Hybrid synchronization of hyperchaotic n-scroll Chua circuit using adaptive backstepping control

In this paper, hybrid synchronization is investigated for n-scroll hyperchaotic Chua circuit using adaptive backstepping control. The theorem on hybrid synchronization for n-scroll hyperchaotic Chua circuit is established using Lyapunov stability theory. The backstepping scheme is recursive procedure that links the choice of Lyapunov function with the design of a controller and guarantees global stability performance of strict-feedback nonlinear systems. The backstepping control method is effective and convenient to hybrid synchronize the hyperchaotic systems which are mainly in this technique that gives the flexibility to construct a control law. Numerical simulations are also given to illustrate and validate the hybrid synchronization results derived in this paper. Subjects: Science; Technology; Systems & Control


PUBLIC INTEREST STATEMENT
Chaos synchronization can be applied in the areas of physics, engineering and biological science. Synchronization has been widely explored in a variety of fields including physical chemical, and ecological systems, secure communications etc. Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator. Because the butterfly effect which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a challenging problem. In most synchronization approaches, the master-slave or drive-response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the slave system so that the output of the response system tracks the output of the master system asymptotically.

Introduction
Synchronization in chaos refers to the tendency of two or more systems which are coupled together to undergo closely related motion, even when the motions are chaotic.
Recently, backstepping method has been developed and designed to control the chaotic systems. A common concept in the method is to synchronize the chaotic system. The backstepping method is based on the mathematical model of the examined system, introducing new variables into a form depending on the state variables, controlling parameters, and stabilizing functions. The difficult work of synchronizing the chaotic system is to remove nonlinearities which were done in the system and influencing the stability of state operation. The use of backstepping method creates an additional nonlinearity and eliminates undesirable nonlinearities from the system (Suresh & Sundarapandian, 2012c;2013;Wang, Zhang, & Guo, 2010;Wang 2011aWang , 2011b.
The uncertainties are commonly in chaos synchronization and other control system problems. The uncertainties are one of the main factors in leading the adaptive-based synchronization. Adaptive control design is a direct aggregation of control methodology with some form of recursive system which identifies the system to determine the control of linear or nonlinear systems.
Adaptive control design is studied and analyzed in theory of unknown, but fixed parameter systems. The controller feedback gain could be depending on the system parameter.
If F i = G i for all i, then the system (1) and (2) are called identical and otherwise they are nonidentical chaotic systems.
The hybrid synchronization error is defined as Then the synchronization error dynamics is obtained as The parameter estimation error is defined as The hybrid synchronization problem basically requires the global asymptotically stability of the error dynamics (4), i.e. for all initial conditions e(0) ∈ R n . (1) Backstepping design procedure is recursive and guarantee global stability performance of strictfeedback chaotic systems. By using the backstepping design, at the ith step, the ith order subsystem is stabilized with respect to a Lyapunov function V i , by the virtual control i , and a control input function u i .
Consider the global asymptotic stability of the system where u 1 is control input, which is the function of the error vector e i , and the state variables x(t) ∈ R n , y(t) ∈ R n . As long as this feedback stabilizes, the system (6) will converge to zero as t → ∞, where e 2 = 1 (e 1 ) is regarded as a virtual controller.
For the design of 1 (e 1 ) is to stabilize the subsystem (6), the Lyapunov function is defined by where P 1 , and R 1 are positive definite matrices.
The derivative of e i is Suppose the derivative of V 1 is where Q 1 , and S 1 are positive definite matrices.
Then V 1 is a negative definite function.
Thus by Lyapunov stability theory, the error dynamics(6) is globally asymptotically stable.
The function 1 (e 1 ) is an estimative function when e 2 is considered as a controller.
The error between e 2 and 1 (e 1 ) is Consider the (e 1 , w 2 ) subsystem given by Let e 3 as a virtual controller in system (11).
Consider the Lyapunov function defined by where P 2 , and R 2 are positive definite matrices.
Suppose the derivative of V 2 (e 1 , w 2 ) is where Q 1 , Q 2 , and S 2 are positive definite matrices.
Then V 2 (e 1 , w 2 ) is a negative definite function.
Thus by Lyapunov stability theory, the error dynamics (11) is globally asymptotically stable. The virtual controller e 3 = 2 (e 1 , w 2 ) and the state feedback input u 2 make the system (11) asymptotically stable.
For the nth state of the error dynamics, define the error variable w n as Considering the (e 1 , w 2 , … , w n ) subsystem given by Consider the Lyapunov function defined by where P n , and R n are positive definite matrices.
Suppose the derivative of V n (e 1 , w 2 , w 3 ....w n ) is where Q 1 , Q 2 , … , Q n , S n are positive definite matrices.
The virtual controller is and the state feedback input u n makes the system (16) globally asymptotically stable.
Hence, the state of master and slave systems are globally and asymptotically synchronized.

