Many researchers have been dealing with synchronizing chaotic dynamical systems extensively [1] [2] [3]. Due to potential applications, many types of chaos synchronization in dynamical systems such as complete, phase, lag, cluster, and anticipated synchronization, etc [4] [5] [6] [7] [8] are widely investigated. Especially, amongst all kinds of chaos synchronization, the projective synchronization reported by Mainieri and Rehacek is one of the most noticeable ones that the drive and response vectors evolve proportionally in a scale matirx [9]. The projective synchronization is usually reported only in partial-linearity systems early. Subsequently, the projective synchronization is extended to non-partially-linear systems [10] - [15].

Because of much more complicated structure and higher unpredictability, hyperchaotic systems have been broadly applied in secure communications [16], lasers [17], optimal control [18], and so on. Many researchers investigate chaos (hyperchaotic) synchronization in continuous-time systems using different methods. However, many mathematical models are defined with discrete-time dynamical systems [19] [20] [21] [22]. Therefore, more and more attention is paid to the synchronization and control in discrete-time chaotic systems, especially in discrete-time hyperchaotic systems.

Backstepping design method [23] [24] plays a very important role in constructing the associated Lyapunov functions and feedback controllers. In this paper, we investigate the function projective synchronization (FPS) by exploring a systematic and automatic algorithm [22], by which the discrete-time drive system and response system, whether is with strict-feedback form or not, can be projectively synchronized via suitable controllers. By means of symbolic-numeric computation, the proposed scheme is used to realize FPS of 3D discrete-time hyperchaotic systems between two identical Rösler systems [25], and two different systems of Rösler system and the Henon system [26], respectively. Moreover, numerical simulations are given to verify the availability of the proposed scheme.

The rest of this paper is organized as follows. In Section 2, the definition of FPS in discrete-time hyperchaotic systems and the Lyapunov stability theory are introduced. In Section 3, we first illustrate the general theory of FPS in two identical Rössler hyperchaotic discrete-time systems, and then give the numerical simulation of the associated results. We also discuss the FPS of the two different discrete-time hyperchaotic systems (the Henon hyperchaotic system and the Rössler hyperchaotic system) in Section 4. Finally, some conclusions and discussions are given in Section 5.

In this section, we give the conception of FPS in discrete-time hyperchaotic dynamical systems as we defined earlier [15].

Consider the two hyperchaotic systems in discrete-time style, which are described as follows: the drive system (a) X ( k + 1 ) = F ( X ( k ) ) , and the response system with controllers (b) Y ( k + 1 ) = G ( Y ( k ) ) + u ( X ( k ) , Y ( k ) ) . Where X ( k ) = ( x 1 ( k ) , x 2 ( k ) , x 3 ( k ) ) , Y ( k ) = ( y 1 ( k ) , y 2 ( k ) , y 3 ( k ) ) , k ∈ Z / Z − , u ( X ( k ) , Y ( k ) ) ∈ R 3 . Additionally, (c) the error system

E ( k ) = ( E 1 ( k ) , E 2 ( k ) , E 3 ( k ) ) = ( x 1 ( k ) − f 1 ( X ( k ) ) y 1 ( k ) , x 2 ( k ) − f 2 ( X ( k ) ) y 2 ( k ) , x 3 ( k ) − f 3 ( X ( k ) ) y 3 ( k ) ) . If there exist suitable controllers

u ( x ( k ) , y ( k ) ) = ( u 1 ( X ( k ) , Y ( k ) ) , u 2 ( X ( k ) , Y ( k ) ) , u 3 ( X ( k ) , Y ( k ) ) ) , satisfying lim k → ∞ ( E ( k ) ) = 0 , one can say that there exists function projective synchronization (FPS) in the above drive (a) and response systems (b).

Furthermore, consider the error discrete-time (c) generated by the drive system (a) and the response system (b). Let L ( E 1 ( k ) , ⋯ , E 4 ( k ) ) | E i ( k ) ≡ 0 ( i = 1 , 2 , 3 ) = 0 , when Δ L ( k ) = L ( k + 1 ) − L ( k ) ≤ 0 , with the equality holding if and only if E i ( k ) ≡ 0 ( i = 1 , 2 , 3 ) , we can say that systems (a) and (b) are function projective synchronized, according to the Lyapunov stability theory.

