A New Matrix Projective Synchronization of Fractional-Order Discrete-Time Systems and its Application in Secure Communication

In this study, it is proposed that a new matrix projective synchronization of fractional-order (FO) chaotic maps in discrete-time. A new synchronization error is introduced and a control law is constructed, which makes the synchronization error converge towards zero in sufficient time under the stability theory of linearization method of FO systems. Numerical simulation results are presented to illustrate the feasibility of the scheme. Finally, a secure communication scheme based on FO discrete-time (FODT) systems was proposed.


I. INTRODUCTION
Chaos theory, in nonlinear dynamical systems, is a very attractive phenomenon, which has been extensively investigated and studied in the last decades. Due to its initial value sensitivity, non-periodic, continuous bandwidth spectrum, trajectory unpredictability and pseudo-random, chaos has wide application in secure communication, cryptography, image encryption, signal processing and other fields [1]- [8].
In the past three centuries, the study of fractional calculus theory has been carried out mainly in the purely theoretical field of mathematics, but in recent decades, fractional differential equations and fractional difference equations have been used more and more to describe optical and thermal systems, mechanics systems, signal processing, system identification, robotics and other applications [9]- [15]. A great number of FO chaotic maps in continuous-time and discrete-time were investigated in recent years, including FO Chen map [16], FO Rossler map [17], FO Lorenz map [18], FO Lu map [19], FO Ikeda map [20], FO Sine map [21], FO cubic Logistic map [22]. In recent years, many scholars have also studied the sliding mode control and circuit implementation of FO chaotic systems [37]- [42].
Chaos synchronization, in nonlinear science, is one of the hot topics. Since Pecora and Carroll proposed the complete The associate editor coordinating the review of this manuscript and approving it for publication was Sun Junwei . synchronization method of chaotic system in 1990s, great strides have been made in chaotic synchronization [23], [24], [24]- [28], [30]. However, there are few references on the synchronization of FODT chaotic systems. In 2014, Hu [31] studied FO Henon map, which has made an unprecedented contribution to the high dimensional FODT chaotic synchronization method. In 2015, Wu and Baleanu [32] proposed synchronization of the FO Logistic map, and this synchronization method is only suitable for master-slave systems with the same dimension. In 2017, Shukla and Sharma [33] studied generalized FO Henon map and proposed active control synchronization, and this synchronization method is also only suitable for master-slave systems with the same dimension. Ouannas et al. [34] proposed a general synchronization of FODT chaotic systems in 2018. Though this synchronization method can be suitable for master-slave systems with different dimensions, it's hard to find the right bijection function f to meet requirement.
In order to better synchronize FODT chaotic systems with different dimensions, a new matrix projective synchronization, in this article, is proposed. This new synchronization method relies on an invertible matrix P and an arbitrary matrix M, so it is called P-M synchronization. Compared with the existing synchronization methods, P-M synchronization, constructs a controller U to make the synchronization error meet the requirements. Secondly, compared with matrix projection synchronization, P-M synchronization can realize synchronization of different dimensions of the same master-slave system. Lastly, compared with one-to-one mapping f , it is easy to construct an invertible matrix P.
This article is organized as follows. In Section 2, some preliminary knowledge of discrete fractional order equations is given. In Section 3, we give the P-M synchronization criterion and the stability theory of discrete fractional order system. Section 4 present some numerical simulation results related to concrete examples to show the feasibility of this synchronization scheme. In Section 5, a secure communication scheme based on fractional-order discrete-time systems was proposed. Finally, a general summary of this article is drawn in Section 5.

II. PRELIMINARIES
The drive and response maps considered in presented article are in the following forms, where the n-tuple vectors X (t) = (x 1 (t), x 2 (t), . . . , x n (t)) T and the m-tuple vectors Y (t) = (y 1 (t), y 2 (t), . . . , y m (t)) T are state vectors of the drive and response maps, respectively, A ∈ R n×n , and B ∈ R m×m are the linear parts of the drive and response maps, respectively, and the map φ : R n → R n , ϕ : R m → R m are the nonlinear functions of the above maps and U = (u 1 , u 2 , . . . , u m ) T is a vector controller, which is to be determined by the control law of the synchronization scheme.
Remark 1: v denotes the order of fractional difference equation, and 0 < v ≤ 1.
Remark 2: N a denotes the set of natural numbers beginning from a. The notation C α a X (t) denotes the Caputo type delta difference of X (t) defined over N a .
Definition 1 [36]: where n = [a] + 1, σ (s) = s + 1, and the notation n s X (s) denotes the v-th fractional sum of x i . Lemma 1 [37]: The Caputo type delta difference in the form of equation (4) below is equivalent to the discrete integral equation in the form of equation (5) where σ (s) = s + 1, t ∈ N a+n , and u 0 (t) =

