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Projective-lag synchronization scheme between two different discrete-time chaotic systems
Journal Pre-proof Projective-lag synchronization scheme between two different discrete-time chaotic systems Cun-Fang Feng, Hai-Jun Yang
PII: DOI: Reference:
S0020-7462(19)30625-0 https://doi.org/10.1016/j.ijnonlinmec.2020.103451 NLM 103451
To appear in:
International Journal of Non-Linear Mechanics
Received date : 2 September 2019 Revised date : 20 December 2019 Accepted date : 11 February 2020 Please cite this article as: C.-F. Feng and H.-J. Yang, Projective-lag synchronization scheme between two different discrete-time chaotic systems, International Journal of Non-Linear Mechanics (2020), doi: https://doi.org/10.1016/j.ijnonlinmec.2020.103451. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Journal Pre-proof Projective-lag synchronization scheme between two different discrete-time chaotic systems Cun-Fang Feng*a, Hai-Jun Yang*b a
School of Electronic and Electrical Engineering, Hubei Engineering and
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Technology Research Center for Functional Fiber Fabrication and Testing, Wuhan Textile University, Wuhan, China b
Department of Preventive Medicine, School of Basic Medical Sciences, Hubei
University of Chinese Medicine, Wuhan, China Abstract:
This
paper
addresses
theoretical
analysis
of
projective-lag
synchronization between two different discrete-time chaotic systems. We overcome some limitations of previous works where projective-lag synchronization can only be achieved in continuous-time chaotic systems. In this paper, we realize projective-lag synchronization in discrete-time chaotic systems with the same order. Based on the Lyapunov stability theory and a nonlinear control scheme, the controllers are designed by using the relevant variables of master and slave systems, and we find sufficient conditions for the stability of the error dynamics. The correctness of theoretical analysis is verified by 2D and 3D discrete-time chaotic systems, respectively.
Keywords: projective-lag synchronization; Lyapunov stability theory; nonlinear discrete-time systems
PACS: 05.45. Gg; 05.45. Pq; 05.45Tp
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1. Introduction
Discrete-time dynamical systems have drawn considerable attention over the past decades since practical systems, such as biological phenomena [1], chemical systems [2,3], physical systems [4] and economic systems [5] etc., are well defined using discrete-time dynamical systems. Considering the measurement is usually carried out at
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specific time intervals and digital simulations can be performed conveniently and quickly, chaos synchronization of discrete-time dynamical systems has attracted much attention due to its practical applications in secure communication and cryptology [6-8]. So, it might play an important role to achieve synchronization in discrete-time chaotic systems.
Recently, more and more attention has been paid to synchronization in discrete-time chaotic systems. Many types of synchronization have been presented for chaotic maps ———————————— *Corresponding authors:
Cun- Fang Feng, Email: [email protected]; Hai-Jun Yang, Email: [email protected]. 1
Journal Pre-proof such as complete synchronization [9], impulsive synchronization [10], projective synchronization [11], inverse full state hybrid projective synchronization [12], generalized synchronization [13], Q-S synchronization [14], outer synchronization [15], anticipating synchronization [16], robust synchronization [17] and so on. Among these
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different types of synchronization defined to date, an interesting synchronization scheme is represented by the so-called projective-lag synchronization [18], which includes projective synchronization and lag synchronizations. There is a proportional relationship between the state variables of the master system and slave system in projective synchronization. In application to secure communication, this feature can be used to extend binary digital to M-nary digital to achieve fast communication [19]. However, in real practical application, the system is inevitably affected by several external factors. One of these factors is that there is a certain time delay when the state variables of master and slave systems are synchronized. The lag is more reasonable in engineering applications because time delays are ubiquitous between communication channels in a practical point of view. The systems introduced with lags are infinite-dimensional systems, which have high randomness and unpredictability. So systems with lags improve the confidentiality and open up a new way for the application of projective synchronization in secure communication. It is much of importance to consider the slave system to synchronize the master system with lag τ when dealing with projective synchronization. From what has been discussed above, it
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is of great significance to study projective-lag synchronization. In this synchronization scheme, the slave system’s output 𝑦𝑦 lags behind the output of the master system 𝑥𝑥
proportionally after a transient time, i.e., 𝑦𝑦(𝑡𝑡) = 𝛼𝛼𝛼𝛼(𝑡𝑡 − 𝜏𝜏), where 𝜏𝜏 is positive real.
Projective-lag synchronization has received more and more attention just recently.
To the best of our knowledge, these works about projective-lag synchronization just
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concentrate on continuous-time chaotic dynamical systems. However, in addition to continuous-time evolution, systems can evolve on discrete intervals. In particular, in comparison with continuous ones, discrete controllers have lower cost. Therefore, research in discrete-time control has attracted more attention in recent years. As of nowadays, the possibility of projective-lag synchronization for discrete-time chaotic systems has not been discussed to the best of our knowledge. On the other hand, the active control as a simple and effective control method is widely utilize to control chaotic systems [20,21]. The main idea of the active control method is cancelling out 2
Journal Pre-proof the nonlinear terms through the feedback design. The system is transformed into a simple and easily controlled linear system and then the controller is designed by using the mature linear system theory. When the parameters are known, the active control method is simple and effective. There is no need to construct the Lyapunov function and
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achieve synchronization in a short time.
Motivated by abovementioned discussions, this paper aims to observe projective-lag synchronization in two different discrete-time chaotic systems with the identical order using active control method, which overcomes the limitations of previous works where projective-lag synchronization observed only in continuous-time chaotic systems [22-29]. The rest of this paper is arranged as follows. In section 2, based on the stability theory of discrete-time dynamical system, a suitable controller is designed using the relevant variables of master and slave systems and sufficient conditions of asymptotic stability of error dynamics are derived. In section 3, two numerical simulations are presented to verify the effectiveness and feasibility of the proposed synchronization method. Finally, some conclusions are drawn. 2. Analytical Results
Assuming there are two different discrete-time chaotic systems with the same order. The master system is given by
(1)
𝑋𝑋(𝑘𝑘 + 1) = 𝐴𝐴𝐴𝐴(𝑘𝑘) + 𝐹𝐹(𝑋𝑋(𝑘𝑘)),
and the controlled slave system is described as 𝑘𝑘 ∈ 𝑁𝑁 is
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where
𝑌𝑌(𝑘𝑘 + 1) = 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺(𝑌𝑌(𝑘𝑘) ) + U (𝑘𝑘), an
iteration
index;
(2)
𝑋𝑋(𝑘𝑘) = (𝑥𝑥1 (𝑘𝑘), … 𝑥𝑥𝑛𝑛 (𝑘𝑘))𝑇𝑇 ∈ 𝑅𝑅𝑛𝑛 , 𝑌𝑌(𝑘𝑘) =
(𝑦𝑦1 (𝑘𝑘), … 𝑦𝑦𝑛𝑛 (𝑘𝑘))𝑇𝑇 ∈ 𝑅𝑅𝑛𝑛 are the state space vectors of systems (1) and (2), respectively;
the real 𝐴𝐴𝑛𝑛×𝑛𝑛 and 𝐵𝐵𝑛𝑛×𝑛𝑛 matrices are known constant matrices with appropriate
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dimensions; 𝐹𝐹 ∈ 𝑅𝑅𝑛𝑛 → 𝑅𝑅𝑛𝑛 and G ∈ 𝑅𝑅𝑛𝑛 → 𝑅𝑅𝑛𝑛 contain the nonlinear parts and the
constant terms of the systems (1) and (2), respectively; and U is a suitable controller
to be designed for projective-lag synchronization between systems (1) and (2). The definition of projective-lag synchronization for systems (1) and (2) is given by Definition. Projective-lag synchronization is said to be achieved between the master system (1) and response system (2) if there exists an effective controller U such that synchronization error 𝑒𝑒(𝑘𝑘) = 𝑌𝑌(𝑘𝑘) − 𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚)
(3)
satisfies that lim ‖𝑒𝑒(𝑘𝑘)‖ → 0 for any initial conditions, where α (α ≠ 0) is scaling 𝑘𝑘→∞
3
Journal Pre-proof factor, 𝑚𝑚 > 0 is time delay and ‖∙‖ is the Euclidean norm. Remark 1.
scaling
factor
Projective-lag synchronization becomes lag synchronization when α = 1;
projective-lag
synchronization
becomes
projective
synchronization when time delay 𝑚𝑚 = 0 ; projective-lag synchronization becomes
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complete synchronization when α = 1 and 𝑚𝑚 = 0.