System description
Recently, theoretical design and hardware implementation of different kinds of chaotic oscillators have attracted increasing attention, aiming real-world applications of many chaos-based technologies and information systems.
The n-scroll hyperchaotic Chua circuit (Yu, Lu, & Chen, 2007) is given by the dynamics The recursive positive switching points z i (i = 2, 3, 4, ..., N − 1) can be deduced as and the k i values are obtained as in which x E i are the positive equilibrium points of g(x 2 − x 1 ).

Case 1 : 2-scroll hyperchaotic attractor
The parameters of the systems (20) are taken in the case of hyperchaotic case as = 2, = 20.

Hybrid Synchronization of n-scroll hyperchaotic Chua circuits via backstepping control with recursive feedback
In this section, the backstepping method with recursive feedback function is applied for the hybrid synchronization of identical hyperchaotic n-scroll Chua circuits (Yu et al., 2007).
The n-scroll hyperchaotic Chua circuit is taken as the master system, which is described by where g(x 2 − x 1 ) is given by where x(t)(i = 1, 2, 3, 4) ∈ 4 are state variables.
The n-scroll hyperchaotic Chua circuit is also taken as the slave system, which is described by where g(y 2 − y 1 ) is given by where y(t)(i = 1, 2, 3, 4) ∈ 4 are state variables.
The hybrid synchronization error is defined by The error dynamics is obtained as The modified error dynamics is defined by Now the objective is to find control law u i , i = 1, 2, 3, 4 and for the parameter update law ̂,̂,̂,̂0 for stabilizing the system (32) at the origin.
First consider the stability of the system where e 3 is regarded as virtual controller.

Consider the Lyapunov function defined by
Let define the parameter estimation error as Differentiating the Equation (36) Differentiate V 1 along with the Equation (37) Assume the controller e 3 = 1 (e 4 ).
The function 1 (e 4 ) is an estimative function when e 3 is considered as a controller.
The error between e 3 and 1 (e 4 ) is Consider the (e 1 , w 2 ) subsystem given by Let e 2 be a virtual controller in system (43).
Assume that when e 2 = 2 (e 4 , w 2 ) and the system (43) is made globally asymptotically stable.
Define the error variable e 2 and 2 (e 4 , w 2 ) as Consider the (e 4 , w 2 , w 3 ) subsystem given by Let e 1 be a virtual controller in system (53).
Assume when it is equal to e 1 = 3 (e 4 , w 2 , w 3 ), the system (53) is made globally asymptotically stable.
Thus, the states of master and slave systems are globally and asymptotically hybrid synchronized.

Theorem
The identical n-scroll hyperchaotic Chua's circuit (27) and (29) are globally and asymptotically hybrid synchronized with the adaptive backstepping controls and with the parameter update laws

Numerical simulation
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the differential Equations (27) and (29) with the backstepping controls u 1 , u 2 , u 3 , and u 4 given by (29).

Case 1: 2-scroll hyperchaotic attractor
The parameters of the systems (27) are taken in the case of hyperchaotic case as = 2, and = 20.

Conclusion
In this paper, the adaptive backstepping control method has been applied to achieve global chaos hybrid synchronization for a family of n-scroll hyperchaotic Chua circuits. The backstepping control is a systematic procedure for hybrid synchronizing hyperchaotic systems and there is no derivative