Generalized Projective Synchronization for Different Hyperchaotic Dynamical Systems

Projective synchronization and generalized projective synchronization have recently been observed in the coupled hyperchaotic systems. In this paper a generalized projective synchronization technique is applied in the hyperchaotic Lorenz system and the hyperchaotic Lü. The sufficient conditions for achieving projective synchronization of two different hyperchaotic systems are derived. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.


Introduction
Chaos is an interesting phenomenon in nonlinear dynamical systems research area.In the last three decades, chaos has been extensively studied within the scientific, engineering, and mathematical communities 1-6 .
A chaotic system is a nonlinear deterministic system that displays complex, noisy-like and unpredictable behavior.These motions are trajectories in which infinite unstable periodic orbits UPOs are embedded.Chaos is generally undesirable in many fields.This irregular and complex phenomenon can lead systems to harmful or even catastrophic situations.In these troublesome cases chaos should be suppressed as much as possible or totally eliminated.Therefore controlling chaos has become one of the most considerable research area in the nonlinear problems ranging from biology, physics and chemistry to economics.
Since Pecora and Carroll 7,8 showed that it is possible to synchronize two identical chaotic systems, chaos synchronization has been intensively and extensively studied due to its potential applications in secure communication, ecological systems, system identification, and so forth.
Among all kinds of chaos synchronizations, projective synchronization is one of the most noticeable ones.This kind of synchronization was first observed in continuous systems 9 where a part of state variables possessed some partial linearity 10 .Its typical feature is that the state variables of the two-coupled system may synchronize up to a scaling factor but the Lyapunov exponents and fractal dimensions remain unchanged.Such synchronization has been relatively understood well 11-14 .In 1999, Mainieri and Rehacek 10 first reported the projective synchronization phenomenon and explained the mechanism of the formation of projective synchronization in three-dimensional systems and further attempted to predict the scaling factor by introducing a vector field.However, they only provided a guideline of predicting the scaling factor rather than a concrete theoretical solution.
Generalized synchronization 15-23 is another interesting chaos synchronization technique.It means that there exists a transformation which is able to map asymptotically the trajectories of the master attractor into those of the slave one.To understand such kind of synchronization needs much mathematics.Till now, there are relatively few publications for generalized synchronization.
A focused problem in the study of chaos synchronization is how to design a physically available and simple controller to guarantee the realization of high-quality synchronization in coupled chaotic systems.Linear feedback is of course a practical technique, but the shortcoming is that it needs to find the suitable feedback constant.Recently, Huang proposed a simple adaptive feedback control method, which dose not need to estimate or find feedback constant, to effectively synchronize two almost arbitrary identical hyperchaotic systems 24-26 .This technique has been adopted by some authors to realize the identical synchronization of almost all kinds of coupled identical neural networks with time-varying delay 27 and the complete synchronization in uncertain complex networks 28 .In this paper, we introduce a new synchronization technique, which is different from projective synchronization, but share the same typical feature of projective synchronization, that is, the Lyapunov exponents and fractal dimensions are also invariant during the synchronization process.To some extent, the synchronization presented here is very similar to the generalized synchronization.
The rest of the paper is organized as follows.In Section 2, a mathematical description of generalized projective synchronization is presented.In Section 3 system description is introduced.In Section 4, the projective synchronization problem of a hyperchaotic Lorenz system is investigated and numerical simulation results are demonstrated in Subsection 4.1.In Section 5, the generalized projective synchronization problem of hyperchaotic L ü system is presented, and numerical simulation results are given in Subsection 5.1.In Section 6, the generalized projective synchronization problem between hyperchaotic Lorenz system and hyperchaotic L ü system is presented, and numerical simulation results are given in Subsection 6.1.Finally, in Section 7 the conclusion of the paper is given.