According to Equation (3), the dynamics of synchronization error between systems
(1) and (2) can be derived as
𝑒𝑒(𝑘𝑘 + 1) = 𝑌𝑌(𝑘𝑘 + 1) − 𝛼𝛼𝛼𝛼(𝑘𝑘 + 1 − 𝑚𝑚)
= 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� + U (𝑘𝑘)
(4)
Our goal is to achieve projective-lag synchronization between master system (1) and slave system (2) via constructing an effective controller U , such that lim ‖𝑒𝑒(𝑘𝑘)‖ →
0. To this end, we design the controller U as follows: where
𝑘𝑘→∞
U (𝑘𝑘) = 𝐻𝐻�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� − 𝑅𝑅�𝑌𝑌(𝑘𝑘)�,
(5)
𝐻𝐻�𝑋𝑋(𝑘𝑘)� = 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘) − 𝛼𝛼Β𝑋𝑋(𝑘𝑘) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘) + 𝛼𝛼𝛼𝛼(𝑋𝑋(𝑘𝑘)), 𝑅𝑅�𝑌𝑌(𝑘𝑘)� = 𝐺𝐺�𝑌𝑌(𝑘𝑘)� + 𝐿𝐿𝐿𝐿(𝑘𝑘),
and L is the undetermined constant control matrix. Hence, we have the following result.
Theorem. If the control matrix 𝐿𝐿 satisfies 𝐼𝐼 − (𝐵𝐵 − 𝐿𝐿)𝑇𝑇 (𝐵𝐵 − 𝐿𝐿) , which is a
positive definite matrix, then the master system (1) and response system (2) are globally Proof. written as
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projective-lag synchronization under the control law (5).
By substituting Eq. (5) into Eq. (4), the synchronization error can be
𝑒𝑒(𝑘𝑘 + 1) = 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚)
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+𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝐿𝐿𝐿𝐿(𝑘𝑘)
= 𝐵𝐵𝐵𝐵(𝑘𝑘) − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝐿𝐿𝐿𝐿(𝑘𝑘)
(6)
= (𝐵𝐵 − 𝐿𝐿)𝑒𝑒(𝑘𝑘)
Consider a Lyapunov function in the form 𝑉𝑉�𝑒𝑒(𝑘𝑘)� = 𝑒𝑒(𝑘𝑘)𝑇𝑇 𝑒𝑒(𝑘𝑘), then we obtain:
𝛥𝛥𝑉𝑉�𝑒𝑒(𝑘𝑘)� = 𝑉𝑉�𝑒𝑒(𝑘𝑘 + 1)� − 𝑉𝑉(𝑒𝑒(𝑘𝑘))
= 𝑒𝑒 𝑇𝑇 (𝑘𝑘)[(𝐵𝐵 − 𝐿𝐿)𝑇𝑇 (𝐵𝐵 − 𝐿𝐿) − 𝐼𝐼]𝑒𝑒(𝑘𝑘) < 0
(7)
From the Lyapunov stability theory [30], it is immediate that lim ‖𝑒𝑒(𝑘𝑘)‖ → 0, 𝑘𝑘→∞
which implies that the origin of the error system (6) is asymptotically stable, so systems (1) and (2) achieve projective-lag synchronization. 4
Journal Pre-proof This completes the proof. Remark 2. The value of scaling factor α has no effect on the error dynamics of
projective-lag synchronization [Eq. (6)], so we can arbitrarily direct the scaling factor α onto any desired value.
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3. Numerical simulation results
To demonstrate the effectiveness of the proposed synchronization schemes, we provide two illustrative examples in this section.
3.1. Two different 2D discrete-time chaotic systems
We achieve the projective-lag synchronization between two 2D different discrete-time systems. The master system is described by the Ushio system [31] 𝑥𝑥 (𝑘𝑘 + 1) = 𝑐𝑐𝑥𝑥1 (𝑘𝑘) − 𝑥𝑥13 (𝑘𝑘) + 𝑥𝑥2 (𝑘𝑘) � 1 𝑥𝑥2 (𝑘𝑘 + 1) = 0.5𝑥𝑥1 (𝑘𝑘).
(8)
Note that it can be expressed into Equation (1), where 𝑐𝑐 0.5
𝐴𝐴 = �
3 1 � and 𝐹𝐹�𝑋𝑋(𝑘𝑘)� = �−𝑥𝑥1 (𝑘𝑘)�. 0 0
(9)
It is well known that Ushio system has a chaotic attractor when 𝑐𝑐 = 1.9. The chaotic
attractor of Ushio system is shown in Fig.1.
The controlled slave system is described as Fold map [32] 𝑦𝑦 (𝑘𝑘 + 1) = 𝑑𝑑𝑦𝑦1 (𝑘𝑘) + 𝑦𝑦2 (𝑘𝑘) + 𝑢𝑢1 (𝑘𝑘) � 1 𝑦𝑦2 (𝑘𝑘 + 1) = 𝑦𝑦12 (𝑘𝑘) + 𝑠𝑠 + 𝑢𝑢2 (𝑘𝑘),
(10)
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where 𝑑𝑑 = −0.1, 𝑠𝑠 = −1.7 are systems constants and U (𝑘𝑘) = (𝑢𝑢1 (𝑘𝑘), 𝑢𝑢2 (𝑘𝑘))𝑇𝑇 is the
vector controller designed by Eq. (5) for achieving projective-lag synchronization. The chaotic attractor of system (10) with 𝑑𝑑 = −0.1, 𝑠𝑠 = −1.7, 𝑢𝑢1 = 0 and 𝑢𝑢2 = 0 is
displayed in Fig. 2. Similarly, the matrix B and the function G of system (10) are given by
0 1 � and 𝐺𝐺�𝑌𝑌(𝑘𝑘)� = � 2 (𝑘𝑘) � . 𝑦𝑦1 + 𝑠𝑠 0
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𝑑𝑑 𝐵𝐵 = � 0
(11)
According to Eq.(5), we take the controllers 𝑢𝑢1 and 𝑢𝑢2 as �
𝑢𝑢1 (𝑘𝑘) 𝑐𝑐 − 𝑑𝑑 + 𝑙𝑙11 � = 𝛼𝛼 � 0.5 + 𝑙𝑙21 𝑢𝑢2 (𝑘𝑘) −�
0
𝑦𝑦12 (𝑘𝑘)
+ 𝑠𝑠
3 (𝑘𝑘 𝑙𝑙12 𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚) �� � + 𝛼𝛼 �−𝑥𝑥1 − 𝑚𝑚)� 𝑙𝑙22 𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚) 0
�−�
𝑙𝑙11 𝑙𝑙21
𝑙𝑙12 𝑦𝑦1 (𝑘𝑘) �� � 𝑙𝑙22 𝑦𝑦2 (𝑘𝑘)
(12)
Thus, the error dynamics between systems (8) and (10) is described by �
𝑒𝑒1 (𝑘𝑘 + 1) −0.1 − 𝑙𝑙11 �=� −𝑙𝑙21 𝑒𝑒2 (𝑘𝑘 + 1)
1 − 𝑙𝑙12 𝑒𝑒1 (𝑘𝑘) �� � −𝑙𝑙22 𝑒𝑒2 (𝑘𝑘)
5
(13)
Journal Pre-proof 0.4 1 �, then the sufficient condition given 0 0.3
If we consider the control matrix 𝐿𝐿 = �
by Theorem is satisfied. In Fig. 3, the slave system 𝑦𝑦1 lags the state of the master system 𝑥𝑥1 with constant lag time 𝑚𝑚 = 10 and the amplitude of the master’s and
𝑇𝑇
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slave’s state variables is correlated by (𝑦𝑦1 (𝑘𝑘), 𝑦𝑦2 (𝑘𝑘))𝑇𝑇 = 2�𝑥𝑥1 (𝑘𝑘 − 10), 𝑥𝑥2 (𝑘𝑘 − 10)� . The time evolution of error systems 𝑒𝑒1 (𝑘𝑘) and 𝑒𝑒2 (𝑘𝑘) are shown in Fig.4 (a) and (b),
respectively. We can see that the synchronization errors converge to zero as time tends to infinity, which implies that projective-lag synchronization between systems (8) and (10) is achieved.
