Rate Estimation of Identical Synchronization by Designing Controllers

This paper investigates the synchronization rate for identical synchronization of chaotic dynamical systems, achieved by using controllers. The paper stresses on the hybrid feedback control technique and the tracking control technique and determines their corresponding maximum, minimum, and average synchronization rates.The results obtained are applied on the Shimizu-Morioka chaotic system, and some necessary and sufficient conditions for synchronization are obtained. Comparison of the two controllers is undertaken on the basis of their synchronization rates, in the context of the Shimizu-Morioka system.The results are analyzed both theoretically and numerically. Moreover, a method of graphical analysis is proposed to completely characterize the set of hybrid controllers for a given system.


Introduction
Chaos control and synchronization has got much attention of the scientists and engineers due to its applicability to various disciplines after the pioneering works by Pecora and Carroll [1] and Ott et al. [2].It is applied in various fields like information processing, secure communication, neural networks, chemical reactions, biological systems, and so on.Sun and Cao [3] proposed the synchronization between two different noise perturbed chaotic systems.Researchers have proposed different synchronization methods and applied them successfully to control chaos and obtain synchronized chaotic system.Notable methods among them are non-linear feedback control [4], active control [5], adaptive control [6], backstepping design [7], hybrid feedback control [8], and so forth.When the two identical chaotic oscillators are mutually coupled or when one of them drives the other, the synchronization that occurs in this case is known as identical synchronization.Let ( 1 (),  2 (), . . .,   ()) and ( 1 (),  2 (), . . .,   ()) denote the variables describing the states of the first and second identical oscillators, respectively, depending on time .For a set of initial conditions ( 1 (0),  2 (0), . . .,   (0)) and ( 1 (0),  2 (0), . . .,   (0)) of the two systems, if lim  → ∞ [  () −   ()] = 0, for  = 1, 2, 3, . . ., , then the identical synchronization occurs.
In this paper, two synchronization schemes for chaotic dynamical systems, using two different controllers [9], are primarily discussed in Section 2. Section 3 deals with measurement of the rate of synchronization [10], when synchronization is achieved using controllers.A quantitative estimate of the rate of convergence of the two systems towards each other in the phase space is obtained.In the following section, the methods stated earlier are applied on coupled Shimizu-Morioka chaotic dynamical system.Synchronization of the chaotic system is achieved both via hybrid feedback control and tracking control, and the results are discussed both analytically and numerically.Finally, the corresponding rates of synchronization are determined and compared.Notably, Section 4.3 explores a method of graphical analysis for determining the possible controller parameters for effective synchronization.It proves to be an efficient and elegant tool for determination of possible controllers where traditional mathematical treatment is rendered inadequate.In fact, it enables us to completely characterize the set of all hybrid controllers for any given problem.

Description of the Hybrid Controller. System of ordinary differential equations can be expressed as
where  ∈ R  ,  ∈ R × ,  ∈ R × , and Φ : R  → R  is the nonlinear part of the system.In the following way, a new system which is identical with the system (1) has been constructed as ( is known as the controller which controls the motion of system (2).Feedback controller  should be chosen appropriately to get the identical synchronization of the systems (1) and (2).The controller  is said to be hybrid if it is a combination of linear controller and non-linear controller.
Here the controller is chosen as  =  1 +  2 , where is the non-linear controller and is the linear controller, with  = diag( 1 ,  2 , . . .,   ) as the feedback matrix.The synchronization error between the systems (1) and ( 2) is defined as  = ( 1 ,  2 , . . .,   )  =  − , where  = ( 1 ,  2 , . . .,   )  and  = ( 1 ,  2 , . . .,   )  .It is easy to observe that the time evolution of the synchronization error obeys the dynamical equation given by where  is the feedback matrix.Yang et al. [8] derived the sufficient condition that if the eigenvalues of the matrix − have negative real parts, then the error dynamical system (5) will be asymptotically stable at origin and the synchronization between the systems (1) and (2) will occur.
It is noted that the error dynamical system is of the form where  ∈ R × .

