Quasi-Matrix and Quasi-Inverse-Matrix Projective Synchronization for Delayed and Disturbed Fractional Order Neural Network

1Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 4Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China


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][2][3][4][5][6][7][8][9][10][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.
Moreover, some scholars have studied the synchronization and quasi-synchronization problems for delayed fractional order memristor-based neural network with uncertain parameters [35][36][37][38][39][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][43][44], the scholars derived some new sufficient conditions and designed the appropriate controllers to guarantee projective synchronization for delayed fractional order memristorbased 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 quasiinverse-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 quasiinverse-matrix projective synchronization types for two 3dimensional 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.
Consider two nonidentical n-dimensional delayed fractional order neural networks, which are subjected to external disturbances, as the drive system and response system, respectively:

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).
. .Fractional Quasi-Matrix Projective Synchronization.Let's first define the quasi-matrix projective synchronization as follows.
Next, let us research the quasi-matrix projective synchronization between systems (7) and (8).Taking Caputo derivative of both sides of error function   =   − ∑  =1 Λ    and substituting into ( 7) and ( 8), the error system can be obtained as Constructing the control function   () as and substituting it into (10), the error system is changed to Because of different external disturbances for two systems,  = 0 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 holds and the following inequality is satisfied: then the drive system ( ) and response system ( ) with control law ( ) will achieve the quasi-matrix projective synchronization with the error bound  1 /( 1 −  1 ) + , where  12), one can get has the same initial values with (), then  0    () ≤  0    ().Using Lemma 2, we have By using properties of Caputo derivative, ( 15) is equivalent to where Because  1 >  1 > 0, based on Lemma 3, we know lim →+∞ S() = 0.So S () =  () − D1 → 0, ( → +∞) .
Taking Caputo derivative of both sides of error function   =   −∑  =1     and substituting into ( 7) and ( 8), the error system is Constructing the control function   (), and substituting it into (23), error system changes to So, quasi-inverse-matrix projective synchronization with error bound  2 /( 2 −  2 ) +  between drive system (7) and response system (8) can be realized.This completes the proof.
Remark .According to Theorems 6 and 9, choosing larger control parameter , the error bound   /(  −   ) + , ( = 1, 2) 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 .When derivate order  = 1, 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 quasimatrix 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.

Some Applications
In this part, two numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed theoretical results. . .Application to Fractional Quasi-Matrix Projective Synchronization.Two nonidentical drive and response systems are considered as where  and   () 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  = 0.92, which can make the two systems generate chaotic attractors as shown in Figures 1(a) and 1(b) with initial conditions According to Definition 5, choosing projective matrix Λ as then error function of quasi-matrix projective synchronization is computed as Choosing  1 = 0,  2 = 0, one can easily obtain  = −3.35< 0 <  = 6.25 which obviously does not satisfy condition (13) of Theorem 6, so the time history of errors  1 ,  2 , ‖()‖ 1 cannot be limited to a bounded area as shown in Figures 2(a   Choosing  1 = 6.35,  2 = 11.35,one can compute 35.By Theorem 6, quasi-matrix projective synchronization with the estimated error bound  = 1.001 between systems (30) and ( 31) can be realized where  1 /( 1 −  1 ) = 1.35/(7.6− 6.25) = 1 and  = 0.001.And there exists  ≥  0 such that |  ()| ≤ ‖()‖ 1 < 1.001 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 where     systems (35) and ( 36), we will select derivative order as  = 0.98, which can make the two systems exhibit the chaotic attractors as depicted in Figures 6(a then error function of quasi-inverse-matrix projective synchronization is computed as Choosing  1 = 0,  2 = 0,  3 = 0, one can easily obtain  = −7.55< 0 <  = 8 which does not satisfy condition (26) of Theorem 9, so the time history of error ‖()‖ 1 cannot be limited to a bounded area as shown in Figures 7(a 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 ).

Conclusions
For synchronizing two nonidentical delayed and disturbed fractional order neural networks, the paper proposes the quasi-matrix and quasi-inverse-matrix projective synchronization and establishes their synchronization criteria.By selecting appropriate control parameters, the synchronization error bound is obtained and can be reduced to the required standard as small as what we need, which is of important significance to practical problem.Two numerical examples verify the feasibility of synchronization analysis.Simply put, Definitions 5 and 8, Theorems 6 and 9 and their proofs, and synchronization analysis of two numerical examples are all new work.This research extends the projective scaling factor to an arbitrary constant matrix and offers a general approach for synchronizing the delayed and disturbed fractional order neural network.Evidently, fractional quasi-matrix and quasiinverse-matrix projective synchronization can guarantee faster and safer image encryption and text encryption in communication and provide new insights for researching the fractional order neural network, which is a meaningful work.The complex-valued neural network with time delay, which is a more general neural network system, is becoming more and more popular, and its finite time stability, boundedness, global robust stability, and global exponential stability have been well researched in [22,28,40,[47][48][49][50][51].So, in future works, by using adaptive control method and sliding mode control method, we will generalize our main results to quasimatrix and quasi-inverse-matrix projective synchronization for delayed fractional order complex-variable neural networks, fractional order memristor-based neural network, and fractional order neural network with unknown parameters.