# Isochronal function projective synchronization between chaotic and time-delayed chaotic systems

- Ranchao Wu
^{1}Email author and - Junlian Liu
^{1}

**2012**:37

https://doi.org/10.1186/1687-1847-2012-37

© Wu and Liu; licensee Springer. 2012

**Received: **7 January 2012

**Accepted: **28 March 2012

**Published: **28 March 2012

## Abstract

Isochronal function projective synchronization between chaotic and time-delayed chaotic systems with unknown parameters is investigated in this article. Based on Lyapunov stability theory, adaptive controllers and parameter updating laws are designed to achieve the isochronal function projective synchronization between chaotic and time-delayed chaotic systems. The scheme is applied to realize the synchronization between time-delayed Lorenz systems and time-delayed hyper-chaotic Chen systems, respectively. Numerical simulations are also presented to show the effectiveness of the proposed method.

**Mathematics Subject Classification 2000**: 34C28; 34D20; 37N35.

### Keywords

isochronal function projective synchronization time-delayed chaotic systems adaptive control## 1. Introduction

In the last few years, chaos synchronization has gained a lot of attention for its potential applications in some engineering applications, such as image processing, chemical and biological systems, information science and in particular secure communication. Since the pioneering work of Pecora and Carroll [1], in which complete synchronization between two identical chaotic systems with different initial conditions was realized, various approaches have been put forward for synchronization of chaotic systems, such as complete synchronization [2], phase synchronization [3], generalized synchronization [4], lag synchronization [5], projective synchronization [6], modified projective synchronization [7, 8] and function projective synchronization [9–11], function projective lag synchronization [12], anti-synchronization [13] and so on.

Among all types of chaos synchronization, projective synchronization phenomenon is of great significance for its potential application in secure communication. In 1999, Mainieri and Rehacek [14] first proposed the concept of projective synchronization, which is characterized that the drive and the response systems could be synchronized up to a scaling factor. Because of the proportionality between its synchronized dynamical states, the feature can be used to *M*-nary digital communication for achieving fast communication. So projective synchronization have attracted increasing attention during recent years and some conditions ensuring projective synchronization have been obtained. Recently, some scholars extended the concept of projective synchronization and proposed modified projective synchronization [15], function projective synchronization [16] and modified function projective synchronization [17, 18], in which master and slave system are synchronized with a scaling function matrix.

As we know, delayed differential equations could exhibit complex dynamical behaviors and have attracted much attention in the field of nonlinear dynamics. Note that research on synchronization between time-delayed chaotic systems has been extensively carried out, see, for example, [19–21]. Recently, dual-anticipating, dual and dual-lag synchronization [22] between two identical time-delayed chaotic systems, lag synchronization [23] were investigated, where lag or anticipatory dynamics occurred, i.e., there existed a lag time or anticipatory time phase shift between state vectors. While results about zero lag time difference between synchronized state shifts, i.e., isochronal synchronization [24, 25] are also obtained between time-delayed systems. Note that projective synchronization [26, 27] between time-delayed chaotic systems was extensively investigated. However, results about isochronal function projective synchronization between chaotic and time-delayed chaotic systems are still few. In this article, isochronal function projective synchronization scheme between chaotic and time-delayed chaotic systems with unknown parameters is proposed. The method is shown to be effective by applying to Lorenz and hyper-chaotic Chen systems.

The remainder of this article is organized as follows. In Section 2 the synchronization scheme is presented. Section 3 is devoted to the application of the proposed scheme to Lorenz and hyper-Chaotic systems, respectively. Numerical simulations are also presented to demonstrate the effectiveness of the method. Some conclusions are drawn in Section 4.

## 2. Statement of the problem

where *x* ∈ *R*^{
n
}denotes the state vector, *F, G* : *R*^{
n
}→ *R*^{n × p}are continuous function matrices, *f* : *R*^{
n
}→ *R*^{
n
}is a continuous nonlinear vector function, *θ, β* ∈ *R*^{
p
}are parameter vectors. Note that many chaotic and hyper-chaotic systems, such as Lorenz system, Chen system, Lü system, Rössler system, hyper-chaotic Chen system, etc, could be described by system (1).

where *u* = (*u*_{1}, *u*_{2},..., *u*_{
n
})^{
T
}∈ *R*^{
n
}is the control input to be determined later.

