Study on Proportional Synchronization of Hyperchaotic Circuit System

Transcription

1 Commun. Theor. Phys. (Beijing, China) 43 (25) pp c International Academic Publishers Vol. 43, No. 4, April 15, 25 Study on Proportional Synchronization of Hyperchaotic Circuit System JIANG De-Ping, 1 LUO Xiao-Shu, 1 WANG Bin-Hong, 2 FANG Jin-Qing, 3 and JIANG Pin-Qun 2 1 College of Physics and Information Engineering, Guangxi Normal University, Guilin 5414, China 2 Department of Modern Physics, University of Science and Technology of China, Hefei 2326, China 3 China Institute of Atomic Energy, Beijing 12413, China (Received September 1, 24) Abstract In this paper, the proportional synchronization between drive system and response system is achieved by using the concept of generalized synchronization. The phase space of all variables in response system can be expanded and compressed flexibly. Meanwhile, the 6-D hyperchaotic chua s circuit is considered as an illustrative example to demonstrate the effectiveness of the proposed approach. Furthermore, focusing on the shortcoming of the long transient behavior during the process of synchronization, a feedback method is adopted to shorten the transitional time of synchronization, which will provide an effective way for speeding up the transmitting velocity of code in chaotic multiple access communication. PACS numbers: a Key words: hyperchaos, generalized synchronization, projective synchronization, proportional synchronization 1 Introduction Synchronization of chaotic system and its application to secure communication have received a great deal of attention in recent years. [1 8] During the former studies on synchronization of chaotic system, identical synchronization, phase synchronization, generalized synchronization, [9,1] and projective synchronization [11,12] were achieved. In this paper, we obtain a new kind of synchronization according to the concept of generalized synchronization, which can make all variables of drive system and response system reach proportional synchronization. This kind of synchronization is different from projective synchronization of chaotic system. Projective synchronization can only make partial corresponding variables between response system and drive system reach proportional relationship, but the proportional synchronization we proposed can make all corresponding variables between drive system and response system satisfy proportional relationship. The phase space of response system can be expanded and compressed flexibly. Meanwhile, aiming at the problem that the synchronous speed is very slow, which will seriously influence the transmitting velocity of code in chaotic multiple access communication and may cause the phenomenon of ISI (inter-symbol interference), a method via state feedback is adopted to promote the synchronous velocity and reduce the synchronous transitional time. Computer simulations indicate that this approach will provide an effective way for solving the problems we mentioned above. 2 Principle of Generalized Synchronization as A nonlinear dynamical system can always be expressed ẋ = Ax + ϕ(x), (1) where A is a constant matrix, A R n n, Ax is the linear part, ϕ : R n R n. The nonlinear function ϕ(x) is taken as drivable signal to drive the response system. Considering synchronization of unidirectional transmission, the response system can be written as ẏ = Ay + Ωϕ(x), (2) where Ω R n n. If Ω and A satisfy AΩ = ΩA and all the eigenvalues of A have negative real parts, the response system and the drive system can reach generalized synchronization as t. [9,1] The corresponding relationship of all variables in the two systems is given by Proof Let z = y Ωx, then y = H(x) = Ωx. (3) ż = ẏ Ωẋ = Ay + Ωϕ(x) Ω[Ax + ϕ(x)] = Ay ΩAx = Az. (4) We can reach a conclusion that if the real parts of all eigenvalues of A are negative, the synchronization error system has a globally asymptotically stable equilibrium point for z =. Only when A and Ω satisfy the abovementioned conditions can generalized synchronization be The project supported by National Natural Science Foundation of China under Grant No , the Guangxi New Century Foundation for Ten, Hundred, and Thousand Talents under Grant No , and Natural Science Foundation of Guangxi Province of China under Grant No