Here we would like to point out that the controller u desponds on the synchronization method chosen. In fact, when E i ( k ) ≡ 0 ( i = 1 , 2 , 3 ) , u = ( f − 1 ) F − G , where f = d i a g ( f 1 ( x ( k ) ) , f 2 ( x ( k ) ) , f 3 ( x ( k ) ) ) . That is to say, u = ( f − 1 ) F − G is the situation when all the error functions equal to zero and the corresponding controller is trivial situation. For E i ( k ) = 0 , we need only to solve the equations

( E 1 ( k ) , E 2 ( k ) , E 3 ( k ) ) = ( x 1 ( k ) − f 1 ( X ( k ) ) y 1 ( k ) , x 2 ( k ) − f 2 ( X ( k ) ) y 2 ( k ) , x 3 ( k ) − f 3 ( X ( k ) ) y 3 ( k ) ) = ( 0 , 0 , 0 )

to obtain the trivial controller “u”. Therefore, here we just regard the general condition lim k → ∞ ( E ( k ) ) = 0 .

In this paper, we would like to propose a systematic and constructive scheme to search the controllers between 3D hyperchaotic discrete-time systems [25] [26] with strict-feed form are function projective synchronized.

In this section, we consider the FPS of two identical Rössler hyperchaotic systems [25]. The discrete-time drive and response systems are described as following:

x 1 ( k + 1 ) = a 3 δ x 2 ( k ) + ( a 4 δ + 1 ) x 1 ( k ) , x 2 ( k + 1 ) = a 2 δ x 3 ( k ) + a 1 δ x 1 ( k ) + x 2 ( k ) , x 3 ( k + 1 ) = a 5 δ + a 6 δ x 2 ( k ) x 3 ( k ) + ( a 7 δ + 1 ) x 3 ( k ) , (1)

and

y 1 ( k + 1 ) = a 3 δ y 2 ( k ) + ( a 4 δ + 1 ) y 1 ( k ) + u 1 , y 2 ( k + 1 ) = a 2 δ y 3 ( k ) + a 1 δ y 1 ( k ) + y 2 ( k ) + u 2 , y 3 ( k + 1 ) = a 5 δ + a 6 δ y 2 ( k ) y 3 ( k ) + ( a 7 δ + 1 ) y 3 ( k ) + u 3 . (2)

As we all know, the dynamic system will have different dynamic behavior when choosing different parameter values. When a 1 = − 1.9 , a 2 = 0.2 , a 3 = 0.5 , a 4 = − 2.3 , a 5 = 2 , a 6 = − 0.6 , a 7 = − 1.9 and δ = 1 , hyperchaos occurs in both the systems (1) and (2) without the controllers. In this section, we choose the values of the parameters in the systems (1) and (2) as the above values. The synchronization process for the above discrete-time dynamical systems, with the powerful Lyapunov stability theory and backstepping design method is introduced in detail as follows. We select ( f 1 ( x ) , f 2 ( x ) , f 3 ( x ) ) = ( 2,1 + tanh ( x 1 ( k ) ) , − 2 ) . So the error states should be E 1 ( k ) = x 1 ( k ) − 2 y 1 ( k ) , E 2 ( k ) = x 2 ( k ) − ( 1 + tanh ( x 1 ( k ) ) ) y 2 ( k ) , E 3 ( k ) = x 3 ( k ) − x 2 ( k ) y 3 ( k ) . Substituting (1) and (2) into the above error states, we can obtain the discrete-time error dynamical system

E 1 ( k + 1 ) = a 3 δ x 2 ( k ) + ( a 4 δ + 1 ) x 1 ( k ) − 2 a 3 δ y 2 ( k )                               − 2 ( a 4 δ + 1 ) y 1 ( k ) − 2 u 1 ( x , y ) , E 2 ( k + 1 ) = − ( 1 + tanh ( a 3 δ x 2 ( k ) + ( a 4 δ + 1 ) x 1 ( k ) ) ) ( a 2 δ y 3 ( k ) + a 1 δ y 1 ( k )                               + y 2 ( k ) + u 2 ( x , y ) ) + a 2 δ x 3 ( k ) + a 1 δ x 1 ( k ) + x 2 ( k ) , E 3 ( k + 1 ) = a 6 δ x 2 ( k ) x 3 ( k ) + ( a 7 δ + 1 ) x 3 ( k ) + 2 a 6 δ y 2 ( k ) y 3 ( k )                               + 2 ( a 7 δ + 1 ) y 3 ( k ) + 3 a 5 δ + 2 u 3 ( x , y ) . (3)