Now we need to design the controller *u* such that the chaotic system (1) could track the trajectory of time-delayed system (2). Define the error *e* = *x* - λ(*t*)*y*, where λ(*t*) = diag(λ_{1}(*t*), λ_{2}(*t*),...,λ_{
n
}(*t*)).

**Definition 1**. It is said that isochronal function projective synchronization occurs between systems (1) and (2) if there exists a diagonal function matrix λ(

*t*) such that

*k*= diag(

*k*

_{1},

*k*

_{2},...,

*k*

_{ n }),

*k*

_{ i }> 0(

*i*= 1, 2,...,

*n*) are constants. Consequently, we get

*θ*

_{ r },

*β*

_{ r }are chosen as

then the synchronization result follows immediately.

**Theorem 1**. *Isochronal function projective synchronization between systems (1) and (2) will occur under the control (4) and parameter updating laws (6)*.

*Proof*. Take the Lyapunov function

*e*

_{ θ }=

*θ*-

*θ*

_{ r },

*e*

_{ β }=

*β*-

*β*

_{ r }. The time derivative of

*V*along the trajectory of error system (5) is

So by Lyapunov stability theory, *e*_{
i
}→ 0 as *t* → ∞, i.e., the synchronization will occur.

**Remark 1**. When λ(*t*) = *I*, -*I*, the complete synchronization and anti-synchronization between (1) and (2) are achieved, respectively. When λ(*t*) = *αI*, diag(λ_{1},..., λ_{
n
}), then generalized projective synchronization and modified projective synchronization between (1) and (2) will happen, respectively.

**Remark 2**. Here structures of systems (1) and (2) are in the same form. Similarly, the synchronization result could also hold between chaotic and delayed chaotic systems with different structures under appropriate controllers and parameter updating laws.

## 3. Applications

### 3.1. FPS between Lorenz and delayed Lorenz systems

*a*

_{ r },

*b*

_{ r },

*c*

_{ r }are uncertain parameters to be estimated. The system exhibits chaotic behaviors when

*a*

_{ r }= 10,

*b*

_{ r }= 28, ${c}_{r}=\frac{8}{3}$ and

*τ*= 0.3, see Figure 1.

*e*

_{ i }=

*x*

_{ i }- λ

_{ i }(

*t*)

*y*

_{ i },

*i*= 1, 2, 3. Choose the controllers as follows

*k*

_{ i }> 0,

*i*= 1, 2, 3, then from systems (9), (10), and (11) one has

Along the way similar to that of Theorem 1, one could arrive the following result.

**Theorem 2**. *Isochronal function projective synchronization between Lorenz system (10) and delayed Lorenz system (9) will be realized under the controllers (11) and parameter updating laws (13)*.

### 3.2. FPS between hyper-chaotic Chen and delayed hyper-chaotic Chen systems

*a*

_{ r }= 35,

*b*

_{ r }= 3,

*c*

_{ r }= 12,

*d*

_{ r }= 7,

*p*

_{ r }= 0.5, and

*τ*= 0.4, then the system displays chaotic behaviors, see Figure 2.

*e*

_{ i }=

*x*

_{ i }- λ

_{ i }(

*t*)

*y*

_{ i },

*i*= 1, 2, 3, 4. Choose the controllers as follows

*k*

_{ i }> 0,

*i*= 1, 2, 3, 4, then from systems (14), (15), and (16) one has

As a result, the synchronization between (14) and (15) will happen.

**Theorem 3**. *Isochronal function projective synchronization between hyper-chaotic Chen system (15) and delayed hyper-chaotic Chen system (14) will be achieved under the controllers (16) and parameter updating laws (18)*.

The proof is similar to that of Theorem 1.

### 3.3. Numerical simulations

*y*

_{1}(0),

*y*

_{2}(0),

*y*

_{3}(0)) = (1,-3,2) and (

*x*

_{1}(0),

*x*

_{2}(0),

*x*

_{3}(0)) = (-1,3,-2), and let

*τ*= 0.3. The scaling functions are λ

_{1}(

*t*) = sin(

*t*), λ

_{2}(

*t*) = cos(

*t*), λ

_{3}(

*t*) = - sin(

*t*). Moreover, (

*k*

_{1},

*k*

_{2},

*k*

_{3}) = (1, 3, 2). The simulation results are shown in Figure 3. Note that the error variables

*e*

_{1},

*e*

_{2},

*e*

_{3}tend to zero and the estimated values of unknown parameters

*a*

_{ r },

*b*

_{ r },

*c*

_{ r }converge to $10,28,\frac{8}{3}$, respectively.