3.2. Two different 3D discrete-time chaotic systems
In the following, we consider the 3D hyperchaotic Hénon system as the drive
system and the controlled 3D Stefanski map as the slave system. The master system is described in [33]
𝑥𝑥1 (𝑘𝑘 + 1) = −𝜇𝜇 ∗ 𝑥𝑥2 (𝑘𝑘) �𝑥𝑥2 (𝑘𝑘 + 1) = 𝑥𝑥3 (𝑘𝑘) + 1 − 𝜐𝜐 ∗ 𝑥𝑥22 (𝑘𝑘) 𝑥𝑥3 (𝑘𝑘 + 1) = 𝜇𝜇 ∗ 𝑥𝑥2 (𝑘𝑘) + 𝑥𝑥1 (𝑘𝑘),
(14)
It can be expressed into the Eq. (1) form, where 0 𝐴𝐴 = �0 1
−𝜇𝜇 0 𝜇𝜇
0 0 1�, 𝐹𝐹�𝑋𝑋(𝑘𝑘)� = �1 − 𝜈𝜈𝑥𝑥22 (𝑘𝑘)�. 0 0
(15)
The hyperchaotic 3D generalized Henon has a chaotic attractor with (μ, ν) = (0.3,
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1.07), which is shown in Fig. 5.
The slave system is given in [33]
𝑦𝑦1 (𝑘𝑘 + 1) = 1 + 𝑦𝑦3 (𝑘𝑘) − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) + 𝑢𝑢1 (𝑘𝑘) �𝑦𝑦2 (𝑘𝑘 + 1) = 1 + 𝛾𝛾𝑦𝑦2 (𝑘𝑘) − 𝛽𝛽𝑦𝑦12 (𝑘𝑘) + 𝑢𝑢2 (𝑘𝑘) 𝑦𝑦3 (𝑘𝑘 + 1) = 𝛾𝛾𝑦𝑦1 (𝑘𝑘) + 𝑢𝑢3 (𝑘𝑘)
(16)
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It can be expressed into the Eq. (2), where 0 𝐵𝐵 = �0 𝛾𝛾
0 𝛾𝛾 0
1 0�, 0
1 − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) 𝐺𝐺�𝑌𝑌(𝑘𝑘)� = �1 − 𝛽𝛽𝑦𝑦12 (𝑘𝑘)� 0
(17)
where 𝑢𝑢(𝑘𝑘) = [𝑢𝑢1 (𝑘𝑘), 𝑢𝑢2 (𝑘𝑘), 𝑢𝑢3 (𝑘𝑘)]𝑇𝑇 is the control vector to be designed later for
achieving projective-lag synchronization. The hyperchaotic Stefanski map has a chaotic attractor when (𝛽𝛽, 𝛾𝛾) = (1.4, 0.2). The chaotic attractors of system (16) is shown in Fig. 6.
According to Eq.(5), we take the controllers 𝑢𝑢1 , 𝑢𝑢2 and 𝑢𝑢3 as 6
Journal Pre-proof 𝑢𝑢1 (𝑘𝑘) 𝑙𝑙11 𝑙𝑙21 �𝑢𝑢2 (𝑘𝑘)� = 𝛼𝛼 � 1 − 𝛾𝛾 + 𝑙𝑙31 𝑢𝑢3 (𝑘𝑘)
−𝜇𝜇 + 𝑙𝑙12 −𝛾𝛾 + 𝑙𝑙22 𝜇𝜇 + 𝑙𝑙32
𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚) −1 + 𝑙𝑙13 1 + 𝑙𝑙23 � �𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚)� 𝑙𝑙33 𝑥𝑥3 (𝑘𝑘 − 𝑚𝑚) 𝑙𝑙12 𝑙𝑙22 𝑙𝑙32
𝑦𝑦1 (𝑘𝑘) 𝑙𝑙13 𝑙𝑙23 � �𝑦𝑦2 (𝑘𝑘)� 𝑙𝑙33 𝑦𝑦3 (𝑘𝑘)
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𝑙𝑙11 1 − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) 0 2 +𝛼𝛼 �1 − 𝜈𝜈𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚)� − �1 − 𝛽𝛽𝑦𝑦12 (𝑘𝑘)� − �𝑙𝑙21 𝑙𝑙31 0 0
(18)
Thus, the error dynamics between systems (14) and (16) is described by 𝑒𝑒1 (𝑘𝑘 + 1) −𝑙𝑙11 𝑒𝑒 (𝑘𝑘 + 1) −𝑙𝑙 � 2 �=� 21 𝛾𝛾 − 𝑙𝑙31 𝑒𝑒3 (𝑘𝑘 + 1)
−𝑙𝑙12 𝛾𝛾 − 𝑙𝑙22 −𝑙𝑙32
−0.3 If we consider the control matrix 𝐿𝐿 = � 0 𝛾𝛾
𝑒𝑒1 (𝑘𝑘) 1 − 𝑙𝑙13 −𝑙𝑙23 � �𝑒𝑒2 (𝑘𝑘)� −𝑙𝑙33 𝑒𝑒3 (𝑘𝑘)
(19)
0 1 0.5 0 �, then the sufficient 0 −0.2
condition given by Theorem is satisfied. In Fig.7, 𝑥𝑥(𝑘𝑘 − 𝑚𝑚) instead of 𝑥𝑥(𝑘𝑘) is
convenient for our study. Fig. 7 shows time series of the master system 𝑥𝑥(𝑘𝑘 − 𝑚𝑚) and the slave system 𝑦𝑦(𝑘𝑘) for α = 2.0. The response system 𝑦𝑦 lags the state of the
master system 𝑥𝑥 with 𝑚𝑚 = 10 and the amplitude of the master’s and slave’s state
variables is correlated by (𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 )𝑇𝑇 = 2(𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚), 𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚), 𝑥𝑥3 (𝑘𝑘 − 𝑚𝑚))𝑇𝑇 . The
errors dynamics between the master system (14) and the slave system (16) are shown in Fig. 8. It is found that all the state variables tend to be synchronized in this proportional relationship. 4. Conclusions
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In this paper, we have investigated projective-lag synchronization scheme between two different coupled of general discrete-time chaotic systems with the same order. The main advantage of our work is that we achieve projective-lag synchronization in discrete-time chaotic systems. We overcome some limitations of previous works where
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projective-lag synchronization can only be achieved in continuous-time chaotic systems. Using Lyapunov stability theory, sufficient conditions are obtained to realize projective-lag synchronization between two different discrete-time chaotic systems. Numerical simulations illustrate the efficiency of our proposed scheme by synchronizing two different 2D discrete-time chaotic systems and two different 3D discrete-time chaotic systems. Projective-lag synchronization in discrete-time chaotic systems not only considers lags of transmitting signals to the receiving end in the signal transmission process, but also could extend binary digital to M-nary digital, achieving faster communication and facilitating digital simulation. Therefore, projective-lag 7
Journal Pre-proof synchronization in discrete-time chaotic systems opens up a new way for the application of projective synchronization in secure communication and cryptology in a practical point of view. In the present work, we concentrate on the realization of projective-lag
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synchronization in discrete-time chaotic systems with the same dimensions. Then, we will extend to projective-lag synchronization in discrete-time chaotic systems with different dimensions. We will concern about the effects of time-varying delays on the projective-lag synchronization in discrete-time chaotic systems as well. Furthermore, the influence of uncertainties and disturbances on the projective-lag synchronization in discrete-time chaotic systems would be expected to be investigated. These will be the topics of our future work and be anticipated to obtain more analytical results. Acknowledgements
This work is supported by the Foundation of Wuhan Textile University under Nos. 193128 and 183005. The authors thank the reviewers and editors for their constructive comments and suggestions on the completeness of this paper. References:
[1] H. Fang. Chaos in a single-species discrete population model with stage structure and birth pulses. Adv. Differ. Equ. 2014 (2014), p.175.
