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Ouannas Adel, Grassi Giuseppe. Inverse full state hybrid projective synchronization for chaotic maps with different dimensions. Chinese Physics B, 2016, 25(9): 090503  
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Inverse full state hybrid projective synchronization for chaotic maps with different dimensions
Ouannas Adel1, †, , Grassi Giuseppe2, ‡,
Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa 12002, Algeria
Università del Salento, Dipartimento Ingegneria Innovazione, 73100 Lecce, Italy

 

† Corresponding author. E-mail: ouannas_adel@yahoo.fr

‡ Corresponding author. E-mail: giuseppe.grassi@unisalento.it

Abstract
Abstract

A new synchronization scheme for chaotic (hyperchaotic) maps with different dimensions is presented. Specifically, given a drive system map with dimension n and a response system with dimension m, the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states. The method, based on the Lyapunov stability theory and the pole placement technique, presents some useful features: (i) it enables synchronization to be achieved for both cases of n < m and n > m; (ii) it is rigorous, being based on theorems; (iii) it can be readily applied to any chaotic (hyperchaotic) maps defined to date. Finally, the capability of the approach is illustrated by synchronization examples between the two-dimensional Hénon map (as the drive system) and the three-dimensional hyperchaotic Wang map (as the response system), and the three-dimensional Hénon-like map (as the drive system) and the two-dimensional Lorenz discrete-time system (as the response system).

PACS: 05.45.–a;05.45.Xt
Keyword:chaotic map;full state hybrid projective synchronization;inverse problem;maps with different dimensions
1. Introduction

The topic of controlling a chaotic drive system and a driven (response) system to get synchronized dynamics has attracted great interest in nonlinear science.[1–31] Most of the methods that have been inspired by the Carroll and Pecora work[1] focus on complete (identical) synchronization, where two identical chaotic systems asymptotically approach the same trajectory.[2–6] Subsequently, different types of synchronization have been proposed in the literature.[8–10] Among them, the projective synchronization provides response system variables that are scaled replicas of the drive system variables.[7,8] Since the scaling factor is not predictable in Ref. [7], control methods have been successively developed for choosing any desired scaling factor.[8–13] A variation of projective synchronization is the so-called full state hybrid projective synchronization (FSHPS).[14–18] In this type of synchronization, the scaling factor can be different for each state variable, meaning that the single scaling parameter originally introduced in Ref. [7] is replaced by a diagonal scaling matrix. Recently, an interesting scheme[19] has been introduced, in which each response system state achieves synchronization with any arbitrary linear combination of drive system states. Since the adoption of an arbitrary scaling matrix represents a generalization of FSHPS (where the scaling matrix is a diagonal one), the method in Ref. [19] has been called arbitrary full-state hybrid projective synchronization (arbitrary FSHPS).

Based on these considerations, this paper presents a new synchronization scheme, with which each drive system state synchronizes with a linear combination of response system states. Since the drive system states and the response system states are inverted with respect to the arbitrary FSHPS, the proposed scheme is called inverse full-state hybrid projective synchronization (IFSHPS). The approach presents some useful features: (i) it enables chaotic (hyperchaotic) maps with different dimensions to be synchronized; (ii) it is rigorous, being based on theorems; (iii) it can be applied to a wide class of chaotic (hyperchaotic) maps.

The paper is organized as follows. In Section 2, the definition of inverse full state hybrid projective synchronization is introduced. In Section 3, two theorems are proved, which enable synchronization between n-dimensional and m-dimensional maps to be achieved for the n < m and n > m cases, respectively. Finally, the capability of the method is illustrated by examples of IFSHPS between the two-dimensional Hénon map (as the drive system) and the three-dimensional hyperchaotic Wang map (as the response system), and the three-dimensional Hénon-like map (as the drive system) and the two dimensional Lorenz discrete-time system (as the response system).

2. Inverse full state hybrid projective synchronization

It is assumed that the drive and response chaotic (hyperchaotic) systems considered herein can be written, respectively, in the form

where X (k) = (xi(k))1≤i≤n and Y (k) = (yi(k))1≤i≤m are the state vectors of the drive system and the response system, respectively, (aij) ∈ ℝn×n, fi : ℝn → ℝ, i = 1,2,…, n, and gi : ℝm → ℝ, i = 1,2,…, m are nonlinear functions whereas ui, i = 1,2,…, m, are controllers.

The definition of inverse full-state hybrid projective synchronization for the coupled chaotic systems given in Eqs. (1) and (2) is given as follows.

Definition 1 The n-dimensional drive system X(k) and the m-dimensional response system Y(k) are in inverse full-state hybrid projective synchronization when, for an initial condition, there exist controllers ui, i = 1,2,…, m and real constants aij, i = 1,2,…, n, j = 1,2,…, m such that the synchronization errors

satisfy limk → +∞ei(k) = 0 for i = 1,2,…, n.

