Rate Estimation of Identical Synchronization by Designing Controllers

Abstract

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.

1. 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 (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) and (𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)) denote the variables describing the states of the first and second identical oscillators, respectively, depending on time 𝑡. For a set of initial conditions (𝑥₁(0),𝑥₂(0),…,𝑥_𝑛(0)) and (𝑦₁(0),𝑦₂(0),…,𝑦_𝑛(0)) of the two systems, if lim_𝑡→∞[𝑦_𝑖(𝑡)−𝑥_𝑖(𝑡)]=0, for 𝑖=1,2,3,…,𝑛, 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.

2. Description of the Controllers

2.1. Description of the Hybrid Controller

System of ordinary differential equations can be expressed as 𝑋̇=𝐴𝑋+𝐵Φ(𝑋), (1) where 𝑋∈ℜ^𝑛, 𝐴∈ℜ^𝑛×𝑛, 𝐵∈ℜ^𝑛×𝑛, and Φ∶ℜ^𝑛→ℜ^𝑛 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 𝑌̇=𝐴𝑌+𝐵[Φ(𝑌)+𝑈]. (2)

𝑈 is known as the controller which controls the motion of system (2). Feedback controller 𝑈 should be chosen appropriately to get the identical synchronization of the systems (1) and (2). The controller 𝑈 is said to be hybrid if it is a combination of linear controller and non-linear controller.

Here the controller is chosen as 𝑈=𝑢₁+𝑢₂, where 𝑢₁=Φ(𝑋)−Φ(𝑌) (3) is the non-linear controller and 𝑢₂=𝐾(𝑋−𝑌) (4) is the linear controller, with 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛) as the feedback matrix.

The synchronization error between the systems (1) and (2) is defined as 𝑒=(𝑒₁,𝑒₂,…,𝑒_𝑛)^𝑇=𝑋−𝑌, where 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 and 𝑌=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇. It is easy to observe that the time evolution of the synchronization error obeys the dynamical equation given by 𝑒̇=(𝐴−𝐵𝐾)𝑒, (5) where 𝐾 is the feedback matrix.

Yang et al. [8] derived the sufficient condition that if the eigenvalues of the matrix 𝐴−𝐵𝐾 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 𝑒̇=𝐻𝑒, (6) where 𝐻∈ℜ^𝑛×𝑛.

2.2. 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 𝑈∈ℜ^𝑛, the controlled response system is taken as 𝑌̇=𝐴𝑌+𝐵Φ(𝑌)+𝑈, (7) where 𝑌∈ℜ^𝑛 is the state vector of the response system. The aim of this technique is to design the controller 𝑈 which will synchronize the states of both the drive and the response systems.

Thus, the error dynamical system has a representation 𝑒̇=𝐴𝑒+𝐵Φ(𝑌)−𝜙(𝑋)+𝑈, (8) where 𝑒=𝑌−𝑋 is the error vector. With a proper choice of controller 𝑈, the previous system can be put in the form 𝑒̇=𝐻𝑒, (9) where 𝐻∈ℜ^𝑛×𝑛. Section 4.5 illustrates a method of choosing the controller suitably.

Let us now construct a Lyapunov function 𝑉(𝑒) of the form 𝑉(𝑒)=(1/2)𝑒^𝑇𝑒. Evidently, 𝑉(𝑒) is positive definite. The controller 𝑈 has to be chosen suitably so that 𝑉̇<0 which implies that lim_𝑡→∞‖𝑒(𝑡)‖=0, and, hence, global synchronization between the states of the drive and response system is achieved asymptotically via the tracking controller 𝑈.

3. 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 𝑒̇=𝐻𝑒, (10) where 𝑒∈ℜ^𝑛 and 𝐻∈ℜ^𝑛×𝑛. Let us define the Euclidean norm for all 𝑛×1 real matrices 𝐺 as 𝐺^𝑇𝐺, to be represented, henceforth, as ‖𝐺‖.