[2] A. Duncan, S. Liao, T. Vejchodský, R. Erban, R. Grima. Noise-induced
multistability in chemical systems: Discrete versus continuum modeling. Phys. Rev. E
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91 (2015), p.042111.
[3] Y. Oono, M. Kohmoto. Discrete model of chemical turbulence. Phys. Rev. Lett. 55 (1985), p.2927.
[4] A.P. Balachandran, L. Chandar. Discrete time from quantum physics. Nucl. Phys. B 428 (1994), pp. 435-448.
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[5] A. J. Zaslavski. Turnpike results for discrete-time optimal control systems arising in economic dynamics. Nonlinear Anal. Theory Methods Appl. 67 (2007), pp. 2024-2049. [6] S. Kassim, H. Hamiche, S. Djennoune, M. Bettayeb. A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems. Nonlinear Dyn. 88 (2017), pp. 2473-2489. [7] R. L. Filali, M. Benrejeb, P. Borne. On observer-based secure communication design using discrete-time hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), pp.1424-1432. 8
Journal Pre-proof [8] M. Lahdir, H. Hamiche, S. Kassim, M. Tahanout, K. Kemih, S. Addouche. A novel robust compression-encryption of images based on SPIHT coding and fractional-order discrete-time chaotic system. Opt. Laser Technol. 109 (2019), pp. 534-546. [9] G. Wu, D.Baleanu, H. Xie, F. Chen. Chaos synchronization of fractional chaotic
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maps based on the stability condition. Physica A 460 (2016), pp. 374-383.
[10] O. Megherbi, H. Hamiche, S. Djennoune, M. Bettayeb. A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems. Nonlinear Dyn. 90 (2017), pp. 1519-1533.
[11] B. Xin, Z. Wu. Projective synchronization of chaotic discrete dynamical systems via linear state error feedback control. Entropy 17 (2015), pp. 2677-2687.
[12] A.Ouannas, G. Grassi. Inverse full state hybrid projective synchronization for chaotic maps with different dimensions. Chin. Phys. B 25 (2016), p. 090503.
[13] A.Ouannas, Z.Odibat. Generalized synchronization of different dimensional chaotic dynamical systems in discrete time.
Nonlinear Dyn. 81(2015), pp.765-771.
[14] Z. Yan. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approach, Chaos 16 (2006), p.013119.
[15] C. Li, C. Xu, W. Sun, J. Xu, J. Kurths. Outer synchronization of coupled discrete-time networks. Chaos 19 (2009), p.013106.
[16] Z. Yan. A nonlinear control scheme to anticipated and complete synchronization in
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discrete-time chaotic (hyperchaotic) systems. Phys. Lett. A 343(2005), pp. 423-431. [17] J. Liang, Z. Wang, Y. Liu, X. Liu. Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks. IEEE Trans. Neural Networks 19 (2008), pp.1910-1921.
[18] T. M. Hoang, M. Nakagawa. A secure communication system using projective-lag
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and/or projective-anticipating synchronizations of coupled multidelay feedback systems. Chaos, Solitons Fractals 38 (2008), pp.1423-1438. [19] C. Chee, D. Xu. Secure digital communication using controlled projective synchronization of chaos. Chaos, Solitons Fractals 23 (2005), pp.1063-1070. [20] C. Huang, J. Cao. Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Physica A 473 (2017), pp. 262-275. [21] Z. Yan. A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems. Chaos 15 (2005), p.013101. 9
Journal Pre-proof [22] G. Li. Projective lag synchronization in chaotic systems. Chaos, Solitons Fractals 41(2009), pp.2630-2634. [23] Y.Chai, L. Q. Chen. Projective lag synchronization of spatiotemporal chaos via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 17 (2012),
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pp.3390-3398.
[24]C. Luo, X. Wang. Modified function projective lag synchronization in fractional-order chaotic (hyperchaotic) systems. J. Vib. Control 20 (2014), pp.1498-1511.
[25] G. Al-mahbashi, M. S. M. Noorani, S. A. Bakar, M. M. Al-sawalha. Robust projective lag synchronization in drive-response dynamical networks via adaptive control. Eur. Phys. J- Spec. Top. 225 (2016), pp. 51-64.
[26] M. Zheng, Z. Wang, L. Li, H. Peng, J. Xiao, Y. Yang, Y. Zhang, C. Feng. Finite-time generalized projective lag synchronization criteria for neutral-type neural networks with delay. Chaos, Solitons Fractals 107 (2018), pp.195-203.
[27] B. Zhang, H. Guo. Universal function projective lag synchronization of chaotic systems with uncertainty by using active sliding mode and fuzzy sliding mode control. Nonlinear Dyn. 81 (2015), pp.867-879.
[28] X. Wu, H. Lu. Projective lag synchronization of the general complex dynamical networks with distinct nodes. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 4417-4429.
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[29] M. M. El-Dessoky, M. A.Khan. Application of fractional calculus to combined modified function projective synchronization of different systems. Chaos 29 (2019), p.013107.
[30] R. He, P. G.Vaidya. Analysis and synthesis of synchronous periodic and chaotic systems. Phys. Rev. A 46 (1992), p.7387.
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[31] Ushio T. Chaotic synchronization and controlling chaos based on contraction mappings. Phys. Lett. A 198 (1995), pp.14-22. [32] Z. Yan. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic hyperchaotic system: a symbolic-numeric computation approach. Chaos 16 (2006), pp. 013111-013119. [33] Z. Yan. Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller. Phys. Lett. A 342 (2005), pp.309-317.
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Fig1. The chaotic attractor of Ushio system with 𝑐𝑐 = 1.9.
Fig 2. Chaotic attractor of Fold map with 𝑑𝑑 = −0.1 and 𝑠𝑠 = −1.7.
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Fig. 3 The time evolution for synchronized states: (a) x1 (k) of master system (8) and y1 (k) of and slave system (10) ; (b) x2 (k) of master system (8) and y2 (k) of and slave system (10) for α = 2 and 𝑚𝑚 = 10 when controller is switched on.
Fig. 4 The time evolution of error systems 𝑒𝑒1 and 𝑒𝑒2 between systems (8) and (10) when controller is switched on.
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Fig.5 Chaotic attractor of Generalized Hénon map with μ = 0.3 and ν = 1.07.
Fig.6 Chaotic attractor of Stefanski map with β = 1.4 and γ = 0.2.
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Fig.7 The time evolution for synchronized states: (a) x1 (k-m) of master system (14) and y1 (k) of and slave system (16) ; (b) x2 (k-m) of master system (14) and y2 (k) of and slave system (16) for α = 2 and 𝑚𝑚 = 10; (c) (b) x3 (k-m) of master system (14) and y3 (k) of and slave system (15) when controller is switched on.
Fig 8. The time evolution of error systems 𝑒𝑒1 , 𝑒𝑒2 and 𝑒𝑒3 between systems (14) and (16) when controller is switched on.
14
Journal Pre-proof • These works about projective-lag synchronization just concentrate on continuoustime chaotic systems. The possibility of the projective-lag synchronization scheme between two different coupled of general discrete-time chaotic systems has been discussed in this paper.
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• A nonlinear discrete-time controller is designed and its feasibility is analyzed by using Lyapunov stability theory, which has lower cost compared with continuous controllers.
Journal Pre-proof Declaration of interests
The authors declare that they have no known competing financial interests or
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personal relationships that could have appeared to influence the work reported in this
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paper.