We now consider FPS between the Henon-like map [26] and the above hyperchaotic system (1). Here we choose Henon-like map as drive system and hyperchaotic Rössler system as response system to realize FPS of two different chaotic dynamic systems with the backstepping method. The drive system and the response system with controllers are rewritten as follows: the drive system

x 1 ( k + 1 ) = 1 + x 3 ( k ) − α x 2 2 ( k ) , x 2 ( k + 1 ) = 1 + β x 2 ( k ) − α x 1 2 ( k ) , x 3 ( k + 1 ) = β x 1 ( k ) . (12)

and the response system

y 1 ( k + 1 ) = a 3 δ y 2 ( k ) + ( a 4 δ + 1 ) y 1 ( k ) + u 1 , y 2 ( k + 1 ) = a 2 δ y 3 ( k ) + a 1 δ y 1 ( k ) + y 2 ( k ) + u 2 , y 3 ( k + 1 ) = a 5 δ + a 6 δ y 2 ( k ) y 3 ( k ) + ( a 7 δ + 1 ) y 3 ( k ) + u 3 . (13)

The projections of the hyperchaotic attrator of systems (12) and (1) are displayed in Figure 4(a) and Figure 4(b), respectively.

Then we also use the backstepping design method to realize the FPS of the two different discrete-time hyperchaotic systems (12) and (1). Here we choose ( f 1 ( x ) , f 2 ( x ) , f 3 ( x ) ) = ( − 1 , − 1 1 + x 1 2 ( k ) , − 1 ) , that is to say

E 1 ( k ) = x 1 ( k ) + y 1 ( k ) , E 2 ( k ) = x 2 ( k ) + 1 1 + x 1 2 ( k ) y 2 ( k ) , E 3 ( k ) = x 3 ( k ) + y 3 ( k ) . (14)

According to (12) and (13), we can obtain the error dynamical system

E 1 ( k + 1 ) = − α x 2 2 ( k ) + x 3 ( k ) + 1 + a 3 δ y 2 ( k ) + ( a 4 δ + 1 ) y 1 ( k ) + u 1 , E 2 ( k + 1 ) = − α x 1 2 ( k ) + β x 2 ( k ) + 1 + a 1 δ y 1 ( k ) + a 2 δ y 3 ( k ) + u 2 + y 2 ( k ) ( − α x 2 ( k ) 2 + x 3 2 ( k ) + 1 ) 2 + 1 , E 3 ( k + 1 ) = β x 1 ( k ) + a 5 δ + a 6 δ y 2 ( k ) y 3 ( k ) + ( a 7 δ + 1 ) y 3 ( k ) + u 3 . (15)

Based on the steps of backstepping methods [27] with the above Equations (4) to (8), we have the controllers

u 1 = − ( a 3 δ x 1 2 ( k ) y 2 ( k ) + a 4 δ x 1 2 ( k ) y 1 ( k ) − α x 1 2 ( k ) x 2 2 ( k ) ) + y 1 ( k ) − y 2 ( k )             + a 4 δ y 1 ( k ) − α x 2 2 ( k ) − x 1 2 ( k ) x 2 ( k ) + x 1 2 ( k ) x 3 ( k ) + x 1 2 ( k ) y 1 ( k ) + x 1 2 ( k )             + a 3 δ y 2 ( k ) − x 2 ( k ) + x 3 ( k ) − c 11 x 1 ( k ) − c 11 y 1 ( k ) − c 11 x 1 2 ( k ) y 1 ( k + 1 )             − c 11 x 1 3 ( k ) ) / ( x 1 2 ( k ) + 1 ) ,

u 2 = − ( 2 − 2 α x 1 2 ( k ) x 2 2 ( k ) + a 2 δ y 3 ( k ) − c 21 x 1 3 ( k ) x 3 2 ( k ) − 2 c 21 x 1 3 ( k ) x 3 ( k )             − 2 c 21 x 1 2 ( k ) y 1 ( k ) − c 21 x 1 ( k ) x 3 2 ( k ) − c 21 x 3 2 ( k ) y 1 ( k ) − 2 c 21 x 3 ( k ) y 1 ( k )             − 2 c 22 x 3 ( k ) y 2 ( k ) − α 2 x 2 4 ( k ) x 3 ( k ) + 2 α x 2 2 ( k ) x 3 2 ( k ) − x 1 2 ( k ) x 3 2 ( k ) y 3 ( k )             + 2 α x 2 2 ( k ) y 3 ( k ) − 2 x 1 2 ( k ) x 3 ( k ) y 3 ( k ) − α 3 x 1 4 ( k ) x 2 4 ( k ) − α 3 x 1 2 ( k ) x 2 4 ( k )             − α x 1 4 ( k ) x 3 2 ( k ) + 2 α 2 x 1 2 ( k ) x 2 2 ( k ) − 2 α x 1 4 ( k ) x 3 ( k ) − 2 α x 1 2 ( k ) x 3 ( k )