Description of the Tracking Controller.
As already noted in Section 2.1, any system of first-order non-linear differential equation can be represented as (1).In the context of synchronization of chaotic systems through introduction of tracking controller, a chaotic system of the form (1) is termed as the drive system.Introducing the control vector  ∈ R  , the controlled response system is taken as where  ∈ R  is the state vector of the response system.The aim of this technique is to design the controller  which will synchronize the states of both the drive and the response systems.Thus, the error dynamical system has a representation where  =  −  is the error vector.With a proper choice of controller , the previous system can be put in the form where  ∈ R × .Section 4.5 illustrates a method of choosing the controller suitably.
Let us now construct a Lyapunov function () of the form () = (1/2)  .Evidently, () is positive definite.The controller  has to be chosen suitably so that V < 0 which implies that lim  → ∞ ‖()‖ = 0, and, hence, global synchronization between the states of the drive and response system is achieved asymptotically via the tracking controller .

Theoretical Analysis: Rate Measure for Identical Synchronization via Controllers
As observed already in Section 2, the proper choice of controllers for the coupled chaotic dynamical systems produces an error dynamical systems of the form where  ∈ R  and  ∈ R × .Let us define the Euclidean norm for all  × 1 real matrices  as √   , to be represented, henceforth, as ‖‖.
Proof.Simple mathematical calculation yields It yields an estimate of ‖()‖ as where () and Σ() are the minimum and the maximum eigenvalues of the symmetric matrix  = ( +   )/2, respectively.
Choosing  1 and  2 such that for all  ≥ 0, we obtain a fairly accurate estimate of the average rate of synchronization  = ( 1 +  2 )/2.
Note.In case  becomes a constant matrix, it is easy to obtain the least upper bound and the greatest lower bound of the eigenvalues of  as follows: whereby we obtain the average rate of synchronization  = ( 1 +  2 )/2.

Application of the Results on the Shimizu-Morioka Chaotic System
4.1.The Shimizu-Morioka Chaotic System.Shimizu-Morioka dynamical system is as follows: where  1 ,  2 ,  3 are the state variables and ,  are the parameters.Studies by Shil'nikov [11] revealed that the system exhibits Lorentz-like attractors for  = 0.608,  = 1.0499 and for  = 0.549,  = 0.605.

Synchronization of Shimizu-Morioka Chaotic System
Using Hybrid Controller.For the feedback matrix  = ( Characteristic equation of the matrix  −  is By the previous Routh-Hurwitz criterion, the roots of the above equation in  will have negative real parts if and only if Using ( 3) and (4), the hybrid controller for the Shimizu-Morioka system is given as  = ( ).
Therefore, the response system is (20)

Complete Characterization of Control Parameters for
Hybrid Controller.The necessary and sufficient conditions for synchronization are given in (19), involving the three control parameters  1 ,  2 , and  3 .Due to the complicated nature of the expressions involved, we adopt an alternate path of graphical analysis for explicit determination of "feasible" control parameters, that is, the set of values of ( 1 ,  2 ,  3 ) for which the inequalities in (19) are simultaneously satisfied.In this sense, we can claim to have completely characterized the set of hybrid controllers for the synchronization of chaotic coupled Shimizu-Morioka system.The same procedure can be applied to any coupled dynamical system, thus completely characterizing the set of hybrid controllers for it.The method presented also has potential practical applications in controller design.
The set of figures presented in Figures 1 and 2 illustrate the effectiveness of the method of graphical analysis.Since the problem involves three parameters, a three-dimensional graphical analysis would have been most general.But for the sake of clarity and understandability, we keep one parameter fixed and generate the two-dimensional graphs.The blue regions in all the graphs are the regions of "controllability." For any set ( 1 ,  2 ,  3 ), values within the blue region produce a hybrid controller that drives the response into synchrony with the drive system.Figure 1 fixes  1 at different values and shows the possible values of  2 and  3 .Clearly,  3 > − always for a successful controller design.However, the lower limit on values of  2 decreases with increase of  1 .In Figure 2, when  2 is fixed, again  3 > − always and  1 assumes much lower values as  2 is increased.Thus, high values of  1 and  2 always increases the region of "controllability, " while  3 is constrained to lie above − always.