Definition 1. If there exist positive constants 𝐶₁, 𝐶₂, 𝑟₁, and 𝑟₂ such that 𝐶₁exp(−𝑟₁𝑡)≤‖𝑒(𝑡)‖≤𝐶₂exp(−𝑟₂𝑡), then 𝑟₁ is defined as the minimum rate of synchronization, 𝑟₂ as the maximum rate of synchronization, and their mean 𝑟=(𝑟₁+𝑟₂)/2 as the average rate of synchronization.
If the Euclidean norm of error, that is, ‖𝑒‖, can be written in the form 𝐶₁exp(−𝑟₁𝑡)≤‖𝑒(𝑡)‖≤𝐶₂exp(−𝑟₂𝑡), where 𝐶₁, 𝐶₂ are positive constants and 𝑟₁, 𝑟₂ are positive, then ‖𝑒(𝑡)‖→0 as 𝑡→∞. Clearly, ‖𝑒(𝑡)‖ tends to zero at least as fast as exp(−𝑟₂𝑡), but not faster than exp(−𝑟₁𝑡). Clearly, for large 𝑟₂, the rate of convergence of ‖𝑒‖ to zero is very rapid while it is much slower for smaller values of 𝑟₂. The trend for 𝑟₁ is also the same. It is noted that if 𝑟₁ and 𝑟₂ are close to each other, then ‖𝑒‖ lies within a narrow window between two exponentially decaying curves. Moreover, its value is approximately the same as that of exp(−𝑟𝑡), where 𝑟=(𝑟₁+𝑟₂)/2. When 𝑟₁ and 𝑟₂ are far apart, then exp(−𝑟𝑡) determines a mean curve and, hence, is a reasonable approximation for ‖𝑒(𝑡)‖. In this sense, we define 𝑟₁ as the minimum rate of synchronization, 𝑟₂ as the maximum rate of synchronization, and 𝑟=(𝑟₁+𝑟₂)/2 as the mean rate of synchronization.

Theorem 2. The average rate of synchronization 𝑟 is given by 𝑟=(𝑟₁+𝑟₂)/2, where 𝑟₁≥sup{−𝜎(𝑡)∣𝑡≥0} and 𝑟₂≤inf{−Σ(𝑡)∣𝑡≥0}, 𝜎(𝑡) and Σ(𝑡) being the minimum and the maximum eigenvalues of the symmetric matrix 𝐺(𝑡)=(𝐻(𝑡)+𝐻(𝑡)^𝑇)/2; 𝐻(𝑡) is defined in (10).

Proof. Simple mathematical calculation yields 12𝑑‖‖𝑑𝑡𝑒(𝑡)²=𝑒^𝑇𝐻𝑒=𝑒^𝑇𝐻^𝑇𝑒. (11)
It yields an estimate of ‖𝑒(𝑡)‖ as 𝑒𝑡₀exp^𝑡_𝑡0𝜎(𝑢)≤‖𝑒(𝑡≤𝑒𝑡𝑑𝑢)‖₀exp^𝑡_𝑡0Σ(𝑢)𝑑𝑢, (12) where 𝜎(𝑡) and Σ(𝑡) are the minimum and the maximum eigenvalues of the symmetric matrix 𝐺=(𝐻+𝐻^𝑇)/2, respectively.
Choosing 𝑟₁ and 𝑟₂ such that −𝑟₁≤𝜎(𝑡)≤Σ(𝑡)≤−𝑟₂ (13) for all 𝑡≥0, we obtain a fairly accurate estimate of the average rate of synchronization 𝑟=(𝑟₁+𝑟₂)/2.

Note. In case 𝐺 becomes a constant matrix, it is easy to obtain the least upper bound and the greatest lower bound of the eigenvalues of 𝐺 as follows: 𝑟₁=sup{−𝜎(𝑡)∣𝑡≥0}≤inf{−Σ(𝑡)∣𝑡≥0}=𝑟₂, (14) whereby we obtain the average rate of synchronization 𝑟=(𝑟₁+𝑟₂)/2.

4. Application of the Results on the Shimizu-Morioka Chaotic System

4.1. The Shimizu-Morioka Chaotic System

Shimizu-Morioka dynamical system is as follows: 𝑥̇₁=𝑥₂,𝑥̇₂=𝑥₁−𝜆𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝛼𝑥₃+𝑥²₁, (15) where 𝑥₁, 𝑥₂, 𝑥₃ are the state variables and 𝜆, 𝛼 are the parameters. Studies by Shil’nikov [11] revealed that the system exhibits Lorentz-like attractors for 𝛼=0.608, 𝜆=1.0499 and for 𝛼=0.549, 𝜆=0.605.

The previous system of (15) can be written as 𝑋̇=𝐴𝑋+𝐵Φ(𝑋) where 𝑋=𝑥₁,𝑥₂,𝑥₃^𝑇,,,Φ(𝑋)=𝐴=0101−𝜆000−𝛼𝐵=1000100010,−𝑥₁𝑥₃,𝑥²₁^𝑇. (16)

4.2. Synchronization of Shimizu-Morioka Chaotic System Using Hybrid Controller

For the feedback matrix 𝐾=^𝑘1000𝑘₂⁰_00𝑘3, we have 𝐴−𝐵𝐾=−𝑘₁000−𝜆−𝑘₂000−𝛼−𝑘₃=𝐻. (17)