Generalized Projective Synchronization of Chaotic Systems
First, both projective and generalized synchronization are introduced.
A partial linear system is often expressed as in which the state vector u is linearly related to u with respect to t, while the matrix M z only depends upon the variable z which is nonlinearly related to the variable u.Projective synchronization often occurs when two identical system are coupled through the variable z in the form as ud M z u d ,

2.2
The subscripts d and r stand for the driver or master and response or slave systems, respectively.
If there exists a constant α ∈ R α / 0 such that lim t → ∞ u r − α u d 0, then the projective synchronization between the drive system and response system is achieved, and we call α as "scaling factor." Consider the following coupled system: where where x d , x r ∈ R n , u : R 2n → R n , g : R 2n → R n , and u 0, 0 0, g x, u 0, 0 f x : R 2k → R k .If there exists a constant α ∈ R α / 0 such that lim t → ∞ x r − α x d 0, then we call them "generalized projective synchronization."Remark 2.1.i This definition is very similar to that of generalized synchronization, see 2.3 and 2.4 .ii The master attractor synchronizes to the slave one up to a scaling factor a. Obviously, the Lyapnove exponents and fractal dimension remain invariant.iii From the last equation of 2.4 , u can be regarded as a feedback controller or "synchronizer" , that is, similar to 7, 10 , and if and only if such feedback controller u is applied to the slave system, generalized projective synchronization may occur.

System Description
Very recently, based on Lorenz system 29, 30 and L ü system 31, 32 , two hyperchaotic systems, we are constructed by introducing state feedback controller function, which were named as hyperchaotic Lorenz system and hyperchaotic L ü system, respectively.The hyperchaotic Lorenz system is described by

Generalized Projective Synchronization for Hyperchaotic Lorenz System
In order to observe generalized projective synchronization between two identical hyperchaotic Lorenz systems, we assume that the drive system with four state variables denoted by the subscript 1 and the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system.The drive and response systems are defined below, respectively, where U u 1 u 2 u 3 u 4 T is the controller functions.The controller U is to be determined for the purpose of projective synchronizing the two identical hyperchaotic Lorenz systems.
In order to get generalized projective synchronization, we define the error system as the difference between the system 4. 2 and then the error dynamical system can be rewritten as ėx a e y − e x e w V 1 ,

4.6
To get the projective synchronization to occur, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as t → ∞.As we know, if all the eigenvalues of the Jacobian matrix of closed-loop system have negative real parts, the system is stable.Based on this theory, we desired the T to guarantee that all the eigenvalues of closed-loop system 4.6 have negative real part.There are of course some other choices of T , but here the choice is very easy and convenient.For simplicity

Numerical Results
By using MAPLE 12, the systems of differential equations 4.1 and 4.2 are solved numerically.The parameters are chosen as a 10, r 28, b 8/3, and d 1.3 in all simulations so that the hyperchaotic Lorenz system exhibits a chaotic behavior if no control is applied see Figure 1 .The initial states of the drive system are x 1 0 0.1, y 1 0 0.1, z 1 0 0.1, and w 1 0 0.1, and initial states of the response system are x 2 0 1, y 2 0 −1, z 2 0 1, and w 2 0 1.Choosing α −2 then the error system 4.4 has the initial values e x 0 1.2, e y 0 −0.8, e z 0 1.2, and e w 0 1.2. Figure 3 shows that the trajectories of e x t , e y t , e z t , and e w t tended to zero after t ≥ 5. Figure 4  Choosing α 5 then the error system 4.4 has the initial values e x 0 0.5, e y 0 −1.5, e z 0 0.5, and e w 0 0.5.Figure 5 shows that the trajectories of e x t , e y t , e z t and e w t tended to zero after t ≥ 5. Figure 6   , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α 5.

Generalized Projective Synchronization for Hyperchaotic L ü System
In order to observe generalized projective synchronization between two identical hyperchaotic L ü systems, we assume that the drive system with four state variables denoted subscript 1 and the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system.The drive and response systems are defined below, respectively, where U u 1 u 2 u 3 u 4 T is the controller functions.The controller U is to be determined for the purpose of projective synchronizing the two identical hyperchaotic L ü systems.
In order to get generalized projective synchronization, we define the error system as the difference between the systems 5.

