasi-matrix and quasi-inverse-matrix projective synchronization between two nonidentical delayed fractional order neural networks subjected to external disturbances. First, the definitions of quasi-matrix and quasi-inverse-matrix projective synchronization are given, respectively. Then, in order to realize two types of synchronization for delayed and disturbed fractional order neural networks, two sufficient conditions are established and proved by constructing appropriate Lyapunov function in combination with some fractional order differential inequalities. And their estimated synchronization error bound is obtained, which can be reduced to the required standard as small as what we need by selecting appropriate control parameters. Because of the generality of the proposed synchronization, choosing different projective matrix and controllers, the two synchronization types can be reduced to some common synchronization types for delayed fractional order neural networks, like quasi-complete synchronization, quasi-antisynchronization, quasi-projective synchronization, quasi-inverse projective synchronization, quasi-modified projective synchronization, quasi-inverse-modified projective synchronization, and so on. Finally, as applications, two numerical examples with simulations are employed to illustrate the efficiency and feasibility of the new synchronization analysis.

#### 1. Introduction

Fractional calculus, which is applied to deal with differentiation and integration of arbitrary noninteger orders, has become an important and powerful tool to research the practical problems in many subjects [1, 2]. Fractional order phenomenon is ubiquitous in the real world and has strong memory and hereditary characteristic, so fractional model can better describe the dynamical properties and internal structure of many classical problems than integer ones. In recent years, many valuable results of fractional order dynamical systems have been obtained and widely applied in many areas, such as mathematical physics [1–11], optimum theory [12], financial problems [13], anomalous diffusion [14], secure communication [15, 16], biological systems [17, 18], and heat transfer process [19]. These research works illustrate the practicality and importance of fractional calculus and promote its development.

Neural network has attracted more and more attention since the introduction of fractional calculus and its dynamical behaviors, such as chaos, hyperchaos, bifurcations [20–22], existence, stability and consensus [23–30], and control and synchronization [31–40], have been widely studied. Recently, its synchronization problem has become a research focus and attracted many researchers. In [32, 33], the authors considered the adaptive pinning synchronization and finite time synchronization for delayed fractional order neural network. In [34], He and his cooperators explored quasi-synchronization problem of heterogeneous dynamic networks via distributed impulsive control. Moreover, some scholars have studied the synchronization and quasi-synchronization problems for delayed fractional order memristor-based neural network with uncertain parameters [35–40]. These studies have promoted the development of fractional order neural networks to some extent, but most of them are aimed at the special complete synchronization problem, so it is still an important problem to propose a more general and practical synchronization type.

Projective synchronization, where the drive and response systems could be synchronized up to a scaling factor, is an important concept in practical applications. Nowadays, many researchers have introduced the projective synchronization into fractional order neural network. In [41], the authors applied LMI-based method to realize the global projective synchronization for fractional order neural network. In [42–44], the scholars derived some new sufficient conditions and designed the appropriate controllers to guarantee projective synchronization for delayed fractional order memristor-based neural network. In [45], by using comparison principle, Zhang and her cooperators designed suitable controllers to reach projective synchronization for delayed fractional order neural network. In [46], Wu and his cooperators introduced new sliding mode control laws to realize projective synchronization for nonidentical fractional order neural network in finite time. And, in [47, 48], researchers explored projective synchronization and quasi-projective synchronization for fractional order neural network in complex domain.

However, in the above research works, the proportion factor of projective synchronization is a fixed constant, while the simple scaling factor like this maybe does not guarantee high security of the image encryption and text encryption in communication. It is an important and meaningful work to extend the scaling factor to an arbitrary constant matrix and propose a more general synchronization type. So a new synchronization type, i.e., matrix projective synchronization, whose scaling factor is a constant matrix, appears and it can realize faster and safer communication. Additionally, another interesting problem is the inverse case of matrix projective synchronization, that is, when each drive system state synchronizes with a linear combination of response system states. Obviously, complexity of the scaling factors in matrix and inverse-matrix projective synchronization can have important effect in applications. Besides, it is well known that time delay is unavoidable due to finite switching speeds of the amplifiers, and it may cause oscillations or instability of dynamic systems. And external disturbances for the fractional order neural network can result in complicated topological structures because of the complexity and uncertainty of fractional nonlinear systems. Therefore, researching two more general synchronization types for delayed fractional order neural network with external disturbances is a meaningful problem.

According to the aforementioned discussions, this work aims to address these problems and present two more general synchronization types, i.e., fractional quasi-matrix and quasi-inverse-matrix projective synchronization, and establish the synchronization criteria for delayed and disturbed fractional order neural network. The remainder of this paper is organized as follows.