Characteristic equation of the matrix 𝐴−𝐵𝐾 is 𝜇³+𝑘₁+𝑘₂+𝑘₃+𝜆+𝛼𝜇²+𝑘₁𝑘₂+𝜆−𝜆+𝑘₁+𝑘₂−𝛼−𝑘₃𝜇+−1−𝛼−𝑘₃−𝑘₁𝜆+𝑘₂−𝛼−𝑘₃=0. (18)

By the previous Routh-Hurwitz criterion, the roots of the above equation in 𝜇 will have negative real parts if and only if 𝑘₁+𝑘₂+𝑘₃𝑘+𝜆+𝛼>0,₁𝑘₂+𝜆−𝜆+𝑘₁+𝑘₂−𝛼−𝑘₃−1>0,−𝛼−𝑘₃+𝑘₁𝜆+𝑘₂𝛼+𝑘₃𝑘>0,₁+𝑘₂+𝑘₃×𝑘+𝜆+𝛼₁𝑘₂+𝜆−𝜆+𝑘₁+𝑘₂−𝛼−𝑘₃−1+𝛼+𝑘₃−𝑘₁𝜆+𝑘₂𝛼+𝑘₃>0. (19)

Using (3) and (4), the hybrid controller for the Shimizu-Morioka system is given as 𝑈=^{𝑘1(𝑥1−𝑦1)−𝑥1𝑥3+𝑦}₁^𝑦₃^{+𝑘2(𝑥2−𝑦}₂⁾_{𝑥21−𝑦21+𝑘3(𝑥3−𝑦3)}. Therefore, the response system is 𝑦̇₁=𝑦₂+𝑘₁𝑥₁−𝑦₁,𝑦̇₂=𝑦₁−𝜆𝑦₂−𝑥₁𝑥₃+𝑘₂𝑥₂−𝑦₂,𝑦̇₃=−𝛼𝑦₃+𝑥²₁+𝑘₃𝑥₃−𝑦₃. (20)

4.3. Complete Characterization of Control Parameters for Hybrid Controller

The necessary and sufficient conditions for synchronization are given in (19), involving the three control parameters 𝑘₁, 𝑘₂, and 𝑘₃. 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 (𝑘₁,𝑘₂,𝑘₃) 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 (𝑘₁,𝑘₂,𝑘₃), values within the blue region produce a hybrid controller that drives the response into synchrony with the drive system. Figure 1 fixes 𝑘₁ at different values and shows the possible values of  𝑘₂ and 𝑘₃. Clearly, 𝑘₃>−𝛼 always for a successful controller design. However, the lower limit on values of 𝑘₂ decreases with increase of 𝑘₁. In Figure 2, when 𝑘₂ is fixed, again 𝑘₃>−𝛼 always and 𝑘₁ assumes much lower values as 𝑘₂ is increased. Thus, high values of 𝑘₁ and 𝑘₂ always increases the region of “controllability,” while 𝑘₃ is constrained to lie above −𝛼 always.

(a) 𝑘1=0.1

(b) 𝑘1=0.2

(c) 𝑘1=0.6

(d) 𝑘1=2.0

Figure 1 

Region of “controllability” for different fixed values of 𝑘1.

(a) 𝑘3=0.55

(b) 𝑘3=6.0

(c) 𝑘2=0.2

(d) 𝑘2=2.0

Figure 2 

Region of “controllability” for different fixed values of 𝑘3 (a), (b) and 𝑘2 (c), (d).

4.4. Rate of Synchronization Using Hybrid Controller

The symmetric matrix 𝐺 is given by −𝑘₁101−(𝜆+𝑘₂)000−𝛼−𝑘₃. It is clear that the problem has become largely simplified because the matrix 𝐺 has become constant because of a suitable controller choice. The characteristic equation for this matrix is 𝛼+𝑘₃+𝜇𝑘₁+𝜇𝜆+𝑘₂+𝜇−1=0 (21) which has roots −𝜇₁, −𝜇₂, and −𝜇₃, where 𝜇₁=𝑘₃𝜇+𝛼,₂=12𝜆+𝑘₁+𝑘₂+𝜆+𝑘₂−𝑘₁²,𝜇+4₃=12𝜆+𝑘₁+𝑘₂−𝜆+𝑘₂−𝑘₁²+4. (22)

With the conditions 𝑘₃+𝛼>0 and 𝑘₁(𝜆+𝑘₂)>1, it is observed that 𝜇₂>𝜇₃. Thus, Σ(𝑡)=−min{𝜇₁,𝜇₃} and 𝜎(𝑡)=−max{𝜇₁,𝜇₂} for all 𝑡≥0. Using Theorem 2,(1)minimum rate of synchronization = 𝑟₁=max{𝜇₁,𝜇₂}; (2)maximum rate of synchronization = 𝑟₂=min{𝜇₁,𝜇₃}; (3)mean rate = 𝑟=(𝑟₁+𝑟₂)/2.