Synchronization of Shimizu-Morioka Chaotic System via
Tracking Controller.The dynamical system (15) is the drive system.Its identical controlled response system is taken as where  = ( 1 ,  2 ,  3 )  is the controller to be determined and  = ( 1 ,  2 ,  3 )  is the response system.Let us define   =   −   ( = 1, 2, 3).Motivated by the observation stated in outline of this controller design, let us choose the controller as Using systems (15), (23), and (24), the error dynamical system takes the form ė = , where Some trite calculations yield V = −( 2 1 +  2 2 +  2 3 ) < 0. Hence, the tracking controller given by (24) leads to global synchronization between the drive and the response systems.In all the cases, average rate of synchronization is always a constant, where  = (1 + )/2; that is,  =   max = (1 + )/2.

Discussion and Conclusion
Numerical simulations are done with the parameter values  = 0.549 and  = 0.605, and the initial conditions are taken as (0) = (1, 3, 4)  and (0) = (−2, 1, 2)  .In hybrid control technique, the control parameters  1 ,  2 ,  3 play an important role for the synchronization of the drive and the response systems.Time evolution of the synchronization errors is plotted in Figure 3.All the errors vanish with time but the rapidity of vanishing of errors depend on the values of   ( = 1, 2, 3).The synchronization rates are depicted in Figures 4 and 5 for hybrid and tracking controllers, respectively.In both figures, the minimum, maximum, and the average rates of synchronization are shown in addition to ‖()‖.
In hybrid control technique, the rate of synchronization is highly dependent on the control parameters   ( = 1, 2, 3) as depicted in Figures 3 and 4.
It is seen from extensive numerical experiments that the average synchronisation rate function, that is, ‖(0)‖ exp(−), lies extremely close to ‖()‖ (refer to Figures 4 and  5).Clealry, the rate estimation is highly accurate for both the tracking controller and the hybrid controller, illustrating the power of our method of rate estimation.Thus, by and large, the average synchronisation rate function ‖(0)‖ exp(−), where  = average synchronisation rate, can be used as a good approximation to ‖()‖ for practical purposes.
Figure 6 compares the rate of synchronization for hybrid control and the tracking control techniques.It is evident that the hybrid controller may be made better than the  tracking controller by adjusting the parameters of the hybrid controller.It is seen from numerical experiments that if  1 ,  2 and  3 have significantly large values, then hybrid controller is by and large better than the tracking controller in the context of synchronizing the chaotic Shimizu-Morioka system.But choice of arbitrarily low values of  1 and  2 from the region of "controllability" (refer to Section 4.3) causes the tracking controller to work better than the hybrid controller.Thus, we can safely conclude that the choice of comparatively larger parameter values is a sufficient condition for the hybrid controller to be better than the tracking controller.The average rate of synchronization for hybrid controller can be modified at will through parameter adjustments.It is this flexibility that makes hybrid control technique important in chaos synchronization.This is in contrast to tracking controller whose average rate of synchronisation is fairly constant.
As already noted, maximum possible average rate of hybrid control synchronization is   max = ( 1 +  2 )/2 and that of tracking control synchronization is   max = (1 + )/2, which is a constant.Thus, a more rigorous sufficient condition for the hybrid controller to be made superior over tracking controller is (26)

Figure 1 :
Figure 1: Region of "controllability" for different fixed values of  1 .

Figure 3 :
Figure 3: Convergence of ‖()‖ for different parameter choices of the hybrid controller.

Figure 6 :
Figure 6: Comparison of the hybrid and the tracking control techniques in the context of chaotic Shimizu-Morioka system.