2 672 JIANG De-Ping, LUO Xiao-Shu, WANG Bin-Hong, FANG Jin-Qing, and JIANG Pin-Qun Vol. 43 obtained between the response system Eq. (2) and the drive system Eq. (1), i.e., y Ωx = H(x). 3 Proportional Synchronization of Hyperchaotic Chua s Circuit The early report [13] proposed that two chua s circuits can be coupled into 6-D hyperchaotic system. The circuit dynamics can be described by the following equations in dimensionless form, ẋ 1 = α[x 2 x 1 f(x 1 )], ẋ 2 = x 1 x 2 + x 3 + m(x 5 x 2 ), ẋ 3 = βx 2, ẋ 4 = α[x 5 x 4 f(x 4 )], ẋ 5 = x 4 x 5 + x 6, ẋ 6 = βx 5, (5) where f(x) = bx + (a b)( x 1 x + 1 )/2, in which α, β, a, and b are constant parameters and m is a coupled coefficient. The circuit will exhibit a hyperchaotic behavior under the following conditions: m =.2, α = 1., β = 14.87, a = 1.27, and b =.68. Here λ 1, λ 2 are the two positive Lyapunov exponents of hyperchaotic system, where λ 1 =.431, λ 2 =.412. [13] In order to implement generalized synchronization in the 6-D hyperchaotic circuit, we introduce an adjustable parameter first, and make the eigenvalues of A lie in the open left-half plane. Secondly we turn Eq. (5) into Eq. (1). x = (x 1, x 2, x 3, x 4, x 5, x 6 ) T are the state variables of drive system, v is an adjustable parameter. After transformation, the corresponding matrix of linear part and nonlinear function are given by v v 1 1 m 1 m β A =, v v β ϕ(x) = (v α)(x 1 x 2 ) αf(x 1 ). (v α)(x 4 x 5 ) αf(x 4 ) Calculation shows the eigenvalues of A are w 1 = , w 2 = 8.841, w 3,4 =.891 ± i, and w 5,6 =.799 ± i when adjustable parameter v is 8.. Obviously, the real parts of all eigenvalues of A are negative, which meets one of the conditions of generalized synchronization. Suppose we select Ω which satisfies this relationship AΩ = ΩA, generalized synchronization can be reached in hyperchaotic chua s circuit in accordance with the concept of generalized synchronization. Therefore, we can draw a conclusion y = H(x) = Ωx. (6) We select Ω = u u u, u u u where u is a constant, clearly AΩ = ΩA. Then according to Eq. (6), the relationship of variables between response system and drive system can be described by lim i ux i =, i = 1, 2,..., 6, (7) t where y = (y 1, y 2, y 3, y 4, y 5, y 6 ) T are the state variables of response system. Generalized synchronization can be turned into proportional synchronization in accordance with Eq. (7). In addition, we can find that it is not partial variables but all variables that satisfy the proportional relationship from Eq. (7). Therefore, the response system implementing proportional synchronization in hyperchaotic chua s circuit can be written as ẏ 1 = v(y 1 y 2 ) + u[(v α)(x 1 x 2 ) αf(x 1 )], ẏ 2 = y 1 (1 + m)y 2 + y 3 + my 5, ẏ 3 = βy 2, ẏ 4 = v(y 4 y 5 ) + u[(v α)(x 4 x 5 ) αf(x 4 )], ẏ 5 = y 4 y 5 + y 6, ẏ 6 = βy 5. (8) If u = 1, proportional synchronization becomes identical synchronization; if u < 1, the range of the phase space shrinks; if u > 1, the range of the phase space amplifies. We carry out computer simulations in order to observe the variation of phase space. A time step is.5, the initial conditions of the system are x = (.56,.1,.1,.56,.1,.1), y = (1., 1., 1., 2.,.5, 1.) (In all of the following simulations, the above parameters are the same).

3 No. 4 Study on Proportional Synchronization of Hyperchaotic Circuit System 673 Fig. 1 The curve of synchronization error (y 1 ux 1), (y 3 ux 3), (y 4 ux 4) respectively with u = 2. Fig. 2 The size of the hyperchaotic attractor of the response system is dramatically compressed to 5 times as small as that of the drive system (u =.2). Figures 1 3 show the results of simulations. Figure 1 shows the curve of synchronization error (y 1 ux 1 ), (y 3 ux 3 ), (y 4 ux 4 ) respectively, which all varies from time to time, where u = 2. Figure 2 shows the comparison between the phase spaces of y 1 and y 3 in the response system and the phase spaces of x 1 and x 3 in original drive system (compressed by five times, u =.2). Figure 3 shows the comparison between the phase spaces of y 4 and y 5 in the response system and the phase spaces of x 4 and x 5 in original drive system (expanded twice, u = 2). We can find that

