A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation
Mahmoud M. El-Borai^{*}, Wagdy G. El-sayed., Aafaf E. Abduelhafid
Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt
Email address:
(M. M. El-Borai)
. (W. G. El-sayed)
(A. E. Abduelhafid)
^{*}Corresponding author
To cite this article:
Mahmoud M. El-Borai, Wagdy G. El-sayed., Aafaf E. Abduelhafid. A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation. American Journal of Theoretical and Applied Statistics. Special Issue: Statistical Distributions and Modeling in Applied Mathematics. Vol. 6, No. 5-1, 2017, pp. 8-12. doi: 10.11648/j.ajtas.s.2017060501.12
Received: January 27, 2017; Accepted: February 3, 2017; Published: February 18, 2017
Abstract: In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic system is considered. Finally, a sufficient condition is given for Lyapunov stability.
Keywords: Lag Synchronization, Parabolic Partial Differential Equation, Uncertain Parameters, Adaptive Technique, Lorenz Chaotic System
1. Introduction
In the past two decades, many schemes for chase synchronization have been proposed, including linear and non-linear, such as in [1, 18-21]. At present, the researchers are concentrating on the following types of synchronization phenomena [23-34]. In this paper, we study generalized adaptive of Lorenz chaotic system with parabolic partial differential equation and with three uncertain coefficients, (see [2-17]).
We investigate the lag synchronization of Lorenz parabolic partial differential chaotic systems with uncertain three coefficients. Based on the generalized adaptive technique, a new controller and coefficient adaptive laws are designed such that coefficients identification is realized and lag synchronization of Lorenz parabolic partial differential chaotic system is achieved simultaneously.
2. A general Chaotic Problem
Let us consider the following generalized chaotic problem:
(1)
,
and are given bounded continuous functions on ,
(a, b, and c are given positive numbers).
The response system is controlled Lorenz Chaotic system as following
(2)
Where and of (2) are unknown functions, which need to be identified in the response system.
It is easy to see that
(3)
Where,
Also,
(4)
Differentiating equation (3) with respect to t, one gets:
(5)
Also:
(6)
Where:
is the controller which should be designed such that two systems can be Lag synchronized.
Let
(7)
Where is the time delay for the error dynamical system. Therefor the goal of parameters identification Lag synchronization is to find an appropriate controller function and parameter adaptive laws and, such that the synchronization errors.
(8)
and the unknown parameters.
(9)
3. Lag Synchronization of Lorenz Chaotic System and the Errors
In This section, we shall study the systems of errors (10) and the appearance of the lag synchronization of systems (1) and (2).
From systems (5) and (6), we get the following errors dynamical systems:
(10)
Obviously, Lag synchronization of system (5) and (6) appears if the errors dynamical system (10) has an asymptotically stable equilibrium point, , where
Theorem1. Assuming that the Lorenz chaotic system (5) derives the controlled Lorenz chaotic system (6), take
(11)
and parameter adaptive laws
(12)
Systems (5) and (6) can realize lag synchronization and the unknown confidents will be identified, i.e.; equation (8) and (9) will be achieved.
Proof Equation (10) can be converted to following form under the controller (11)
(13)
Consider a Lyapunov function as
Where,
.
Obviously, is a positive definite function. Taking its time derivative along with the trajectories of equation (12) and (13) leads to
Where . It is obvious that if and only if
Namely the set
is the largest invariant set contained in for equation (13). So according to the LaSalle’s invariance principle [22], starting with arbitrary initial values of equation (13), the trajectory converges asymptotically to the set ,
i.e.; and as.
This indicates that the lag synchronization of Lorenz chaotic system is achieved and the unknown parameters and can be successfully identified by using controller (11) and parameter adaptive laws (12). This completes the proof of the theorem, (Comp. [34-39].
4. Conclusion
This paper investigates the synchronization problem of coupled nonlinear diffusion systems. The lag synchronization of diffusion Lorenz chaotic system with uncertain coefficients is studied. The controller and coefficients adaptive laws are designed such that coefficient identification is realized and lag synchronization of the diffusion Lorenz system is achieved. The Lyapunov stability is also studied.
Acknowledgments
We would like to thank the referees for their careful reading of the paper and their valuable comments.
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