5.4
Let To get the projective synchronization, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as t → ∞.As we know, if all the eigenvalues of the Jacobian matrix of closed-loop system have negative real parts, the system is stable.Based on this theory, we desired the V 1 , V 2 , V 3 , V 4 T to guarantee that all the eigenvalues of closed-loop system 5.6 have negative real part.There are of course some other choices of T , but here the choice is very easy and convenient.For simplicity, choose System 5.6 has four negative eigenvalues −36, −20, −3, and −1.3 when setting a 1 36, b 1 3, c 1 20, and d 1 1.3.That is to say, the error states e x , e x , e z , and e w converge to zero as t → ∞.So the generalized projective synchronization is achieved.

Numerical Results
By using MAPLE 12, the systems of differential equations 5.1 and 5.2 are solved numerically.The parameters are chosen as a 1 36, b 1 3, c 1 20, and d 1 1.3 in all simulations so that the hyperchaotic L ü system exhibits a chaotic behavior if no control is applied see Figure 2 .The initial states of the drive system are x 1 0 −7, y 1 0 −12, z 1 0 7, and w 1 0 11 and initial states of the response system are x 2 0 −4, y 2 0 −6, z 2 0 1, and w 2 0 1.Choosing α −2 then the error system 5.4 has the initial values e x 0 −18, e y 0 −30, e z 0 −13, and e w 0 21. Figure 7 shows that the trajectories of e x t , e y t , e z t , and e w t tended to zero after t ≥ 5. Figure 8 shows the evaluation of the ratios log |x 2 /x 1 |, log |y 2 /y 1 |, log |z 2 /z 1 |, and log |w 2 /w 1 | whose limits are equal to log 2 0.693.
Choosing α 5 then the error system 5.4 has the initial values e x 0 31, e y 0 54, e z 0 −34, and e w 0 −54. Figure 9 shows that the trajectories of e x t , e y t , e z t , and e w t tended to zero after t ≥ 5. Figure 10 shows the evaluation of the ratios log |x 2 /x 1 |, log |y 2 /y 1 |, log |z 2 /z 1 |, and log |w 2 /w 1 | whose limits are equal to log 5 1.609.

Generalized Projective Synchronization between Hyperchaotic Lorenz System and Hyperchaotic L ü System
In order to observe generalized projective synchronization between hyperchaotic Lorenz system and hyperchaotic L ü system, we assume that hyperchaotic Lorenz system is the drive system and hyperchaotic L ü system is the response system.The drive system with four state variables denoted by the subscript 1 and the response system with four state variables denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system.The drive and response systems are defined below, respectively, where U u 1 u 2 u 3 u 4 T is the controller functions.The controller U is to be determined for the purpose of projective synchronizing between hyperchaotic Lorenz system and hyperchaotic L ü system.In order to get generalized projective synchronization, we define the error system as the difference between the systems 6.2 and 6.1 .Set e x x 2 − αx 1 , e y y 2 − αy 1 , e z z 2 − αz 1 , e w w 2 − αw 1 , 6.3 then one obtains the error dynamical system between 6.2 and 6.1  then the error dynamical system can be rewritten as ėx a e y − e x e w V 1 ėy re x − e y V 2 ėz −be z V 3 ėw −de w V 4 .

6.6
To get the projective synchronization, the zero solutions of error system must be stable, that is to say, the error evolution of the drive system and response system tends to zero as t → ∞.As we know, if all the eigenvalues of the Jacobian matrix of closed loop system have negative real parts, the system is stable.Based on this theory, we desired the V 1 , V 2 , V 3 , V 4 T to guarantee that all the eigenvalues of closed-loop system 6.7 have negative real part.There are of course some other choices of T , but here the choice is very easy and convenient.For simplicity, choose V 1 , V 2 , V 3 , V 4 T as follows:   e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α 5.

