Generalized Synchronization with Uncertain Parameters of Nonlinear Dynamic System via Adaptive Control

An adaptive control scheme is developed to study the generalized adaptive chaos synchronization with uncertain chaotic parameters behavior between two identical chaotic dynamic systems. This generalized adaptive chaos synchronization controller is designed based on Lyapunov stability theory and an analytic expression of the adaptive controller with its update laws of uncertain chaotic parameters is shown. The generalized adaptive synchronization with uncertain parameters between two identical new Lorenz-Stenflo systems is taken as three examples to show the effectiveness of the proposed method. The numerical simulations are shown to verify the results.


Introduction
The chaos synchronization phenomenon has the following feature: the trajectories of the master and the slave chaotic system are identical in spite of starting from different initial conditions or different nonlinear dynamic system. However, slight differentiations of initial conditions, for chaotic dynamical systems, will lead to completely different trajectories [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The issue may be treated as the control law design for observer of slave chaotic system using the master chaotic system so as to ensure that the controlled receiver synchronizes with the master chaotic system. Hence, the slave chaotic system completely traces the dynamics of the master chaotic system in the course of time [15][16][17][18][19]. The key technique of chaos synchronization for secret communication has been widely investigated. Until now, a wide variety of approaches have been proposed for control and synchronization of chaotic systems, such as adaptive control [20,21], backstepping control [22][23][24][25], sliding mode control [26][27][28], and fuzzy control [29][30][31], just to name a few. The forenamed strategies and many other existing skills of synchronization mainly concern the chaos synchronization of two identical chaotic systems with known parameters or identical unknown parameters [32][33][34][35][36][37][38].
Among many kinds of chaos synchronizations, the generalized synchronization is widely studied. This means that there exists a given functional relationship between the states of the master system and that of the slave system = ( ). In this paper, a new generalized synchronization with uncertain parameters,̇= ( , , ( )) , is studied, where , are the state vectors of the master and slave system, respectively, and the ( ) is uncertain chaotic parameters in . The ( ) may be given a regular/chaotic dynamical system. The rest of the paper is organized as follows. In Section 2, by the Lyapunov asymptotic stability theorem, the generalized synchronization with uncertain chaotic parameters by adaptive control scheme is given. In Section 3, various adaptive controllers and update laws are designed for the generalized synchronization with uncertain parameters of the identical Lorenz-Stenflo systems. The numerical simulation of three examples is also given in Section 3. Finally, some concluding remarks are given in Section 4.
Our goal is to design a controller ( ) and an adaptive laẇso that the state vector of the slave system equation (3) asymptotically approaches the state vector of the master system equation (2) Introduce (2) Its derivative along the solution of (7) iṡ where ( ) anḋare chosen so thaṫ= 1 +̃2̃, 1 and 2 are negative constants, anḋis a negative definite function of 1 , 2 , . . . , and̃1,̃2, . . . ,̃. When the generalized adaptive synchronization with uncertain parameters is obtained.

Results of Numerical Simulation
In this section, a mathematical proof is provided for the three cases' results of numerical, adaptive synchronization, generalized adaptive synchronization, and generalized adaptive synchronization with uncertain parameters.
Define an error vector function The Scientific World Journal From the error functions, we get the error dynamicṡ Choose a Lyapunov function candidate in the form of a positive definite function and its time derivative iṡ Choose the parameters estimation update laws as follows: The initial values of estimates for uncertain parameters arê(0) = 0,̂(0) = 0,̂(0) = 0, and̂(0) = 0. Through (16) and (17), the appropriate controllers can be designed as Substituting (18) and (17) into (16), we obtaiṅ Since the Lyapunov function The Scientific World Journal  negative definite in the neighborhood of the zero solutions for (12a) and (12b), according to the Lyapunov stability theory, the zero solutions of error states dynamic and parameters error vector are asymptotically stable; namely, the slave system equation (11) can asymptotically converge to its master system equation (10) with the adaptive control law equation (18) and the estimation parameter update law equation (17). The adaptive synchronization concept proof had to be completed. The numerical simulation results are shown in Figures 2, 3, and 4. 3.2. Case II Generalized Adaptive Synchronization. The given functional system for generalized synchronization is also a new Lorenz-Stenflo system but with different initial conditions: 1 (0) = 25, 2 (0) = 25, 3 (0) = 25, and 4 (0) = 25: When the time approaches infinite, the error functions approach zero. The generalized adaptive synchronization can be accomplished as where the error functions here can be defined as = + − , ( = 1, 2, 3, 4) .

Case III Generalized Adaptive Synchronization with
Uncertain Parameters. Consider that the master system is the new Lorenz-Stenflo system with uncertain chaotic parame-terṡ1 where 1 ( ), 2 ( ), 3 ( ), and 4 ( ) are uncertain chaotic parameters. The uncertain parameters are given as where 1 , 2 , 3 , and 4 are arbitrary positive constants. Positive constants are 1 = 2 = 3 = 4 = 0.005. The chaotic signals 1 , 2 , 3 , and 4 are given as the states of system as follows:̇1 The initial constants of the chaotic signals are 1 (0) = 0.7, 2 (0) = 0.7, 3 (0) = 0.7, and 4 (0) = 0.7. The new Lorenz-Stenflo system with uncertain chaotic parameters of master system will exhibit a more complex dynamic behavior since the parameters of the system change over time.

Conclusion
A generalized adaptive synchronization with uncertain chaotic parameters is new chaos synchronization concept.    The theoretical analysis and numerical simulation results of three cases, adaptive synchronization, generalized adaptive synchronization, and generalized adaptive synchronization with uncertain parameters, are shown in the corresponding figures which imply that the adaptive controllers and update laws we designed are feasible and effective. In this paper, the three examples can be used to increase the security of secret communication system.