Simple calculations yield the maximum possible value of 𝑟 given by 𝑟^𝐻_max=(𝜇₁+𝜇₂)/2. It is possible to attain 𝑟=𝑟^𝐻_max if the following inequality holds: 𝜇₂>𝜇₃>𝜇₁.

4.5. 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 𝑦̇₁=𝑦₂+𝑢₁,𝑦̇₂=𝑦₁−𝜆𝑦₂−𝑦₁𝑦₃+𝑢₂,𝑦̇₃=−𝛼𝑦₃+𝑦²₁+𝑢₃, (23) where  𝑈=(𝑢₁,𝑢₂,𝑢₃)^𝑇 is the controller to be determined and 𝑌=(𝑦₁,𝑦₂,𝑦₃)^𝑇 is the response system. Let us define 𝑒_𝑖=𝑦_𝑖−𝑥_𝑖(𝑖=1,2,3). Motivated by the observation stated in the outline of this controller design, let us choose the controller as 𝑢₁=−𝑒₁−𝑒₂,𝑢₂=−𝑒₁+𝑦₁𝑦₃−𝑥₁𝑥₃,𝑢₃=−(−𝛼+1)𝑒₃−𝑦²₁+𝑥²₁. (24)

Using systems (15), (23), and (24), the error dynamical system takes the form 𝑒̇=𝐻𝑒, where 𝐻=−1000−𝜆000−1. (25)

Some trite calculations yield 𝑉̇=−(𝑒²₁+𝜆𝑒²₂+𝑒²₃)<0. Hence, the tracking controller given by (24) leads to global synchronization between the drive and the response systems.

4.6. Rate of Synchronization Using Tracking Controller

The symmetric matrix 𝐺 is given by −1000−𝜆000−1. It is clear that the problem has again become largely simplified because the matrix 𝐺 is a constant diagonal matrix due to proper controller choice. Here, Σ(𝑡)=−min{1,𝜆} and 𝜎(𝑡)=−max{1,𝜆} for all 𝑡≥0. By Theorem 2,(i)Case I: 0<𝜆<1, 𝑟₁=𝜆, and 𝑟₂=1;(ii)Case II: 𝜆=1, 𝑟₁=1, and 𝑟₂=1;(iii)Case III: 𝜆>1, 𝑟₁=1, and 𝑟₂=𝜆.

In all the cases, average rate of synchronization is always a constant, where 𝑟=(1+𝜆)/2; that is, 𝑟=𝑟^𝑇_max=(1+𝜆)/2.

5. Discussion and Conclusion

Numerical simulations are done with the parameter values 𝛼=0.549 and 𝜆=0.605, and the initial conditions are taken as 𝑋(0)=(1,3,4)^𝑇 and 𝑌(0)=(−2,1,2)^𝑇. In hybrid control technique, the control parameters 𝑘₁, 𝑘₂, 𝑘₃ 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 𝑘_𝑖(𝑖=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 ‖𝑒(𝑡)‖. In hybrid control technique, the rate of synchronization is highly dependent on the control parameters 𝑘_𝑖(𝑖=1,2,3) as depicted in Figures 3 and 4.

(a)

(b)

Figure 3 

Convergence of ‖𝑒(𝑡)‖ for different parameter choices of the hybrid controller.

(a)

(b)

Figure 4 

Comparison of the proposed “rate” curves with ‖𝑒(𝑡)‖ for synchronization using the hybrid control technique (𝑐=‖𝑒(0)‖).

Figure 5 

Comparison of the proposed “rate” curves with ‖𝑒(𝑡)‖ for synchronization using the tracking control technique (𝑐=‖𝑒(0)‖).

It is seen from extensive numerical experiments that the average synchronisation rate function, that is, ‖𝑒(0)‖exp(−𝑟𝑡), lies extremely close to ‖𝑒(𝑡)‖ (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 ‖𝑒(0)‖exp(−𝑟𝑡), where 𝑟= average synchronisation rate, can be used as a good approximation to ‖𝑒(𝑡)‖ 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 𝑘₁, 𝑘₂ and 𝑘₃ 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 𝑘₁ and 𝑘₂ 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.

(a)

(b)

(c)

(d)

Figure 6 

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

As already noted, maximum possible average rate of hybrid control synchronization is 𝑟^𝐻_max=(𝜇₁+𝜇₂)/2 and that of tracking control synchronization is 𝑟^𝑇_max=(1+𝜆)/2, which is a constant. Thus, a more rigorous sufficient condition for the hybrid controller to be made superior over tracking controller is 𝜇₁+𝜇₂>1+𝜆. (26)

Acknowledgment

The authors acknowledge the valuable insights and constructive criticism of the anonymous reviewers, which have played a significant role in giving the paper its final form.