*y*

_{1}(0),

*y*

_{2}(0),

*y*

_{3}(0),

*y*

_{4}(0)) = (-3,4,-2,-2) and (

*x*

_{1}(0),

*x*

_{2}(0),

*x*

_{3}(0),

*x*

_{4}(0)) = (3,-4,2,2), and let

*τ*= 0.3. The scaling functions are λ

_{1}(

*t*) = sin(

*t*), λ

_{2}(

*t*) = -cos(

*t*), λ

_{3}(

*t*) = cos(

*t*), λ

_{4}(

*t*) = -sin(

*t*). Moreover, the control gains are chosen as (

*k*

_{1},

*k*

_{2},

*k*

_{3},

*k*

_{4}) = (7, 5, 5, 5). The simulation results are shown in Figures 4 and 5. Note that the error variables tend to zero and the estimated values of unknown parameters

*a*

_{ r },

*b*

_{ r },

*c*

_{ r },

*d*

_{ r },

*p*

_{ r }converge to 35, 3, 12, 7, 0.5, respectively.

## 4. Conclusions

In this article, function projective synchronization between chaotic and time-delayed chaotic systems with unknown parameters is investigated. Adaptive synchronization scheme is proposed by designing appropriate controllers and parameter updating laws. Based on Lyapunov stability theory, synchronization results are obtained. The method is applied to Lorenz and hyper-chaotic Chen systems, respectively. Corresponding numerical simulations show the effectiveness of the method proposed. In existing literatures results about synchronization between chaotic and time-delayed chaotic systems are still few. So the obtained will be helpful in synchronizing chaotic and time-delayed chaotic systems.

## Declarations

### Acknowledgements

This study is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20093401120001), the Natural Science Foundation of An-hui Province (No. 11040606M12) and the Natural Science Foundation of Anhui Education Bureau (No. KJ2010A035), the 211 project of Anhui University(No. KJJQ1102).