Numerical Results
By using MAPLE 12, the systems of differential equations 6.1 and 6.  e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α 1. b It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic L ü system error system tends to zero as t tends to 5 when the scaling factor α 1. c It shows that the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α 1. w 1 0 0.1, and initial states of the response system are x 2 0 −7, y 2 0 −12, z 2 0 7, and w 2 0 11.Choosing α −2, then the error system has the initial values e x 0 −6.8, e y 0 −11.8, e z 0 7.2, and e w 0 11.2. Figure 11 shows that the trajectories of e x t , e y t , e z t , and e w t tended to zero after t ≥ 5. Figure 12 shows the evaluation of the ratios log |x 2 /x 1 |, log |y 2 /y 1 |, log |z 2 /z 1 |, and log |w 2 /w 1 | whose limits are equal to log 2 0.693.
Choosing α 5, then the error system has the initial values e x 0 −7.5, e y 0 −12.5, e z 0 6.5, and e w 0 10.5. Figure 13 shows that the trajectories of e x t , e y t , e z t ,  e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α −1.b It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic L ü system error system tends to zero as t tends to 5 when the scaling factor α −1.c It shows that the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α −1.
and e w t tended to zero after t ≥ 5. Figure 14 shows the evaluation of the ratios log

Conclusion
This paper shows that the generalized projective synchronizations for the hyperchaotic Lorenz system and the hyperchaotic L ü system can be easily achieved by using the fundamental feedback techniques, where the scaling factors can be arbitrarily manipulated amplified or reduced .We note that the driver and response systems achieve complete synchronization when α is equal to 1 see Figure 15 .Further, if α is equal to −1, then the two systems are said to be in anti-synchronization see Figure 16 .Numerical simulations are used to verify the effectiveness of the proposed generalized projective synchronization techniques.

Figure 1 :
Figure 1: It shows the attractor of hyperchaotic Lorenz dynamical system at a 10, r 28, b 8/3, and d 1.3 in x, y, z subspace.
6 has four negative eigenvalues −10, −1, −8/3, and −1.3 when setting a 10, r 28, b 8/3, and d 1.3.That is to say, the error states e x , e x , e z , and e w converge to zero as t → ∞.So the generalized projective synchronization is achieved.

Figure 3 :|x 2 /x 1 | log |y 2 /y 1 | log |z 2 /z 1 | log |w 2 /w 1 |Figure 4 :
Figure3: It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α −2.

Figure 5 :
Figure5: It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α 5.

5 . 5 then the error dynamical system can be rewritten as ėx a 1 e y − e x e w V 1 ,Figure 7 :
Figure7: It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic L ü system error system tends to zero as t tends to 5 when the scaling factor α −2.

Figure 11 :
Figure11: It shows the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α −2.
7 has four negative eigenvalues −10, −1, −8/3, and −1.3 when setting a 10, r 28, b 8/3, and d 1.3.That is to say, the error states e x , e x , e z , and e w converge to zero as t → ∞.So the generalized projective synchronization is achieved.

Figure 13 :
Figure13: It shows that the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α 5.

Figure 15 :
Figure15:a It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α 1. b It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic L ü system error system tends to zero as t tends to 5 when the scaling factor α 1. c It shows that the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α 1.

Figure 16 :
Figure16: a Its shows the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic Lorenz system error system tends to zero as t tends to 5 when the scaling factor α −1.b It shows that the behaviour of the trajectories e x , e y , e z , and e w of the hyperchaotic L ü system error system tends to zero as t tends to 5 when the scaling factor α −1.c It shows that the behaviour of the trajectories e x , e y , e z , and e w of the error system between hyperchaotic Lorenz system and hyperchaotic L ü system tends to zero as t tends to 5 when the scaling factor α −1.
and g : R 2k → R k .When μ 0, y r evolves independently and has no relation to x d , and we assume that both systems are chaotic.When μ / 0, the chaotic trajectories of the two systems are said to be generalized synchronization if there exists a transformation ϕ : x d → y r which is able to map asymptotically the trajectories of the master attractor into those of the slave attractor