In Section 2, some lemmas of fractional calculus are introduced and* n*-dimensional delayed and disturbed fractional order neural network is constructed. In Section 3, fractional quasi-matrix and quasi-inverse-matrix projective synchronization are defined and the sufficient criteria for realizing two synchronization types of the delayed and disturbed fractional order neural networks are derived by means of Lyapunov function and some fractional order properties. In Section 4, as applications, quasi-matrix projective synchronization for two 2-dimensional and quasi-inverse-matrix projective synchronization types for two 3-dimensional delayed and disturbed fractional order neural networks are realized, respectively. And numerical simulations demonstrate the feasibility of synchronization analysis. Conclusions are given in Section 5.

#### 2. Preliminaries and System Description

The Caputo derivative of order for a function is defined as [1]where and are the limits of Caputo derivative operation , is Euler’s Gamma function, that is, , represents the th-order derivative of , and is the positive integer satisfying . When , the operation coincides with the integer order derivative .

Lemma 1 (see [25]). *If denotes a continuously differentiable function, the following inequality holds almost everywhere*

Lemma 2 (see [29]). *Consider the following fractional order differential inequality with time delayand linear fractional order differential system with time delaywhere and are continuous and nonnegative in ; if and , then*

Lemma 3 (see [32]). *Let be a continuously differentiable and nonnegative function and satisfywhere If for all , then *

Consider two nonidentical* n*-dimensional delayed fractional order neural networks, which are subjected to external disturbances, as the drive system and response system, respectively:andwhere and is the number of units in a neural network. and denote the state variables. are self-regulating parameters of neurons. ,, , and express neuron activation functions at time and . and denote synaptic connection weight of unit to unit . , are different bounded external disturbances and . is the controller to be designed later.

*Assumption 4. *Neuron activation functions are continuous and satisfy Lipschitz condition on with Lipschitz constants as

#### 3. Main Results

In this section, by using the active control method, we will focus on designing the suitable controllers to realize the quasi-matrix and quasi-inverse-matrix projective synchronization types between systems (7) and (8).

##### 3.1. Fractional Quasi*-*Matrix Projective Synchronization

Let’s first define the quasi-matrix projective synchronization as follows.

*Definition 5. *Systems (7) and (8) are said to be quasi-matrix projective synchronization with error bound , if there exists such that, for all and initial values , the synchronization error satisfies . Here means an arbitrary constant projective matrix, , , and and are the initial values of systems (7) and (8).

Next, let us research the quasi-matrix projective synchronization between systems (7) and (8). Taking Caputo derivative of both sides of error function and substituting into (7) and (8), the error system can be obtained asConstructing the control function asand substituting it into (10), the error system is changed to

Because of different external disturbances for two systems, is not the equilibrium point of system (12). So the complete synchronization between systems (7) and (8) cannot be realized. However, the quasi-matrix projective synchronization can be investigated.

Theorem 6. *Suppose Assumption 4 holds and the following inequality is satisfied:then the drive system (7) and response system (8) with control law (11) will achieve the q uasi-matrix projective synchronization with the error bound , where , , and is an arbitrary small constant.*

*Proof. *Construct the Lyapunov function as ; then . According to Lemma 1 and Assumption 4, taking Caputo derivative of along trajectory of error equation (12), one can getNext, if systemhas the same initial values with , then . Using Lemma 2, we haveBy using properties of Caputo derivative, (15) is equivalent towhere . Letting , system (17) becomes Because , based on Lemma 3, we know . SoAccording to (16) and (19), we get , when , i.e.,So, quasi-matrix projective synchronization with error bound between drive system (7) and response system (8) can be realized. This completes the proof.

*Remark 7. *Substitute (11) into (8) and activate controller (11); response system (8) becomes According to error system (12), drive system (7), and controlled response system (22), we can explore the quasi-matrix projective synchronization behaviors between fractional order neural networks (7) and (8).

##### 3.2. Fractional Q*uasi-Inverse*-Matrix Projective Synchronization

Next, let us define the quasi-inverse-matrix projective synchronization.

*Definition 8. *Systems (7) and (8) are said to be quasi-inverse-matrix projective synchronization with error bound , if there exists such that, for all and initial values , the synchronization error satisfies , where means an arbitrary invertible projective matrix.

Taking Caputo derivative of both sides of error function and substituting into (7) and (8), the error system isConstructing the control function ,and substituting it into (23), error system changes to

Theorem 9. *Suppose Assumption 4 holds and the following inequality is satisfied:and then drive system (7) and response system (8) with control law (24) will achieve the q uasi-inverse-matrix projective synchronization with the error bound , where and .*

*Proof. *Choose the Lyapunov function ; then . By using Lemma 1 and Assumption 4, taking Caputo derivative of along trajectory of error system (25), one can getThen, by referring to (15)–(20) of the proof for Theorem 6, similarly we can knowSo, quasi-inverse-matrix projective synchronization with error bound between drive system (7) and response system (8) can be realized. This completes the proof.

*Remark 10. *Substitute (24) into (8) and activate controller (24); response system (8) becomes Then, according to error system (25), drive system (7), and controlled response system (29), we can explore quasi-inverse-matrix projective synchronization behaviors between fractional order neural networks (7) and (8).