PII: DOI: Reference:
S0020-7462(19)30625-0 https://doi.org/10.1016/j.ijnonlinmec.2020.103451 NLM 103451
To appear in:
International Journal of Non-Linear Mechanics
Received date : 2 September 2019 Revised date : 20 December 2019 Accepted date : 11 February 2020 Please cite this article as: C.-F. Feng and H.-J. Yang, Projective-lag synchronization scheme between two different discrete-time chaotic systems, International Journal of Non-Linear Mechanics (2020), doi: https://doi.org/10.1016/j.ijnonlinmec.2020.103451. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Journal Pre-proof Projective-lag synchronization scheme between two different discrete-time chaotic systems Cun-Fang Feng*a, Hai-Jun Yang*b a
School of Electronic and Electrical Engineering, Hubei Engineering and
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Technology Research Center for Functional Fiber Fabrication and Testing, Wuhan Textile University, Wuhan, China b
Department of Preventive Medicine, School of Basic Medical Sciences, Hubei
University of Chinese Medicine, Wuhan, China Abstract:
This
paper
addresses
theoretical
analysis
of
projective-lag
synchronization between two different discrete-time chaotic systems. We overcome some limitations of previous works where projective-lag synchronization can only be achieved in continuous-time chaotic systems. In this paper, we realize projective-lag synchronization in discrete-time chaotic systems with the same order. Based on the Lyapunov stability theory and a nonlinear control scheme, the controllers are designed by using the relevant variables of master and slave systems, and we find sufficient conditions for the stability of the error dynamics. The correctness of theoretical analysis is verified by 2D and 3D discrete-time chaotic systems, respectively.
Keywords: projective-lag synchronization; Lyapunov stability theory; nonlinear discrete-time systems
PACS: 05.45. Gg; 05.45. Pq; 05.45Tp
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1. Introduction
Discrete-time dynamical systems have drawn considerable attention over the past decades since practical systems, such as biological phenomena [1], chemical systems [2,3], physical systems [4] and economic systems [5] etc., are well defined using discrete-time dynamical systems. Considering the measurement is usually carried out at
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specific time intervals and digital simulations can be performed conveniently and quickly, chaos synchronization of discrete-time dynamical systems has attracted much attention due to its practical applications in secure communication and cryptology [6-8]. So, it might play an important role to achieve synchronization in discrete-time chaotic systems.
Recently, more and more attention has been paid to synchronization in discrete-time chaotic systems. Many types of synchronization have been presented for chaotic maps ———————————— *Corresponding authors:
Cun- Fang Feng, Email: [email protected]; Hai-Jun Yang, Email: [email protected]. 1
Journal Pre-proof such as complete synchronization [9], impulsive synchronization [10], projective synchronization [11], inverse full state hybrid projective synchronization [12], generalized synchronization [13], Q-S synchronization [14], outer synchronization [15], anticipating synchronization [16], robust synchronization [17] and so on. Among these
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different types of synchronization defined to date, an interesting synchronization scheme is represented by the so-called projective-lag synchronization [18], which includes projective synchronization and lag synchronizations. There is a proportional relationship between the state variables of the master system and slave system in projective synchronization. In application to secure communication, this feature can be used to extend binary digital to M-nary digital to achieve fast communication [19]. However, in real practical application, the system is inevitably affected by several external factors. One of these factors is that there is a certain time delay when the state variables of master and slave systems are synchronized. The lag is more reasonable in engineering applications because time delays are ubiquitous between communication channels in a practical point of view. The systems introduced with lags are infinite-dimensional systems, which have high randomness and unpredictability. So systems with lags improve the confidentiality and open up a new way for the application of projective synchronization in secure communication. It is much of importance to consider the slave system to synchronize the master system with lag τ when dealing with projective synchronization. From what has been discussed above, it
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is of great significance to study projective-lag synchronization. In this synchronization scheme, the slave system’s output 𝑦𝑦 lags behind the output of the master system 𝑥𝑥
proportionally after a transient time, i.e., 𝑦𝑦(𝑡𝑡) = 𝛼𝛼𝛼𝛼(𝑡𝑡 − 𝜏𝜏), where 𝜏𝜏 is positive real.
Projective-lag synchronization has received more and more attention just recently.
To the best of our knowledge, these works about projective-lag synchronization just
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concentrate on continuous-time chaotic dynamical systems. However, in addition to continuous-time evolution, systems can evolve on discrete intervals. In particular, in comparison with continuous ones, discrete controllers have lower cost. Therefore, research in discrete-time control has attracted more attention in recent years. As of nowadays, the possibility of projective-lag synchronization for discrete-time chaotic systems has not been discussed to the best of our knowledge. On the other hand, the active control as a simple and effective control method is widely utilize to control chaotic systems [20,21]. The main idea of the active control method is cancelling out 2
Journal Pre-proof the nonlinear terms through the feedback design. The system is transformed into a simple and easily controlled linear system and then the controller is designed by using the mature linear system theory. When the parameters are known, the active control method is simple and effective. There is no need to construct the Lyapunov function and
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achieve synchronization in a short time.
Motivated by abovementioned discussions, this paper aims to observe projective-lag synchronization in two different discrete-time chaotic systems with the identical order using active control method, which overcomes the limitations of previous works where projective-lag synchronization observed only in continuous-time chaotic systems [22-29]. The rest of this paper is arranged as follows. In section 2, based on the stability theory of discrete-time dynamical system, a suitable controller is designed using the relevant variables of master and slave systems and sufficient conditions of asymptotic stability of error dynamics are derived. In section 3, two numerical simulations are presented to verify the effectiveness and feasibility of the proposed synchronization method. Finally, some conclusions are drawn. 2. Analytical Results
Assuming there are two different discrete-time chaotic systems with the same order. The master system is given by
(1)
𝑋𝑋(𝑘𝑘 + 1) = 𝐴𝐴𝐴𝐴(𝑘𝑘) + 𝐹𝐹(𝑋𝑋(𝑘𝑘)),
and the controlled slave system is described as 𝑘𝑘 ∈ 𝑁𝑁 is
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where
𝑌𝑌(𝑘𝑘 + 1) = 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺(𝑌𝑌(𝑘𝑘) ) + U (𝑘𝑘), an
iteration
index;
(2)
𝑋𝑋(𝑘𝑘) = (𝑥𝑥1 (𝑘𝑘), … 𝑥𝑥𝑛𝑛 (𝑘𝑘))𝑇𝑇 ∈ 𝑅𝑅𝑛𝑛 , 𝑌𝑌(𝑘𝑘) =
(𝑦𝑦1 (𝑘𝑘), … 𝑦𝑦𝑛𝑛 (𝑘𝑘))𝑇𝑇 ∈ 𝑅𝑅𝑛𝑛 are the state space vectors of systems (1) and (2), respectively;
the real 𝐴𝐴𝑛𝑛×𝑛𝑛 and 𝐵𝐵𝑛𝑛×𝑛𝑛 matrices are known constant matrices with appropriate
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dimensions; 𝐹𝐹 ∈ 𝑅𝑅𝑛𝑛 → 𝑅𝑅𝑛𝑛 and G ∈ 𝑅𝑅𝑛𝑛 → 𝑅𝑅𝑛𝑛 contain the nonlinear parts and the
constant terms of the systems (1) and (2), respectively; and U is a suitable controller
to be designed for projective-lag synchronization between systems (1) and (2). The definition of projective-lag synchronization for systems (1) and (2) is given by Definition. Projective-lag synchronization is said to be achieved between the master system (1) and response system (2) if there exists an effective controller U such that synchronization error 𝑒𝑒(𝑘𝑘) = 𝑌𝑌(𝑘𝑘) − 𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚)
(3)
satisfies that lim ‖𝑒𝑒(𝑘𝑘)‖ → 0 for any initial conditions, where α (α ≠ 0) is scaling 𝑘𝑘→∞
3
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scaling
factor
Projective-lag synchronization becomes lag synchronization when α = 1;
projective-lag
synchronization
becomes
projective
synchronization when time delay 𝑚𝑚 = 0 ; projective-lag synchronization becomes
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complete synchronization when α = 1 and 𝑚𝑚 = 0.