            + 2 β x 1 2 ( k ) x 2 ( k ) + β x 2 ( k ) x 3 2 ( k ) + 2 β x 2 ( k ) x 3 ( k ) + 2 α x 1 2 ( k ) x 2 2 ( k ) x 3 2 ( k )             + 2 β x 2 ( k ) − 2 α x 1 2 ( k ) − α 2 c 21 x 1 2 ( k ) x 2 4 ( k ) y 1 ( k ) + 2 α c 21 x 1 3 ( k ) x 2 2 ( k ) x 3 ( k )             + 2 α c 21 x 1 ( k ) x 2 2 ( k ) x 3 ( k ) + 2 α c 21 x 2 2 ( k ) x 3 ( k ) y 1 ( k )             + 2 α c 22 x 1 2 ( k ) x 2 3 ( k ) x 3 ( k ) + 2 α x 1 2 ( k ) x 2 2 ( k ) x 3 ( k ) y 3 ( k )             − 2 α β x 1 2 ( k ) x 2 3 ( k ) x 3 ( k ) − α 2 c 21 x 1 ( k ) x 2 4 ( k ) − α 2 c 21 x 2 4 ( k ) y 1 ( k )

            + 2 α c 21 x 1 3 ( k ) x 2 2 ( k ) − c 21 x 1 2 ( k ) x 3 2 ( k ) y 1 ( k ) − 2 c 21 x 1 3 ( k )             − 2 c 21 x 1 2 ( k ) x 3 ( k ) y 1 ( k ) − α 2 c 22 x 1 2 ( k ) x 2 5 ( k ) − α 2 c 22 x 2 4 ( k ) y 2 ( k ) − 2 α x 1 4 ( k )             − c 22 x 1 2 ( k ) x 2 ( k ) x 3 2 ( k ) + 2 α c 22 x 2 2 ( k ) y 2 ( k ) − 2 c 22 x 1 2 ( k ) x 2 ( k ) x 3 ( k )             − 2 c 22 y 2 ( k ) − α 2 x 1 2 ( k ) x 2 4 ( k ) y 3 ( k ) + 2 α x 1 2 ( k ) x 2 2 ( k ) y 3 ( k )             + 2 α x 2 2 ( k ) x 3 ( k ) y 3 ( k ) − 2 c 22 x 2 ( k ) + α 2 β x 1 2 ( k ) x 2 5 ( k ) − 2 α β x 1 2 ( k ) x 2 3 ( k )

            − 2 α β x 2 3 ( k ) x 3 ( k ) + β x 1 2 ( k ) x 2 ( k ) x 3 2 ( k ) + a 1 δ x 1 2 ( k ) y 1 ( k )             + a 2 δ x 1 2 ( k ) y 3 ( k ) − x 3 3 ( k ) − x 3 2 ( k ) − 2 x 1 2 ( k ) y 3 ( k ) − 2 c 21 x 1 ( k ) − 2 c 21 y 1 ( k )             − x 1 2 ( k ) x 3 3 ( k ) − x 1 2 ( k ) x 3 2 ( k ) + 2 α 2 x 1 2 ( k ) x 2 2 ( k ) x 3 ( k ) − x 3 2 ( k ) y 3 ( k )             − 2 x 3 ( k ) y 3 ( k ) + α 2 x 2 4 ( k ) + x 1 2 ( k ) y 2 ( k ) + a 1 δ y 1 ( k ) + y 2 ( k )             − 2 c 22 x 2 ( k ) x 3 ( k ) − 2 c 21 x 1 ( k ) x 3 ( k ) − α 2 x 2 4 ( k ) y 3 ( k ) + 2 α 2 x 1 4 ( k ) x 2 2 ( k )