III. STABILITY CRITERIA FOR DISCRETE FRACTIONAL ORDER LINEAR SYSTEMS
In this section, we will introduce a new matrix projective synchronization based on FODT chaotic system. Definition 1: Matrix projective synchronization is said to be achieved between drive system (1) and response system (2) if there exists a matrix M ∈ R m×n such that the dynamic synchronization error Based on matrix projection synchronization, we propose a new synchronization scheme, namely, P-M synchronization. Definition 2: P-M synchronization is said to be achieved between drive system (1) and response system (2) if there exists an invertible matrix P ∈ R n×n and a matrix M ∈ R m×n such that the dynamic synchronization error satisfies the condition lim t→∞ e (t) = 0 . The v order Caputo fractional difference of equation (7) is as the following form.
Equation (8) can be further derived as follows: where C ∈ R m×m is a control matrix and Lemma 1 [38]: The zero equilibrium of the linear fractional order discrete system: is asymptotically stable, if where e (t) ∈ R n , 0 < v ≤ 1, M ∈ R n×n , ∀t ∈ N a+1−v and λ is the eigenvalues of the matrix M . Based on Lemma 3.1, P-M synchronization can be achieved if there exists a controller U such that the eigenvalues of the coefficient matrix of the error function e (t) meet the requirements of equation (12). Next, we are now ready to present the P-M synchronization. Theorem 1: system (1) and (2) can be achieved P-M synchronization if where matrix Q is the inverse of matrix P and the control matrix C is chosen such that all the eigenvalues λ of the Proof. By substituting equation (10) into equation (9), the synchronization error system (8) reduces to Furthermore, according to the equation (14), it is easy to see that all eigenvalues of the matrix (B − C) satisfy

IV. NUMERICAL EXAMPLES
In this section, we consider some numerical examples to show the validity of synchronization scheme illustrated above. The 2D FO Henon map as expressed in [32] is given by: when parameter v = 0.984 and (a 1 , b 1 ) = (1.4, 0.3), The FO Henon map exhibits chaotic behavior. The phase portrait with the initial values (x 1 (0) , x 2 (0)) = (0, 0) is depicted in Fig.1. It can be seen from Fig. 1 that the 2D FO Henon map become chaotic under certain parameters. The 3D FO generalized Henon map as expressed in [34] is given as when the parameters (a 2 , b 2 ) = (0.99, 0.2) and the order v = 0.984, system (18) is chaotic system. The chaotic trajectories is shown in Fig.2 with the initial values (x 1 (0) , x 2 (0) , x 3 (0)) = (0.1, 0.2, 0.5). It can be seen from Fig. 2 that the 3D FO Henon map become chaotic under certain parameters.

A. P-M SYNCHRONIZATION IN 3D
In order to unify the notation, equation (17) can be rewritten as the following expression: where .
The 3D FO Henon map is selected as response system. Map (18) can be written as where Given the approach presented in last section, the error system is given by where matrix P 1 and M 1 are selected as followings: Based on our approach proposed in Section 3, a control matrix C 1 is selected such that the whole eigenvalues of matrix (B − C 1 ) satisfy the condition of Theorem 3.1. Matrix C 1 is selected as following: then, the error system is given by It easy to say that the whole eigenvalues of matrix (B − C 1 ) satisfy the condition of Theorem 3.1. The initial value of the error iteration is given The synchronization error of system (17) and (18) is depicted in Fig.3. It is clear to see that the synchronization error quickly converges to a very small value that satisfies the requirement, which indicate that drive system (17) and response system (18) can achieve P-M synchronization in 3 dimensions. VOLUME 8, 2020

B. P-M SYNCHRONIZATION IN 2D
The 3D FO Henon map is selected as drive system, and the 2D FO Henon map is selected as response system. We are just interchanging the matrix A and B, and the mapping φ and ϕ, so we are not dividing the linear part from the nonlinear part.
Given the approach presented in last section, the error system is given by where matrix P 2 and M 2 are selected as followings: Matrix C 2 is selected as following: then, the error system is given by The initial value of the error iteration is given as follows:   The synchronization error of system (18) and (17) is depicted in Fig.4. It is clear to see that the synchronization error quickly converges to a very small value that satisfies the requirement, which indicate that drive system (18) and response system (17) can achieve P-M synchronization in 2 dimensions.