According to Equation (3), the dynamics of synchronization error between systems
(1) and (2) can be derived as
𝑒𝑒(𝑘𝑘 + 1) = 𝑌𝑌(𝑘𝑘 + 1) − 𝛼𝛼𝛼𝛼(𝑘𝑘 + 1 − 𝑚𝑚)
= 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� + U (𝑘𝑘)
(4)
Our goal is to achieve projective-lag synchronization between master system (1) and slave system (2) via constructing an effective controller U , such that lim ‖𝑒𝑒(𝑘𝑘)‖ →
0. To this end, we design the controller U as follows: where
𝑘𝑘→∞
U (𝑘𝑘) = 𝐻𝐻�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� − 𝑅𝑅�𝑌𝑌(𝑘𝑘)�,
(5)
𝐻𝐻�𝑋𝑋(𝑘𝑘)� = 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘) − 𝛼𝛼Β𝑋𝑋(𝑘𝑘) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘) + 𝛼𝛼𝛼𝛼(𝑋𝑋(𝑘𝑘)), 𝑅𝑅�𝑌𝑌(𝑘𝑘)� = 𝐺𝐺�𝑌𝑌(𝑘𝑘)� + 𝐿𝐿𝐿𝐿(𝑘𝑘),
and L is the undetermined constant control matrix. Hence, we have the following result.
Theorem. If the control matrix 𝐿𝐿 satisfies 𝐼𝐼 − (𝐵𝐵 − 𝐿𝐿)𝑇𝑇 (𝐵𝐵 − 𝐿𝐿) , which is a
positive definite matrix, then the master system (1) and response system (2) are globally Proof. written as
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projective-lag synchronization under the control law (5).
By substituting Eq. (5) into Eq. (4), the synchronization error can be
𝑒𝑒(𝑘𝑘 + 1) = 𝐵𝐵𝐵𝐵(𝑘𝑘) + 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚)
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+𝛼𝛼𝛼𝛼�𝑋𝑋(𝑘𝑘 − 𝑚𝑚)� − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝐺𝐺�𝑌𝑌(𝑘𝑘)� − 𝐿𝐿𝐿𝐿(𝑘𝑘)
= 𝐵𝐵𝐵𝐵(𝑘𝑘) − 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) + 𝛼𝛼𝛼𝛼𝛼𝛼(𝑘𝑘 − 𝑚𝑚) − 𝐿𝐿𝐿𝐿(𝑘𝑘)
(6)
= (𝐵𝐵 − 𝐿𝐿)𝑒𝑒(𝑘𝑘)
Consider a Lyapunov function in the form 𝑉𝑉�𝑒𝑒(𝑘𝑘)� = 𝑒𝑒(𝑘𝑘)𝑇𝑇 𝑒𝑒(𝑘𝑘), then we obtain:
𝛥𝛥𝑉𝑉�𝑒𝑒(𝑘𝑘)� = 𝑉𝑉�𝑒𝑒(𝑘𝑘 + 1)� − 𝑉𝑉(𝑒𝑒(𝑘𝑘))
= 𝑒𝑒 𝑇𝑇 (𝑘𝑘)[(𝐵𝐵 − 𝐿𝐿)𝑇𝑇 (𝐵𝐵 − 𝐿𝐿) − 𝐼𝐼]𝑒𝑒(𝑘𝑘) < 0
(7)
From the Lyapunov stability theory [30], it is immediate that lim ‖𝑒𝑒(𝑘𝑘)‖ → 0, 𝑘𝑘→∞
which implies that the origin of the error system (6) is asymptotically stable, so systems (1) and (2) achieve projective-lag synchronization. 4
Journal Pre-proof This completes the proof. Remark 2. The value of scaling factor α has no effect on the error dynamics of
projective-lag synchronization [Eq. (6)], so we can arbitrarily direct the scaling factor α onto any desired value.
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3. Numerical simulation results
To demonstrate the effectiveness of the proposed synchronization schemes, we provide two illustrative examples in this section.
3.1. Two different 2D discrete-time chaotic systems
We achieve the projective-lag synchronization between two 2D different discrete-time systems. The master system is described by the Ushio system [31] 𝑥𝑥 (𝑘𝑘 + 1) = 𝑐𝑐𝑥𝑥1 (𝑘𝑘) − 𝑥𝑥13 (𝑘𝑘) + 𝑥𝑥2 (𝑘𝑘) � 1 𝑥𝑥2 (𝑘𝑘 + 1) = 0.5𝑥𝑥1 (𝑘𝑘).
(8)
Note that it can be expressed into Equation (1), where 𝑐𝑐 0.5
𝐴𝐴 = �
3 1 � and 𝐹𝐹�𝑋𝑋(𝑘𝑘)� = �−𝑥𝑥1 (𝑘𝑘)�. 0 0
(9)
It is well known that Ushio system has a chaotic attractor when 𝑐𝑐 = 1.9. The chaotic
attractor of Ushio system is shown in Fig.1.
The controlled slave system is described as Fold map [32] 𝑦𝑦 (𝑘𝑘 + 1) = 𝑑𝑑𝑦𝑦1 (𝑘𝑘) + 𝑦𝑦2 (𝑘𝑘) + 𝑢𝑢1 (𝑘𝑘) � 1 𝑦𝑦2 (𝑘𝑘 + 1) = 𝑦𝑦12 (𝑘𝑘) + 𝑠𝑠 + 𝑢𝑢2 (𝑘𝑘),
(10)
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where 𝑑𝑑 = −0.1, 𝑠𝑠 = −1.7 are systems constants and U (𝑘𝑘) = (𝑢𝑢1 (𝑘𝑘), 𝑢𝑢2 (𝑘𝑘))𝑇𝑇 is the
vector controller designed by Eq. (5) for achieving projective-lag synchronization. The chaotic attractor of system (10) with 𝑑𝑑 = −0.1, 𝑠𝑠 = −1.7, 𝑢𝑢1 = 0 and 𝑢𝑢2 = 0 is
displayed in Fig. 2. Similarly, the matrix B and the function G of system (10) are given by
0 1 � and 𝐺𝐺�𝑌𝑌(𝑘𝑘)� = � 2 (𝑘𝑘) � . 𝑦𝑦1 + 𝑠𝑠 0
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𝑑𝑑 𝐵𝐵 = � 0
(11)
According to Eq.(5), we take the controllers 𝑢𝑢1 and 𝑢𝑢2 as �
𝑢𝑢1 (𝑘𝑘) 𝑐𝑐 − 𝑑𝑑 + 𝑙𝑙11 � = 𝛼𝛼 � 0.5 + 𝑙𝑙21 𝑢𝑢2 (𝑘𝑘) −�
0
𝑦𝑦12 (𝑘𝑘)
+ 𝑠𝑠
3 (𝑘𝑘 𝑙𝑙12 𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚) �� � + 𝛼𝛼 �−𝑥𝑥1 − 𝑚𝑚)� 𝑙𝑙22 𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚) 0
�−�
𝑙𝑙11 𝑙𝑙21
𝑙𝑙12 𝑦𝑦1 (𝑘𝑘) �� � 𝑙𝑙22 𝑦𝑦2 (𝑘𝑘)
(12)
Thus, the error dynamics between systems (8) and (10) is described by �
𝑒𝑒1 (𝑘𝑘 + 1) −0.1 − 𝑙𝑙11 �=� −𝑙𝑙21 𝑒𝑒2 (𝑘𝑘 + 1)
1 − 𝑙𝑙12 𝑒𝑒1 (𝑘𝑘) �� � −𝑙𝑙22 𝑒𝑒2 (𝑘𝑘)
5
(13)
Journal Pre-proof 0.4 1 �, then the sufficient condition given 0 0.3
If we consider the control matrix 𝐿𝐿 = �
by Theorem is satisfied. In Fig. 3, the slave system 𝑦𝑦1 lags the state of the master system 𝑥𝑥1 with constant lag time 𝑚𝑚 = 10 and the amplitude of the master’s and
𝑇𝑇
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slave’s state variables is correlated by (𝑦𝑦1 (𝑘𝑘), 𝑦𝑦2 (𝑘𝑘))𝑇𝑇 = 2�𝑥𝑥1 (𝑘𝑘 − 10), 𝑥𝑥2 (𝑘𝑘 − 10)� . The time evolution of error systems 𝑒𝑒1 (𝑘𝑘) and 𝑒𝑒2 (𝑘𝑘) are shown in Fig.4 (a) and (b),
respectively. We can see that the synchronization errors converge to zero as time tends to infinity, which implies that projective-lag synchronization between systems (8) and (10) is achieved.
