Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters

doi:10.1016/j.physleta.2009.08.027

Physics Letters A

Volume 373, Issue 41, 5 October 2009, Pages 3743-3748

https://doi.org/10.1016/j.physleta.2009.08.027 Get rights and content

Abstract

This Letter investigates modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system using adaptive method. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two hyperchaotic systems modified function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.

Introduction

Chaos synchronization, an important topic in nonlinear science, has been developed and studied extensively in the last few years. Since Pecora and Carroll [1] introduced a method to synchronize two identical systems with different initial conditions, a variety of approaches have been proposed for the synchronization of chaotic systems which include complete synchronization [1], phase synchronization [2], generalized synchronization [3], lag synchronization [4], intermittent lag synchronization [5], time scale synchronization [6], intermittent generalized synchronization [7], projective synchronization [8] and modified projective synchronization [9], [10].

Projective synchronization is characterized that the drive and response system could be synchronized up to a scaling factor where as in modified projective synchronization (MPS) the responses of the synchronized dynamical states synchronize up to a constant scaling matrix. Recently a more general form of projective synchronization called function projective synchronization [11], [12] in which drive and response systems are synchronized up to a desired scaling function has attracted much attention of scientists and engineers as it provides more secure communication in applications to secure communication.

Most of the works mentioned so far involved mainly with low-dimensional chaotic systems with only one positive Lyapunov exponent. Hyperchaotic systems possessing at least two positive Lyapunov exponents have more complex behaviour and abundant dynamics than chaotic systems and are more suitable for some engineering applications such as secure communication. Hence how to realize synchronization of hyperchaotic systems is an interesting and challenging work. Fortunately, some existing methods of synchronizing low-dimensional chaotic systems like adaptive control, active control, active backstepping control, sliding mode control methods can be generalized to synchronize hyperchaotic systems [13], [14], [15], [16], [17], [18], [19]. In practical situations parameters are probably unknown and may change time to time. Therefore, how to effectively synchronize two hyperchaotic systems with unknown parameters is an important problem for theoretical research and practical applications. Among different methods of synchronizing two hyperchaotic systems, adaptive control method is an effective one for achieving synchronization of hyperchaotic systems with full unknown parameters [20].

Recently L. Runzi [21] presented function projective synchronization (FPS) of two identical Rossler hyperchaotic systems using adaptive control method. As we all know, the synchronization of between different hyperchaotic systems is very important in engineering applications because they have different complex dynamic behaviours. However FPS between two different hyperchaotic systems is seldom reported.

An important point to be considered with regard to synchronization is that the signal effecting upon the synchronized system is very small in comparison with the amplitude of proper oscillations of the system but, nevertheless, enough to change the system behaviour and impose the rhythm of external influence to it. So we have to make sure that the external influence U applied on the response system is really small and, correspondingly, it is correct to speak about synchronization phenomenon.

Motivated by the above discussions, in this Letter, we propose modified function projective synchronization (MFPS) of two different hyperchaotic systems. Modified function projective synchronization, a more general definition of function projective definition, can provide more security in secure communications and allow us to vary the magnitude of the control signal so that the external influence on the response system imposing synchronization is really small. To illustrate the effectiveness of the proposed MFPS scheme the modified function projective synchronization between hyperchaotic Lorenz and hyperchaotic Lu systems is investigated using adaptive control method. To our knowledge, function projective synchronization between these systems is not explored.

The organization of the rest of this Letter is as follows. Section 2 discusses modified function projective synchronization scheme. Section 3 gives a brief description of hyperchaotic Lorenz and hyperchaotic Lu systems. In Section 4, we present the adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system. In Section 5, a numerical example is given to demonstrate the effectiveness of the proposed method. Finally some concluding remarks are given.

Section snippets

Adaptive modified function projective scheme

Consider the following master and slave system $\dot{x} = g (t, x),$ $\dot{y} = h (t, y) + u (t, x, y),$ where $x, y \in R^{n}$ are the state vectors, $g, h : R^{n} \to R^{n}$ are continuous nonlinear vector functions, $u (t, x, y)$ is the vector controller. We define the error system as $e (t) = x - M f (t) y,$ where M is a constant diagonal matrix $M = diag (m_{1}, m_{2}, \dots, m_{n}) \in R^{n \times n}$ and $f (t)$ a continuous differentiable function with $f (t) \neq 0$ for all t.