## Authors’ Affiliations

## References

- Pecora LM, Carroll TL: Synchronization in chaotic systems.
*Phys Rev Lett*1990, 64: 821–824. 10.1103/PhysRevLett.64.821MATHMathSciNetView ArticleGoogle Scholar - Zhang J, Li CG, Zhang HB, Yu JB: Chaos synchronization using single variable feedback based on backstepping mathod.
*Chaos Solitons Fract*2004, 21: 1183–1193. 10.1016/j.chaos.2003.12.079MATHMathSciNetView ArticleGoogle Scholar - Shuai JW, Durand DM: Phase synchronization in two coupled chaotic neurons.
*Phys Lett A*1999, 264: 289–297. 10.1016/S0375-9601(99)00816-6MATHMathSciNetView ArticleGoogle Scholar - Ge ZM, Chang CM: Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function.
*Nonlinear Anal TMA*2009, 71: 5301–5312. 10.1016/j.na.2009.04.020MATHMathSciNetView ArticleGoogle Scholar - Xu YH, Zhou WN, Fang JA, Sun W: Adaptive lag synchronization and parameters adaptive lag identification of chaotic systems.
*Phys Lett A*2010, 374: 3441–3446. 10.1016/j.physleta.2010.06.064MATHView ArticleGoogle Scholar - Ghosh D, Bhattacharya S: Projective synchronization of new hyperchaotic system with fully unknown paramaters.
*Nonlinear Dyn*2010, 61: 11–21. 10.1007/s11071-009-9627-4MATHMathSciNetView ArticleGoogle Scholar - Park JH: Adaptive control for modified projective synchronization of a four-dimentional chaotic system with uncertain parameters.
*J Comput Appl Math*2008, 213: 288–293. 10.1016/j.cam.2006.12.003MATHView ArticleGoogle Scholar - Park JH: Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter.
*Chaos Solitons Fract*2007, 34: 1552–1559. 10.1016/j.chaos.2006.04.047MATHView ArticleGoogle Scholar - Luo RZ, Wei ZM: Adaptive function projective synchronization of unified chaotic systems with uncertain paramaters.
*Chaos Solitons Fract*2009, 42: 1266–1272. 10.1016/j.chaos.2009.03.076MATHMathSciNetView ArticleGoogle Scholar - Du HY, Zeng QS, Wang CH, Ling MX: Function projective synchronization in coupled chaotic systems.
*Nonlinear Anal RWA*2010, 11: 705–712. 10.1016/j.nonrwa.2009.01.016MATHMathSciNetView ArticleGoogle Scholar - Park JH: Further results on functional projective synchronization of Genesio-Tesi chaotic system.
*Modern Phys Lett B*2009, 24: 1889–1895. 10.1142/S0217732309031302View ArticleGoogle Scholar - Lee TH, Park JH: Adaptive funcrional projective lag synchronization of hyperchaotic Rössler system.
*Chin Phys Lett*2009, 26: 090507. 10.1088/0256-307X/26/9/090507View ArticleGoogle Scholar - Shi XR, Zuo LW: Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters.
*Appl Math Comput*2009, 11: 1711–1717.View ArticleGoogle Scholar - Mainieri R, Rehacek J: Projective synchronization in three-dimensional chaotic systems.
*Phys Rev Lett*1999, 82: 3042–3045. 10.1103/PhysRevLett.82.3042View ArticleGoogle Scholar - Li GH: Modified projective synchronization of chaotic system.
*Chaos Solition Fract*2007, 32: 1786–1790. 10.1016/j.chaos.2005.12.009MATHView ArticleGoogle Scholar - Chen Y, Li X: Function projective synchronization between two identical chaotic systems.
*Int J Modern Phys C*2007, 18: 883–888. 10.1142/S0129183107010607MATHView ArticleGoogle Scholar - Du HY, Zeng QS, Wang CH: Modified function projective synchronization of chaotic system.
*Chaos Solitions Fract*2009, 42: 2399–2404. 10.1016/j.chaos.2009.03.120MATHView ArticleGoogle Scholar - Sudheer KS, Sabir M: Adaptive modified function projective synchronization between hyper-chaotic Lorenz system and hyperchaotic Lü system with uncertain parameters.
*Phys Lett A*2009, 373: 3743–3748. 10.1016/j.physleta.2009.08.027MATHMathSciNetView ArticleGoogle Scholar - Li LX, Peng HP, Yang YX, Wang XD: On the chaotic synchronization of 694 Lorenz systems with the time-delayed lags.
*Chaos Solitons Fract*2009, 41: 783–794. 10.1016/j.chaos.2008.03.014MATHView ArticleGoogle Scholar - Feng CF, Zhang Y, Sun JT, Qi W, Wang YH: Generalized projective synchronization in time-delayed chaotic systems.
*Chaos Solitons Fract*2008, 38: 743–747. 10.1016/j.chaos.2007.01.037MATHView ArticleGoogle Scholar - Sudheer KS, Sabir M: Adaptive modified function projective synchronization of multiple time-delayed chaotic Rössler system.
*Phys Lett A*2011, 375: 1176–1178. 10.1016/j.physleta.2011.01.028MATHView ArticleGoogle Scholar - Ghosh D, Chowdhury AR: Dual-anticipating, dual and dual-lag synchronization in modulated time-delayed systems.
*Phys Lett A*2010, 374: 3425–3436. 10.1016/j.physleta.2010.06.050MATHView ArticleGoogle Scholar - Pourdehi S, Karimaghaee P, Karimipour D: Adaptive controller design for lag-synchronization of two non-identical time-delayed chaotic systems with unknown parameters.
*Phys Lett A*2011, 375: 1769–1775. 10.1016/j.physleta.2011.02.008MATHView ArticleGoogle Scholar - Grzybowski JMV, Macau EEN, Yoneyama T: Isochronal synchronization of time delay and delay-coupled chaotic systems.
*J Phys A Math Theory*2011, 44(17):175103. 10.1088/1751-8113/44/17/175103MathSciNetView ArticleGoogle Scholar - IIIing L, Panda CD, Shareshian L: Isochronal chaos synchronization of delay-coupled optoelectronic oscillators.
*Phys Rev E Stat Nonlinear Soft Matter Phys*2011, 84: 016213.View ArticleGoogle Scholar - Ghosh D: Projective synchronization in multiple modulated time-delayed systems with adaptive scaling factor.
*Nonlinear Dyn*2010, 62: 751–759. 10.1007/s11071-010-9759-6MATHView ArticleGoogle Scholar - Feng CF: Projective synchronization between two different time-delayed chaotic systems using active control approach.
*Nonlinear Dyn*2010, 62: 453–459. 10.1007/s11071-010-9733-3MATHView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.