V. A SECURE COMMUNICATION SCHEME BASED ON FODT SYSTEMS
A. THE SPEECH ENCRYPTION SCHEME Secure communications can be classified into three major security technologies: chaotic masking technology, chaotic parameter modulation technology and chaotic keying technology. Chaos masking technology belongs to chaos analog communication, chaos parameter modulation and chaos keying technology belongs to chaos digital communication technology. Compared with chaotic parameter modulation technology and chaotic keying technology, chaos masking is easy to implement, short encryption time and low complexity because it is simple addition and subtraction. The speech encryption scheme is mainly based on chaos masking, and the block diagram of communication is shown in Fig.9.
The FO discrete-time chaotic map (19) is used as master system, and map (21) is used as slave system. Speech signal s (t) and state variable C v a X (t) are masked into the signal E (t), then, the signal E (t) is sent to the receiver end via the public channel. However, the transmission channel will be contaminated by chaotic noise. Suppose the noise is White Gaussian Noise w (t).At receiver end, the recovered signal s (t) can be obtained through the signal E 2 (t) and state variable C v a Y (t), where E 2 (t) is the signal processed by a filter.
Let the initial values of the systems (19) and (21) Fig.5, Fig.6, Fig.7, Fig.8, respectively. The initial keys of this encryption scheme are the initial values of master system x 0 , y 0 and the order of master system v, respectively Floating point accuracy is 10 −15 , then the key space is 10 45 , which is much larger than 100 2 Therefore, this scheme can provide security in terms of key space.

2) SENSITIVITY ANALYSIS OF KEYS
According to the principle of cryptography, the stronger the sensitivity of the key, the stronger the ability to resist differential attacks Generally speaking, the small difference of the same plaintext encryption key can cause a huge change of the ciphertext sequence; the slight difference of the same ciphertext decryption key can cause the decryption results are quite different First, the encryption key r = (x 1 (0) , x 2 (0)) = (0, 0) is used to complete the encryption operation of the encryption scheme, and then the error decryption key r = (r 1 , r 2 , r 3 ) of the small difference is selected, where r 1 = 0 + 10 −15 , 0 , r 2 = 0, 0 + 10 −15 and r 3 = 0 + 10 −15 , 0 + 10 −15 . Let the original speech signal be p (i) and the decrypted speech signal be p (i) , then their mean square error is as The mean square error of the r decrypted signal and the original speech signal calculated by '' (36)'' is shown in Table 1 It can be seen that the decryption key has a slight error, and the decryption result will be completely different, that is, the key sensitivity of the algorithm is numerically proved.

3) PHASE SPACE RECONSTRUCTION ATTACK
This scheme can resist the phase space reconstruction attack Because the mixed signal that the original speech signal after chaos masking E(t) is transmitted in the common channel, the phase space reconstructed from a series of values of E(t) is not completely topologically equivalent to the real chaotic system.

4) ROBUSTNESS
The robustness of this scheme is poor, since the chaotic system is sensitive to the initial value and system parameters If the system parameters are changed, the output of the chaotic system will change Furthermore, from the perspective of secure communication, if the system has better robustness, it will lose confidentiality.

VI. CONCLUSION
In this article, we proposed a new matrix projective of fractional-order discrete-time chaotic maps of different dimensions. Compared with matrix projection synchron-ization, this synchronization can synchronize any n-dimensional chaotic map (drive system) with m-dimensional chaotic system (response system) by selecting an invertible matrix P ∈ R n×n and an arbitrary matrix M ∈ R m×n . Numerical simulation results in Section 4 have shown the validity of this synchronization scheme. In Section 5, a secure communication scheme based on fractional-order discrete-time systems was proposed, and through numerical simulation results confirmed the feasibility of the scheme. In the future work, we intend to implement the secure communication scheme mentioned in this article on the hardware circuit.
WENHAO YAN received the B.S. degree in information and computing science from Northeast Petroleum University, China, in 2018. He is currently pursuing the M.S. degree in information and communication engineering from Heilongjiang University, Harbin. His research interests include nonlinear dynamics, chaos synchronization, and chaotic secure communication.
QUN DING received the Ph.D. degree in instrument science and technology from the Harbin Institute of Technology, China. She is currently a Professor with the College of Electronic Engineering, Heilongjiang University, China. She has published two books and more than 100 scientific articles in refereed journals and proceedings. Hers research interests include nonlinear dynamics and control, chaos pseudo-random sequence generator, and chaotic secure communication.