The system (1), (2) are said to be in modified function projective synchronization, if there exists a constant diagonal

System description

The hyperchaotic Lorenz system is described as follows [22]: $\dot{x} = α (y - x),$ $\dot{y} = β x + y - x z - w,$ $\dot{z} = x y - γ z,$ $\dot{w} = θ y z,$ where x, y, z and w are state variables and α, β, γ, θ are parameters. When $α = 10$ , $β = 28$ , $γ = 8 / 3$ and $θ = 0.1$ , the system (4) exhibits hyperchaotic behaviour (Fig. 1(a)–(b)).

By using a feedback controller, a novel hyperchaotic Lu system was constructed based on the original three-dimensional Lu system which is given by the following equations [23]: $\dot{x} = a (y - x) + w,$ $\dot{y} = - x z + c y,$ $\dot{z} = x y - b z,$ $\dot{w} = x z + d w,$ where x, y, z

Adaptive modified function projective synchronization between hyperchaotic Lorenz system and Lu system

In order to achieve the behaviour of modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system, we assume that the hyperchaotic Lorenz system is the drive system whose four variables are denoted by subscript 1 and the hyperchaotic Lu system is the response system whose variables are denoted by subscript 2. The drive and response systems are described, respectively, by the following equations: ${\dot{x}}_{1} = α (y_{1} - x_{1}),$ ${\dot{y}}_{1} = β x_{1} + y_{1} - x_{1} z_{1} - w_{1},$ ${\dot{z}}_{1} = x_{1} y_{1} - γ z_{1},$ ${\dot{w}}_{1} = θ y_{1} z_{1},$ $x$

Numerical simulations

Numerical simulations are presented to demonstrate the effectiveness of the proposed synchronization controller. Fourth-order Runge–Kutta method is used to solve systems (6), (7), (10) with time step size 0.001. The parameters are chosen to be $α = 10$ , $β = 28$ , $γ = 8 / 3$ , $θ = 0.1$ , $a = 36$ , $b = 3$ , $c = 20$ and $d = 1$ in all simulations so that the hyperchaotic Lorenz system and Lu system exhibit chaotic behaviours if no control inputs are applied. The initial conditions of the drive system are $x_{1} (0) = - 1$ , $y_{1} (0) = - 1$ , $z_{1} (0) =$

Conclusion

This Letter investigated the modified function projective synchronization between the hyperchaotic Lorenz system and the Lu system with fully uncertain parameters. On the basis of Lyapunov stability theory, we design adaptive synchronization controllers with corresponding parameter update laws to synchronize the two systems. All the theoretical results are verified by numerical simulations to demonstrate the effectiveness of the proposed synchronization schemes.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. We express our thanks to the University Grants Commission, India, for providing financial support through the DSA scheme.

References (23)

G.H. Li
Chaos Solitons Fractals
(2007)
G.H. Li
Chaos Solitons Fractals
(2007)
H. Du et al.
Phys. Lett. A
(2008)
K.S. Sudheer et al.
Phys. Lett. A
(2009)
Q. Jia
Phys. Lett. A
(2007)
X. Zhou et al.
Appl. Math. Comput.
(2008)
M.T. Yassen
Chaos Solitons Fractals
(2008)
U.E. Vincent
Chaos Solitons Fractals
(2008)
H. Zhang
Chaos Solitons Fractals
(2005)
M. Jang et al.
Chaos Solitons Fractals
(2002)

J. Huang

Phys. Lett. A

(2008)

Cited by (65)