3.2. Two different 3D discrete-time chaotic systems
In the following, we consider the 3D hyperchaotic Hénon system as the drive
system and the controlled 3D Stefanski map as the slave system. The master system is described in [33]
𝑥𝑥1 (𝑘𝑘 + 1) = −𝜇𝜇 ∗ 𝑥𝑥2 (𝑘𝑘) �𝑥𝑥2 (𝑘𝑘 + 1) = 𝑥𝑥3 (𝑘𝑘) + 1 − 𝜐𝜐 ∗ 𝑥𝑥22 (𝑘𝑘) 𝑥𝑥3 (𝑘𝑘 + 1) = 𝜇𝜇 ∗ 𝑥𝑥2 (𝑘𝑘) + 𝑥𝑥1 (𝑘𝑘),
(14)
It can be expressed into the Eq. (1) form, where 0 𝐴𝐴 = �0 1
−𝜇𝜇 0 𝜇𝜇
0 0 1�, 𝐹𝐹�𝑋𝑋(𝑘𝑘)� = �1 − 𝜈𝜈𝑥𝑥22 (𝑘𝑘)�. 0 0
(15)
The hyperchaotic 3D generalized Henon has a chaotic attractor with (μ, ν) = (0.3,
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1.07), which is shown in Fig. 5.
The slave system is given in [33]
𝑦𝑦1 (𝑘𝑘 + 1) = 1 + 𝑦𝑦3 (𝑘𝑘) − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) + 𝑢𝑢1 (𝑘𝑘) �𝑦𝑦2 (𝑘𝑘 + 1) = 1 + 𝛾𝛾𝑦𝑦2 (𝑘𝑘) − 𝛽𝛽𝑦𝑦12 (𝑘𝑘) + 𝑢𝑢2 (𝑘𝑘) 𝑦𝑦3 (𝑘𝑘 + 1) = 𝛾𝛾𝑦𝑦1 (𝑘𝑘) + 𝑢𝑢3 (𝑘𝑘)
(16)
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It can be expressed into the Eq. (2), where 0 𝐵𝐵 = �0 𝛾𝛾
0 𝛾𝛾 0
1 0�, 0
1 − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) 𝐺𝐺�𝑌𝑌(𝑘𝑘)� = �1 − 𝛽𝛽𝑦𝑦12 (𝑘𝑘)� 0
(17)
where 𝑢𝑢(𝑘𝑘) = [𝑢𝑢1 (𝑘𝑘), 𝑢𝑢2 (𝑘𝑘), 𝑢𝑢3 (𝑘𝑘)]𝑇𝑇 is the control vector to be designed later for
achieving projective-lag synchronization. The hyperchaotic Stefanski map has a chaotic attractor when (𝛽𝛽, 𝛾𝛾) = (1.4, 0.2). The chaotic attractors of system (16) is shown in Fig. 6.
According to Eq.(5), we take the controllers 𝑢𝑢1 , 𝑢𝑢2 and 𝑢𝑢3 as 6
Journal Pre-proof 𝑢𝑢1 (𝑘𝑘) 𝑙𝑙11 𝑙𝑙21 �𝑢𝑢2 (𝑘𝑘)� = 𝛼𝛼 � 1 − 𝛾𝛾 + 𝑙𝑙31 𝑢𝑢3 (𝑘𝑘)
−𝜇𝜇 + 𝑙𝑙12 −𝛾𝛾 + 𝑙𝑙22 𝜇𝜇 + 𝑙𝑙32
𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚) −1 + 𝑙𝑙13 1 + 𝑙𝑙23 � �𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚)� 𝑙𝑙33 𝑥𝑥3 (𝑘𝑘 − 𝑚𝑚) 𝑙𝑙12 𝑙𝑙22 𝑙𝑙32
𝑦𝑦1 (𝑘𝑘) 𝑙𝑙13 𝑙𝑙23 � �𝑦𝑦2 (𝑘𝑘)� 𝑙𝑙33 𝑦𝑦3 (𝑘𝑘)
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𝑙𝑙11 1 − 𝛽𝛽𝑦𝑦22 (𝑘𝑘) 0 2 +𝛼𝛼 �1 − 𝜈𝜈𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚)� − �1 − 𝛽𝛽𝑦𝑦12 (𝑘𝑘)� − �𝑙𝑙21 𝑙𝑙31 0 0
(18)
Thus, the error dynamics between systems (14) and (16) is described by 𝑒𝑒1 (𝑘𝑘 + 1) −𝑙𝑙11 𝑒𝑒 (𝑘𝑘 + 1) −𝑙𝑙 � 2 �=� 21 𝛾𝛾 − 𝑙𝑙31 𝑒𝑒3 (𝑘𝑘 + 1)
−𝑙𝑙12 𝛾𝛾 − 𝑙𝑙22 −𝑙𝑙32
−0.3 If we consider the control matrix 𝐿𝐿 = � 0 𝛾𝛾
𝑒𝑒1 (𝑘𝑘) 1 − 𝑙𝑙13 −𝑙𝑙23 � �𝑒𝑒2 (𝑘𝑘)� −𝑙𝑙33 𝑒𝑒3 (𝑘𝑘)
(19)
0 1 0.5 0 �, then the sufficient 0 −0.2
condition given by Theorem is satisfied. In Fig.7, 𝑥𝑥(𝑘𝑘 − 𝑚𝑚) instead of 𝑥𝑥(𝑘𝑘) is
convenient for our study. Fig. 7 shows time series of the master system 𝑥𝑥(𝑘𝑘 − 𝑚𝑚) and the slave system 𝑦𝑦(𝑘𝑘) for α = 2.0. The response system 𝑦𝑦 lags the state of the
master system 𝑥𝑥 with 𝑚𝑚 = 10 and the amplitude of the master’s and slave’s state
variables is correlated by (𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 )𝑇𝑇 = 2(𝑥𝑥1 (𝑘𝑘 − 𝑚𝑚), 𝑥𝑥2 (𝑘𝑘 − 𝑚𝑚), 𝑥𝑥3 (𝑘𝑘 − 𝑚𝑚))𝑇𝑇 . The
errors dynamics between the master system (14) and the slave system (16) are shown in Fig. 8. It is found that all the state variables tend to be synchronized in this proportional relationship. 4. Conclusions
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In this paper, we have investigated projective-lag synchronization scheme between two different coupled of general discrete-time chaotic systems with the same order. The main advantage of our work is that we achieve projective-lag synchronization in discrete-time chaotic systems. We overcome some limitations of previous works where
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projective-lag synchronization can only be achieved in continuous-time chaotic systems. Using Lyapunov stability theory, sufficient conditions are obtained to realize projective-lag synchronization between two different discrete-time chaotic systems. Numerical simulations illustrate the efficiency of our proposed scheme by synchronizing two different 2D discrete-time chaotic systems and two different 3D discrete-time chaotic systems. Projective-lag synchronization in discrete-time chaotic systems not only considers lags of transmitting signals to the receiving end in the signal transmission process, but also could extend binary digital to M-nary digital, achieving faster communication and facilitating digital simulation. Therefore, projective-lag 7
Journal Pre-proof synchronization in discrete-time chaotic systems opens up a new way for the application of projective synchronization in secure communication and cryptology in a practical point of view. In the present work, we concentrate on the realization of projective-lag
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synchronization in discrete-time chaotic systems with the same dimensions. Then, we will extend to projective-lag synchronization in discrete-time chaotic systems with different dimensions. We will concern about the effects of time-varying delays on the projective-lag synchronization in discrete-time chaotic systems as well. Furthermore, the influence of uncertainties and disturbances on the projective-lag synchronization in discrete-time chaotic systems would be expected to be investigated. These will be the topics of our future work and be anticipated to obtain more analytical results. Acknowledgements
This work is supported by the Foundation of Wuhan Textile University under Nos. 193128 and 183005. The authors thank the reviewers and editors for their constructive comments and suggestions on the completeness of this paper. References:
[1] H. Fang. Chaos in a single-species discrete population model with stage structure and birth pulses. Adv. Differ. Equ. 2014 (2014), p.175.