Remark 1 According to condition (3), it follows that in IFSHPS, each drive system state xi (k) synchronizes with a linear combination of response system states ai1y1 (k) +ai2y2 (k) + ··· + aimym (k) for i = 1,2,…, n. Recall that in arbitrary FSHPS each response system state achieved synchronization with a linear combination of drive system states. By comparing the definitions of the two schemes, it can be deduced that herein the roles of the drive system states and response system states have been inverted with respect to those in the arbitrary FSHPS.[19] This clarifies the reason why the proposed scheme has been called inverse full-state hybrid projective synchronization.

The error system (3) between the drive system (1) and the response system (2) can be derived as

Two different cases will now be discussed, i.e., n < m and n > m.

2.1. Case 1: n < m

In this case, the error system described by Eq. (4) can be written as

where (lij)1<i,j<n are the control constants and

Rewrite the error system in a compact form as

where e (k) = (ei (k))1<i<n, A = (aij)n×n, L = (lij)n×n, R = (Ri)1≤i≤n, α = (αij)n×m, and U = (u1,u2,…,um)T

In order to achieve IFSHPS between systems (1) and (2), the controller U is chosen as

By substituting Eq. (7) into Eq. (6), the error system (6) can be written as

where α1 = (αij)n×n and U1 = (ui)1<i<n.

Theorem 1 For an invertible matrix α1, IFSHPS between the drive system (1) and the response system (2) will occur if the following conditions are satisfies:

, where is the inverse matrix of α1;

(A − L)T(A − L) − I is a negative definite matrix.

Proof By substituting the control law (i) into Eq. (8), the error dynamics are described by

By selecting the candidate Lyapunov function in the form

it follows that

By using (ii), we obtain ΔV(e (k)) < 0. Thus, from the Lyapunov stability theory, it can be readily shown that limk→+∞ei (k) = 0, i = 1, 2,…,n, indicating that the zero solution of the error system (9) is globally asymptotically stable. As a consequence, systems (1) and (2) are globally IFSHP synchronized.

2.2. Case 2: m < n

Now, the error system between the drive system (1) and the response system (2) can be described as

where (cij)1≤i,j≤n are the control constants,

and are new controllers (i = 1,2,…,n). Then, the error system (11) can be rewritten in a compact form as

where

and αij (i = 1, 2,…,n; j = m + 1, m + 2,…,n) are assumed not all equal to zero.

Theorem 2 IFSHPS between the drive system (1) and the response system (2) will occur if the following conditions are satisfied:

U2 = T

all eigenvalues of A − C are strictly inside the unit circle.

Proof By substituting the control law (i) into Eq. (6), the error system can be written as

By taking into account the stability theory of linear discrete-time systems, if condition (ii) is satisfied, it follows that all the solutions of the error system (13) go to zero as k → ∞. As a consequence, systems (1) and (2) are globally IFSHP synchronized.

3. Examples

In this section, two examples are used to show the effectiveness of the proposed approach.

3.1. Example 1: n < m

Herein, the chaotic two-dimensional Hénon map is selected as the drive system and the hyperchaotic three-dimensional Wang system is chosen as the response system. The Hénon map[32] is described by

and has a chaotic attractor, plotted in Fig. 1, when (a,b) = (1.4, 0.2).

Fig. 1. Chaotic attractor of Hénon map in the (x1, x2) plane when (a, b) = (1.4, 0.2).

System (14) can be rewritten in the form X (k + 1) = AX (k) + f(X (k)) where

The hyperchaotic Wang system[33] can be described by

where u1, u2, and u3 are the synchronization controllers. The uncoupled Wang system (15), obtained for u1 = u2 = u3 = 0, has a hyperchaoticattractor[33] (plotted in Fig. 2) when a1 = 1.9, a2 = 0.2, a3 = 0.5, a4 = 2.3, a5 = 2, a6 = 0.6, and a7 = 1.9.

Fig. 2. Hyperchaotic attractor of the Wang system when (a1, a2, a3, a4, a5, a6, a7) = (1.9, 0.2, 0.5, 2.3, 2, 0.6, 1.9, 1).

In this example, according to Eq. (15) and condition (i) of Theorem 1, the controllers ui, 1 ≤ i ≤ 3, are obtained as follows:

where

The controllers (16) can be rewritten as

Note that, according to Theorem 1, it can be readily verified that (A − L)T(A − L) − I is a negative definite matrix. Since Theorem 1 holds, it follows that the error system (9) is globally asymptotically stable and is described by e1 (k + 1) = −0.4e1 (k) and e2 (k + 1) = 0.1e2 (k). Thus, IFSHPS is achieved between the drive system (14) and the response system (15). Finally, note that in the proposed example of IFSHPS the errors e1 (k) and e2 (k) are

Since e1 (k) and e2 (k) asymptotically approach zero (see Fig. 3), equation (18) highlights that each drive system state is synchronized with a linear combination of response systems states, indicating that IFSHPS has been successfully achieved.