Modified function projective feedback control for time-delay chaotic Liu system synchronization and its application to secure image transmission
2017, Optik
This paper is devoted to synchronize time-delay Liu chaotic system in order to develop a secure image transmission system. An adaptive modified function projective synchronization (MFPS) method is proposed to control the behavior of the uncertain Liu chaotic system in the presence of time-delay state variables and uncertainty in system parameters. The stability and convergence of the present method are analyzed by means of Lyapunov stability theorem. Numerical simulations were used to validate the effectiveness of the proposed MFPS method. Furthermore, a novel image cryptosystem is generated for secure image transmission. Image data is embedded within the response signals and is transmitted to the receiver. The image data can be extracted at the receiver by using of the drive signals. The proposed image cryptosystem was applied to a plain image, and also, performance of the image cryptosystem was evaluated by some mathematical tools and the results were satisfactory, in both cases.
Generalized matrix projective outer synchronization of non-dissipatively coupled time-varying complex dynamical networks with nonlinear coupling functions
2017, Neurocomputing
This paper investigates the generalized matrix projective outer synchronization (GMPOS) between two non-dissipatively coupled time-varying complex dynamical networks via open-plus- closed-loop dynamical compensation controllers. To be more consistent with the real-world networks, besides non-dissipatively couplings, the drive and response networks in our paper can possess different nodes, different time-varying outer coupling configuration matrices and different nonlinear inner coupling functions. Thus, our network models are more general and extensive than almost all of those in the existing literatures about outer synchronization of networks. In order to make our drive and response networks realize the GMPOS, the open-plus-closed-loop dynamical compensation controllers are designed based on the Lyapunov stability theory and Barbalat's lemma. Moreover, the speed of achieving the GMPOS between two networks can be improved by adjusting the parameters in our dynamical compensation systems. A simulation example with different hyperchaotic nodes is given to verify the effectiveness and feasibility of our theoretical results.
Delay-dependent synchronization for non-diffusively coupled time-varying complex dynamical networks
2015, Applied Mathematics and Computation
This paper investigates the delay-dependent synchronization schemes for the non-diffusively coupled time-varying complex dynamical networks. The outer coupling configuration matrix in our network model may be non-diffusive, time-varying, uncertain, asymmetric and irreducible. Different time-varying coupling delays for different nodes are also put into consideration in this paper. Besides, the nodes may have different state dimensions. Furthermore, only the common bound of the outer coupling coefficients (CBOCC) is used to design the synchronization controllers. If the CBOCC is known, our delay-dependent synchronization scheme can guarantee the network achieving exponential synchronization. And when the CBOCC is uncertain, the adaptive synchronization scheme, where only one adaptive law is needed, is proposed to guarantee the network realizing asymptotic synchronization. Simulation examples are provided to verify the effectiveness and feasibility of our theoretical results.
Partial switched modified function projective synchronization of unknown complex nonlinear systems
2015, Optik
Citation Excerpt :
In the past decades, chaos synchronization has become a hot subject in the field of nonlinear science due to its great potential applications in physical systems, biological networks, information sciences and secure communications, etc. Up to now, a variety of synchronization approaches have been proposed which include complete synchronization [1], phase synchronization [2], generalized synchronization [3], lag synchronization [4], projective synchronization [5], modified projective synchronization [6,7], function projective synchronization [8–10] and modified function projective synchronization [11–16]. Modified function projective synchronization is the more general type of synchronization, which means that the master and slave systems could be synchronized up to a scaling function matrix.
Many studies on the switched modified function projective synchronization of nonlinear real dynamic systems have been carried out, whereas the switched modified function projective synchronization of chaotic complex systems has not been studied extensively. In this paper, the concept of the partial switched modified function projective synchronization of complex nonlinear systems with fully unknown parameters is proposed. Partial switched synchronization of chaotic systems means that the state variables of the drive system synchronize with partial different state variables of the response system. Based on the parameter modulation and the adaptive control technique, the adaptive control laws and the parameter update laws are derived to make the states of two different complex nonlinear systems asymptotically synchronized up to a desired scaling function matrix. The given scaling function can be an equilibrium point, a periodic orbit, or even a chaotic attractor in the phase space. Finally, numerical simulations are performed to verify the feasibility and effectiveness of the proposed controllers and identifiers.
Multivariable Function Projective Synchronization of High Dimensional Chaotic Systems and Its Secure Communication Scheme
2023, Jisuan Wuli/Chinese Journal of Computational Physics
An Efficient and Reliable Chaos-Based IoT Security Core for UDP/IP Wireless Communication
2022, IEEE Access

View all citing articles on Scopus

View full text

Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters

Abstract

Introduction

Section snippets

Adaptive modified function projective scheme

System description

Adaptive modified function projective synchronization between hyperchaotic Lorenz system and Lu system

Numerical simulations

Conclusion

Acknowledgements

Chaos Solitons Fractals

Chaos Solitons Fractals

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Appl. Math. Comput.

Chaos Solitons Fractals

Chaos Solitons Fractals

Chaos Solitons Fractals

Chaos Solitons Fractals

Phys. Lett. A