[2] A. Duncan, S. Liao, T. Vejchodský, R. Erban, R. Grima. Noise-induced
multistability in chemical systems: Discrete versus continuum modeling. Phys. Rev. E
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91 (2015), p.042111.
[3] Y. Oono, M. Kohmoto. Discrete model of chemical turbulence. Phys. Rev. Lett. 55 (1985), p.2927.
[4] A.P. Balachandran, L. Chandar. Discrete time from quantum physics. Nucl. Phys. B 428 (1994), pp. 435-448.
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[5] A. J. Zaslavski. Turnpike results for discrete-time optimal control systems arising in economic dynamics. Nonlinear Anal. Theory Methods Appl. 67 (2007), pp. 2024-2049. [6] S. Kassim, H. Hamiche, S. Djennoune, M. Bettayeb. A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems. Nonlinear Dyn. 88 (2017), pp. 2473-2489. [7] R. L. Filali, M. Benrejeb, P. Borne. On observer-based secure communication design using discrete-time hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), pp.1424-1432. 8
Journal Pre-proof [8] M. Lahdir, H. Hamiche, S. Kassim, M. Tahanout, K. Kemih, S. Addouche. A novel robust compression-encryption of images based on SPIHT coding and fractional-order discrete-time chaotic system. Opt. Laser Technol. 109 (2019), pp. 534-546. [9] G. Wu, D.Baleanu, H. Xie, F. Chen. Chaos synchronization of fractional chaotic
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maps based on the stability condition. Physica A 460 (2016), pp. 374-383.
[10] O. Megherbi, H. Hamiche, S. Djennoune, M. Bettayeb. A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems. Nonlinear Dyn. 90 (2017), pp. 1519-1533.
[11] B. Xin, Z. Wu. Projective synchronization of chaotic discrete dynamical systems via linear state error feedback control. Entropy 17 (2015), pp. 2677-2687.
[12] A.Ouannas, G. Grassi. Inverse full state hybrid projective synchronization for chaotic maps with different dimensions. Chin. Phys. B 25 (2016), p. 090503.
[13] A.Ouannas, Z.Odibat. Generalized synchronization of different dimensional chaotic dynamical systems in discrete time.
Nonlinear Dyn. 81(2015), pp.765-771.
[14] Z. Yan. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approach, Chaos 16 (2006), p.013119.
[15] C. Li, C. Xu, W. Sun, J. Xu, J. Kurths. Outer synchronization of coupled discrete-time networks. Chaos 19 (2009), p.013106.
[16] Z. Yan. A nonlinear control scheme to anticipated and complete synchronization in
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discrete-time chaotic (hyperchaotic) systems. Phys. Lett. A 343(2005), pp. 423-431. [17] J. Liang, Z. Wang, Y. Liu, X. Liu. Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks. IEEE Trans. Neural Networks 19 (2008), pp.1910-1921.
[18] T. M. Hoang, M. Nakagawa. A secure communication system using projective-lag
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and/or projective-anticipating synchronizations of coupled multidelay feedback systems. Chaos, Solitons Fractals 38 (2008), pp.1423-1438. [19] C. Chee, D. Xu. Secure digital communication using controlled projective synchronization of chaos. Chaos, Solitons Fractals 23 (2005), pp.1063-1070. [20] C. Huang, J. Cao. Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Physica A 473 (2017), pp. 262-275. [21] Z. Yan. A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems. Chaos 15 (2005), p.013101. 9
Journal Pre-proof [22] G. Li. Projective lag synchronization in chaotic systems. Chaos, Solitons Fractals 41(2009), pp.2630-2634. [23] Y.Chai, L. Q. Chen. Projective lag synchronization of spatiotemporal chaos via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 17 (2012),
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pp.3390-3398.
[24]C. Luo, X. Wang. Modified function projective lag synchronization in fractional-order chaotic (hyperchaotic) systems. J. Vib. Control 20 (2014), pp.1498-1511.
[25] G. Al-mahbashi, M. S. M. Noorani, S. A. Bakar, M. M. Al-sawalha. Robust projective lag synchronization in drive-response dynamical networks via adaptive control. Eur. Phys. J- Spec. Top. 225 (2016), pp. 51-64.
[26] M. Zheng, Z. Wang, L. Li, H. Peng, J. Xiao, Y. Yang, Y. Zhang, C. Feng. Finite-time generalized projective lag synchronization criteria for neutral-type neural networks with delay. Chaos, Solitons Fractals 107 (2018), pp.195-203.
[27] B. Zhang, H. Guo. Universal function projective lag synchronization of chaotic systems with uncertainty by using active sliding mode and fuzzy sliding mode control. Nonlinear Dyn. 81 (2015), pp.867-879.
[28] X. Wu, H. Lu. Projective lag synchronization of the general complex dynamical networks with distinct nodes. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 4417-4429.
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[29] M. M. El-Dessoky, M. A.Khan. Application of fractional calculus to combined modified function projective synchronization of different systems. Chaos 29 (2019), p.013107.
[30] R. He, P. G.Vaidya. Analysis and synthesis of synchronous periodic and chaotic systems. Phys. Rev. A 46 (1992), p.7387.
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[31] Ushio T. Chaotic synchronization and controlling chaos based on contraction mappings. Phys. Lett. A 198 (1995), pp.14-22. [32] Z. Yan. Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic hyperchaotic system: a symbolic-numeric computation approach. Chaos 16 (2006), pp. 013111-013119. [33] Z. Yan. Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller. Phys. Lett. A 342 (2005), pp.309-317.
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Fig1. The chaotic attractor of Ushio system with 𝑐𝑐 = 1.9.
Fig 2. Chaotic attractor of Fold map with 𝑑𝑑 = −0.1 and 𝑠𝑠 = −1.7.
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Fig. 3 The time evolution for synchronized states: (a) x1 (k) of master system (8) and y1 (k) of and slave system (10) ; (b) x2 (k) of master system (8) and y2 (k) of and slave system (10) for α = 2 and 𝑚𝑚 = 10 when controller is switched on.
Fig. 4 The time evolution of error systems 𝑒𝑒1 and 𝑒𝑒2 between systems (8) and (10) when controller is switched on.
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Fig.5 Chaotic attractor of Generalized Hénon map with μ = 0.3 and ν = 1.07.
Fig.6 Chaotic attractor of Stefanski map with β = 1.4 and γ = 0.2.
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Fig.7 The time evolution for synchronized states: (a) x1 (k-m) of master system (14) and y1 (k) of and slave system (16) ; (b) x2 (k-m) of master system (14) and y2 (k) of and slave system (16) for α = 2 and 𝑚𝑚 = 10; (c) (b) x3 (k-m) of master system (14) and y3 (k) of and slave system (15) when controller is switched on.
Fig 8. The time evolution of error systems 𝑒𝑒1 , 𝑒𝑒2 and 𝑒𝑒3 between systems (14) and (16) when controller is switched on.
14
Journal Pre-proof • These works about projective-lag synchronization just concentrate on continuoustime chaotic systems. The possibility of the projective-lag synchronization scheme between two different coupled of general discrete-time chaotic systems has been discussed in this paper.
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• A nonlinear discrete-time controller is designed and its feasibility is analyzed by using Lyapunov stability theory, which has lower cost compared with continuous controllers.
Journal Pre-proof Declaration of interests
The authors declare that they have no known competing financial interests or
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personal relationships that could have appeared to influence the work reported in this
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paper.