4 674 JIANG De-Ping, LUO Xiao-Shu, WANG Bin-Hong, FANG Jin-Qing, and JIANG Pin-Qun Vol. 43 the expanding and compressing phenomena of phase space are very obvious from Figs. 2 and 3. We also can observe that the synchronous transitional time is very long from Fig. 1, which will seriously influence transitional velocity of code in chaotic multiple access communication and cause ISI (inter-symbol interference). So the information signal may not recover without distortion, which does no good to the designation of communication system. Therefore, we are determined to promote the synchronous velocity of proportional synchronization by adding state feedback. Fig. 3 The size of the hyperchaotic attractor of the response system is dramatically amplified to 2 times as large as that of the drive system (u = 2). 4 Speeding up Proportional Synchronization via State Feedback The shorter the synchronous transitional time, the faster the synchronous velocity and the longer the maintainable time, then the better the quality of information signal recovers in chaotic secure communication system. The above-mentioned synchronous speed is very slow, which is inappropriate for designation of the communication scheme, so we can speed up the above proportional synchronization by adopting the method of state variables feedback. The response system is designed as ẏ = Ay + Ωϕ(x) KCy + KCΩx, (9) where C R 1 n, K R n 1. Ω is the same as mentioned above. The response system Eq. (9) has an additional item ( KCy + KCΩx) compared with Eq. (2). Then from the following demonstration, the synchronous speed can be accelerated by adjusting the elements of K and C in this additional item. Proposition If AΩ = ΩA, Ω is the same as we mentioned above and matrix (A KC) is stable, the response system (9) can obtain proportional synchronization with the drive system (1). Furthermore we can speed up the synchronous velocity freely according to the variation of K and C. Proof Assume z = y Ωx, then ż = Ay + Ωϕ(x) KCy + KCΩx Ω[Ax + ϕ(x)] = Ay ΩAx KC(y Ωx) = Ay AΩx KC(y Ωx) = (A KC)(y Ωx) = (A KC)z, (1) where K = (k 1, k 2, k 3, k 4, k 5, k 6 ) T, we select k = (2.8, 3.29, 15.4,.18, 2.59,.4) T, C = (1,,, 1,, ). Therefore, z will be asymptotically stable if the real parts of all the eigenvalues of A KC are negative, that is to say the response system (9) will proportionally synchronize the drive system (1). Meanwhile, we can select appropriate K and C to speed up the velocity of proportional synchronization. In order to demonstrate the effectiveness of our approach in improving the synchronous speed of the above-mentioned proportional synchronization, we design the response system using the 6-D hyperchaotic chua s circuit over again. The corresponding response system is rewritten as ẏ 1 = v(y 1 y 2 ) + u[(v α)(x 1 x 2 ) αf(x 1 )] k 1 ((y 1 + y 4 ) u(x 1 + x 4 )), ẏ 2 = y 1 (1 + m)y 2 + y 3 + my 5 k 2 ((y 1 + y 4 ) u(x 1 + x 4 )), ẏ 3 = βy 2 k 3 ((y 1 + y 4 ) u(x 1 + x 4 )), ẏ 4 = v(y 4 y 5 ) + u[(v α)(x 4 x 5 ) αf(x 4 )] k 4 ((y 1 + y 4 ) u(x 1 + x 4 )),

5 No. 4 Study on Proportional Synchronization of Hyperchaotic Circuit System 675 ẏ 5 = y 4 y 5 + y 6 k 5 ((y 1 + y 4 ) u(x 1 + x 4 )), ẏ 6 = βy 5 k 6 ((y 1 + y 4 ) u(x 1 + x 4 )). Here all eigenvalues of A KC are w 1 = , w 2 = , w 3,4 = ±3.881i, w 5,6 =.851±3.6692i. The drive system (5) and the response system (11) can reach proportional synchronization according to Eq. (1). Fig. 4 The curve of synchronization error (y 1 ux 1), (y 3 ux 3), (y 4 ux 4) respectively after speeding up via state feedback method, with u = 2. Obviously the eigenvalues of A KC is more negative than that of A, which indicates that the synchronous velocity will be faster than ever based on the above discussion. The simulation results can be described by Fig. 4. Figure 4 shows the curve of synchronization error (y 1 ux 1 ), (y 3 ux 3 ), (y 4 ux 4 ) respectively which all varies from time to time after speeding up via state feedback method, where u = 2. Compared with Fig.1 correspondingly, the synchronous velocity increases and the synchronous time decreases obviously, which prove that our method in improving the synchronous speed is very effective. Meanwhile, this approach will establish a better foundation for the project of chaotic digital secure communication. 5 Conclusions We obtain proportional synchronization between response system and drive system in accordance with the principle of generalized synchronization, which can make all corresponding variables in the two systems meet the proportional relationship. Moreover, synchronous velocity is promoted and synchronous time is decreased greatly by adding state feedback, which will have a great significance in the designation for the scheme of chaotic secure communication. This kind of method also can improve transmitting speed of code in chaotic digital communication and effectively avoid generating phenomenon of ISI (inter-symbol interference) during the transmission of digital code, and benefit for the recovery of information signal without distortion as better as possible. The further studies of our work on proportional synchronization for chaotic multiply access communication will be reported later.

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