            − 2 α x 2 2 ( k ) − 2 α β x 2 3 ( k ) − α x 1 2 ( k ) x 3 2 ( k ) + α 2 β x 2 5 ( k ) + α 2 x 1 2 ( k ) x 2 4 ( k )             + 2 x 1 2 ( k ) + 2 α c 21 x 1 2 ( k ) x 2 2 ( k ) y 1 ( k ) + 2 α c 22 x 2 2 ( k ) x 3 ( k ) y 2 ( k ) − 2 y 3 ( k )             − α 2 c 21 x 1 3 ( k ) x 2 4 ( k ) + 2 α c 21 x 1 ( k ) x 2 2 ( k ) + 2 α c 21 x 2 2 ( k ) y 1 ( k )             − α 2 x 1 2 ( k ) x 2 4 ( k ) x 3 ( k ) + 2 α 2 x 1 4 ( k ) x 2 2 ( k ) x 3 ( k ) + 2 β x 1 2 ( k ) x 2 ( k ) x 3 ( k )             + 2 α c 21 x 1 2 ( k ) x 2 2 ( k ) x 3 ( k ) y 1 ( k ) ) / ( x 1 2 ( k ) + 1 ) ,

u 3 = − ( a 6 δ x 1 2 ( k ) y 2 ( k ) y 3 ( k ) + a 7 δ x 1 2 ( k ) y 3 ( k ) + a 5 δ x 1 2 ( k ) + β x 1 3 ( k )             − c 31 x 1 3 ( k ) + a 6 δ y 2 ( k ) y 3 ( k ) − c 31 x 1 2 ( k ) y 1 ( k ) − c 32 x 1 2 ( k ) x 2 ( k )             − c 33 x 1 2 ( k ) x 3 ( k ) + a 7 δ y 3 ( k ) − c 33 x 1 2 ( k ) y 3 ( k ) + x 1 2 ( k ) y 3 ( k ) + a 5 δ             + β x 1 ( k ) − c 31 x 1 ( k ) − c 31 y 1 ( k ) − c 32 x 2 ( k ) − c 32 y 2 ( k ) − c 33 x 3 ( k )             − c 33 y 3 ( k ) + y 3 ( k ) + 2 α c 22 x 1 2 ( k ) x 2 3 ( k ) + 2 α c 22 x 2 3 ( k ) x 3 ( k ) ) (16)

Here numerical simulations are also used to verify the effectiveness of the obtained controllers u ( x , y ) above. Without losing generality, we take the same parameter values of c 11 , c 21 , c 22 , c 31 , c 32 , c 33 , d 1 , d 2 in section 3, and take the initial values as [ x 1 ( 0 ) = 0.1 , x 2 ( 0 ) = 0.2 , x 3 ( 0 ) = 0.3 ] and [ y 1 ( 0 ) = − 0.5 , y 2 ( 0 ) = 0.2 , y 3 ( 0 ) = 0.1 ], respectively. The figures of the error states are shown in Figures 5(a)-(c). That is easy to see, all the three orbits of E i ( i = 1 , 2 , 3 ) converge to zero finally after the controllers u 1 , u 2 , u 3 are activated. So all the state

variables tend to be synchronized in proportion ( − 1, − 1 1 + x 1 2 ( k ) , − 1 ) . The attractors of the two systems with controllers tending to a function scaling factor, are displayed in Figure 6.

In conclusion, the conception of the function projective synchronization between discrete-time hyperchaotic dynamical systems is presented. According to backstepping design method with controllers, a systematic, automatic and constructive scheme is explored in order to investigate FPS between the discrete-time drive systems and response systems, whether is in strict-feedback forms or not. Additionally, the proposed scheme is used to realize the function of projective synchronization between the identical discrete-time hyperchaotic systems from Rössler system and two different hyperchaotic systems which are the hyperchaotic Rössler system and the Henon map, respectively. With the aid of symbolic computation Maple, the numerical simulations are shown to perform the process of the synchronization and the effectiveness of the above-designed controller successfully.

This work is supported by Natural Science Foundation for the Youth of Jiangsu Province (No. BK20181030), National Natural Science Foundation for the Youth of China (No. 12101084), and Jiangsu Overseas Visiting Scholar Program for University Prominent Young and Middle-aged Teachers and Presidents.

The author declares no conflicts of interest regarding the publication of this paper.

Li, X. (2022) Function Projective Synchronization between Two Discrete-Time Hyperchaotic Systems Using Backstepping Method. Applied Mathematics, 13, 178-190. https://doi.org/10.4236/am.2022.132014