Fig. 3. Time evolution of the errors e1 (k) and e2 (k) between systems (14) and (15). According to Theorem 1, the errors asymptotically approach zero, indicating that IFSHPS is effectively achieved.
3.2. Example 2: n > m

Now, the hyperchaotic three-dimensional Hénon-like map is selected as the drive system and the chaotic two-dimensional Lorenz discrete-time system is chosen as the response system. The three-dimensional Hénon-like map[34] is described by

and has a hyperchaotic attractor (plotted in Fig. 4) when (a,b) = (1.4, 0.2).

Fig. 4. Hyperchaotic attractor of the Hénon-like map (19) when (a,b) = (1.4, 0.2).

System (19) can be rewritten in the form X (k + 1) = AX (k) + f (X (k)), where

The controlled Lorenz discrete-time system[35] can be described as

where U = (u1, u2)T is the vector controller. The uncoupled Lorenz discrete-time system, obtained for u1 = u2 = 0, has a chaotic attractor[35,36] (plotted in Fig. 5) when (a1, a2) = (1.25, 0.75).

Fig. 5. Chaotic attractor of the Lorenz discrete-time system in the (y1, y2)) plane when (a1, a2) = (1.25, 0.75).

In this example, the scaling constant matrix α = (αij)3×2 and the control matrix C = (cij)3×3 are chosen as

Then, according to condition (i) of Theorem 2, the control law is given by

where

So, the control input can be rewritten as

According to Theorem 2, it can be readily verified that all the eigenvalues of (A − C) are strictly inside the unit circle. Since Theorem 2 holds, it follows that the error system (13) is globally asymptotically stable and is described by e1 (k + 1) = −0.1e1 (k), e2 (k + 1) = 0.2e2 (k), and e3 (k + 1) = −0.3e3 (k). Thus, IFSHPS is achieved between the drive system (19) and the response system (20). Finally, note that in the proposed example of IFSHPS, the errors e1 (k), e2 (k), and e3 (k) are

Since e1 (k), e2 (k), and e3 (k) asymptotically approach zero (see Fig. 6), equation (23) highlights that each drive system state is synchronized with a linear combination of response systems states, indicating that IFSHPS has been successfully achieved.

Fig. 6. Time evolution of the errors e1 (k), e2 (k), and e3 (k) between systems (19) and (20). According to Theorem 2, the errors asymptotically approach zero, indicating that IFSHPS is effectively achieved.
4. Discussion

Referring to chaotic discrete-time systems, herein comparisons are carried out between the proposed approach and a number of synchronization methods in the literature, with the aim to show the advantages and the novelty of the conceived scheme.

In Ref. [27], attention was mainly focused on the full-state hybrid projective synchronization and related issues, with application to 2D discrete-time systems. Even though the theoretical approach developed in Ref. [27] is interesting, being based on nonlinear control method and Lyapunov stability, the proofs of the synchronization schemes are limited to 2D discrete-time systems only. This is further confirmed by simulation results, which focus on 2D systems only, i.e., the Fold discrete-time system (drive system) and the Lorenz discrete-time system (response system), which are both two-dimensional systems.[27] On the other hand, the technique proposed herein enables inverse full-state hybrid projective synchronization to be achieved for n-dimensional drive systems and m-dimensional response systems. As a consequence, the approach developed herein is much more general than the one illustrated in Ref. [27].

In Ref. [28], the issue of generalized synchronization (GS), along with its inverse problem, was discussed with reference to discrete-time systems. In GS, two chaotic systems are said to be synchronized if there exists a functional relationship Φ between the drive system states and the response system states.[28] However, the synchronization schemes discussed in Ref. [28] are very different from those proposed herein. Namely, in this paper, we do not consider any functional relationship Φ between drive and response systems. According to the IFSHPS schemes developed herein, each n-dimensional drive system state synchronizes with a linear combination of m-dimensional response system states. This holds for both cases of n < m and m > n.

Finally, in Ref. [29], the problem of inverse matrix projective synchronization (IMPS) for different dimensional chaotic systems in discrete-time was investigated. In this type of synchronization, the drive system and the response system synchronize up to a special scaling matrix, as shown through the examples reported in Ref. [29]. It can be argued that, while the synchronization criteria reported in Ref. [29] are restricted to specific scaling matrices, in this paper a general IFSHPS approach is developed, where each drive system state synchronizes with any arbitrary linear combination of response system states. In conclusion, the approach developed herein is much more general than the one illustrated in Ref. [29], since arbitrary scaling matrices can be exploited to achieve IFSHPS.

5. Conclusion

We have illustrated a new scheme to achieve chaos (hyperchaos) synchronization in discrete-time between a drive system of dimension n and a response system of dimension m. The proposed approach, which enables each drive system state to be synchronized with a linear response combination of response system states, has shown some remarkable features. Namely, by exploiting the Lyapunov stability theory and the pole placement technique, synchronization is rigorously proved to be achievable for any chaotic (hyperchaotic) map defined to date, including the two cases n < m and n > m. Finally, the effectiveness of the method has been illustrated by synchronizing a two-dimensional chaotic Hénon map with a three-dimensional hyperchaotic Wang map, and a three-dimensional hyperchaotic Hénon-like map with a two-dimensional chaotic discrete-time Lorenz system.

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