*Remark 11. *According to Theorems 6 and 9, choosing larger control parameter , the error bound will become smaller. Therefore, by selecting appropriate control parameters, the synchronization error bound can be reduced to the required standard as small as what we need, which is of important and practical significance in nonlinear control and chaos synchronization for fractional order neural network.

*Remark 12. *When derivate order , systems (7) and (8) are reduced to the integer order neural networks, from Theorems 6 and 9 and their proof; then we can obtain the quasi-matrix and quasi-inverse-matrix projective synchronization criteria for the disturbed and delayed integer order neural networks.

From the above, Theorems 6 and 9 and their proof process constitute the quasi-matrix and quasi-inverse-matrix projective synchronization method for synchronizing two disturbed and delayed fractional order neural networks. Additionally, it is particular to point out that the above two synchronization types are of general significance. Choosing different projective matrix and controller, they can be reduced to some special synchronization cases as in Remark 13.

*Remark 13. *(1)Choosing projective matrix , systems (7) and (8) can achieve the quasi-complete synchronization.(2)Choosing projective matrix , they can achieve the quasi-antisynchronization.(3)Choosing projective matrix , they can achieve the quasi-projective synchronization (or quasi-inverse projective synchronization).(4)Choosing projective matrix , they can achieve the quasi-modified projective synchronization (or quasi-inverse-modified projective synchronization).(5)If external disturbances , they can achieve the complete matrix and inverse-matrix projective synchronization.

#### 4. Some Applications

In this part, two numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed theoretical results.

##### 4.1. Application to Fractional Quasi-Matrix Projective Synchronization

Two nonidentical drive and response systems are considered aswhereand is the controller. In the following numerical analysis, in order to research the chaotic synchronization between systems (30) and (31), we will select derivative order as , which can make the two systems generate chaotic attractors as shown in Figures 1(a) and 1(b) with initial conditions .

**(a)**

**(b)**

According to Definition 5, choosing projective matrix asthen error function of quasi-matrix projective synchronization is computed as

Choosing , one can easily obtain which obviously does not satisfy condition (13) of Theorem 6, so the time history of errors cannot be limited to a bounded area as shown in Figures 2(a) and 2(b).

**(a)**

**(b)**

Choosing , one can compute , , . By Theorem 6, quasi-matrix projective synchronization with the estimated error bound between systems (30) and (31) can be realized where and . And there exists such that for all .

If a smaller error bound needs to be controlled to , then the larger control parameters should be designed as . Next, one can calculate . By Theorem 6, the quasi-matrix projective synchronization with the smaller error bound between systems (30) and (31) can be realized where and . And there exists such that for all .

Next, numerical simulation is given to verify the analysis of fractional quasi-matrix projective synchronization. The initial conditions for drive system, response system, and errors system are and . Figures 3(a)–3(d) show the time trajectories of quasi-matrix projective synchronization with projective matrix . Figures 4(a)-4(b) depict the time history of synchronization error with error bound under control parameters and can be acquired as Figures 5(a)-5(b) show the time evolution of synchronization error with a smaller error bound under larger control parameters and can be acquired as , which verifies Remark 11 that the larger the control parameter , the smaller the error bound .

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(a)**

**(b)**

##### 4.2. Application to Fractional Quasi-Inverse-Matrix Projective Synchronization

Two nonidentical 3-dimensional drive and response systems are constructed asandwhereand is the controller. In the following numerical analysis, in order to research the chaotic synchronization between systems (35) and (36), we will select derivative order as , which can make the two systems exhibit the chaotic attractors as depicted in Figures 6(a) and 6(b) with initial conditions .

**(a)**

**(b)**

According to Definition 8, selecting the projective matrix and its inverse-matrix asthen error function of quasi-inverse-matrix projective synchronization is computed as

Choosing , one can easily obtain which does not satisfy condition (26) of Theorem 9, so the time history of error cannot be limited to a bounded area as shown in Figures 7(a) and 7(b).

**(a)**

**(b)**

Choosing , one can compute , , . By Theorem 9, the quasi-inverse-matrix projective synchronization with the estimated error bound between systems (35) and (36) can be realized where and . And there exists such that for all .

If a smaller error bound needs to be controlled to , then larger control parameters should be designed as . Next, one can calculate . By Theorem 9, the quasi-inverse-matrix projective synchronization with the smaller estimated error bound between systems (35) and (36) is realized where and . And there exists such that for all .

Next, numerical simulation is given to verify the analysis of fractional quasi-inverse-matrix projective synchronization. The initial conditions for drive-response system and error system are and . Figures 8(a)–8(f) show the time trajectories of synchronization for systems (35) and (36) with projective matrix . Figures 9(a)-9(b) depict the time history of synchronization error with error bound under control parameters and can be acquired as Figures 10(a)-10(b) show the time evolution of synchronization error with a smaller error bound under larger control parameters , , and and