Commun. Theor. Phys. (Beijing, China) 44 (25) pp. 72 78 c International Acaemic Publishers Vol. 44, No. 1, July 15, 25 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal YAN Zhen-Ya Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, the Chinese Acaemy of Sciences, Beijing 18, China (Receive September 28, 24) Abstract In this paper, a systematic an powerful scheme is propose to aress a generalize-type synchronization of a class of continuous-time systems, which inclues generalize lag synchronization, generalize anticipate synchronization, an generalize synchronization. The presente scheme is use to investigate the generalize-type synchronization of the 4D hyperchaotic oscillator an the hyperchaotic oscillator with gyrators. Numerical simulations are use to verify the effectiveness of the propose scheme. The scheme is more powerful than the scalar signal scheme ue to Grassi an Mascolo. PACS numbers: 5.45.Xt Key wors: hyperchaotic system, generalize-type synchronization, 4D hyperchaotic oscillator, hyperchaotic oscillator with gyrators, numerical simulation 1 Introuction In recent years, chaos control an synchronization have attracte increasing attention ue to its potential applications incluing secure communication. [1 4] Since the pioneering work of Pecora an Carroll, [5,6] many control methos have been presente to apply the synchronization of chaotic systems. So far, there exist several types of synchronization that have been researche for chaotic systems namely, complete synchronization (CS), [5,6] phase synchronization (PS), [4,7] generalize synchronization(gs), [7 9], lag synchronization(ls), [7,1] anticipate synchronization(as), [11] anti-phase synchronization (APS). [12] To increase the level of security, the aoption of hyperchaotic systems, characterize by more than one positive Lyapunov exponents, is more avantageous than the application of chaotic systems with only one positive Lyapunov exponent. Up to now, many hyperchaotic systems have been reporte an were synchronize by means of ifferential methos. Recently, Grassi an Mascolo [13,14] presente a nonlinear observer esign to synchronize hyperchaotic systems via a scalar transmitte signal an applie it to the 4D hyperchaotic oscillator, [15] ẋ 1 =.7x 1 x 2 x 3, ẋ 2 = x 1, ẋ 3 = 3(x 1 x 4 ), ẋ 4 = 3x 3 3(x 4 1)Q(x 4 1). (1) where Q(z) is the Heavisie function efine by {, z <, Q(z) = 1, z, an the hyperchaotic oscillator with gyrators [16] ẋ 1 =.55x 1 x 2 4(x 1 x 3 1)Q(x 1 x 3 1), (2) ẋ 2 = x 1, ẋ 3 = 1.31 x 4 + 4.31 (x 1 x 3 1)Q(x 1 x 3 1), ẋ 4 = 1.33 x 3. (3) More recently, Li et al. [17] extene the above metho to investigate lag synchronization of hyperchaotic Rossler system [18] an MCK circuit with hyperchaos. [19] but the metho was only use to the case that all uncontrollable eigenvalues of the error system ha negative real parts. But as they sai, the esign was only use to the case that all uncontrollable eigenvalues of the error systems ha negative real parts. In this paper, we will present a systematic nonlinear vector signal esign to investigate the new type of generalize-type synchronization (GTS) of hyperchaotic systems. The GTS contains CS, LS, AS, GS, generalize lag synchronization (GLS), generalize anticipate synchronization (GAS), which is efine as the presence of certain relationship between the states of the rive an response systems, i.e., there exists a smooth vector function H such that y(t) = H(x(t τ)) with τ R. The propose metho is base on active control theory. [2] An then the propose scheme is applie to stuy the GTS of the 4D hyperchaotic oscillator (1) an the hyperchaotic oscillator with gyrators (3). Numerical simulations are use to verify the effectiveness of the propose scheme. 2 Nonlinear Observe Design for Generalize (Lag, Anticipate, an Complete) Synchronization Consier these two chaotic systems: ẋ = Ax + F (x) + C, (rive system), (4) ẏ = By + G(y) + P, (response system) (5) The project supporte by National Natural Science Founation of China uner Grant No. 14139, the National Key Basic Research Project of China uner Grant No. 24CB318, an the Scientific Research Founation for the Returne Overseas Chinese Scholars, the Ministry of Eucation of China E-mail: zyyan@mmrc.iss.ac.cn
No. 1 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal 73 where x, y R n are the state vectors, F (x), G(y) : R n R n are continuous nonlinear vector functions, C, P R n are the external input constant vectors, A = (a ij ) n n, B = (b ij ) n n R n R n. Many known chaotic systems belong to system (4) such as Lorenz system, Rössler system, Chen system, Duffing system, Chua s circuit, transforme Rossler system, hyperchaotic Rossler system, etc. Definition The rive system an response system are sai to be generalize lag synchronize (τ > ), generalize anticipate synchronize (τ < ), or generalize synchronize (τ = ) if the error state e(t) = y(t) H(x(t τ)) as t +, where τ R is a constant, H(x(t τ)) = (H 1 (x(t τ)),..., H n (x(t τ))) T is a smooth vector function. Remark 1 When H( ) is an ientical operator, the above-mentione synchronization becomes lag synchronization (τ > ), anticipate synchronization (τ < ), or complete synchronization (τ = ), respectively. For the given chaotic system (4) with the vector output s(h(x(t))) R n, the ynamical system ẏ(t) = Ay(t) + F (y(t)) + C + g(s(h(x(t τ))) s(y(t)), (6) is sai to be a nonlinear generalize observer of the rive system (4) if lim t + [y(t) H(x(t τ))] =, where g : R n R n is a properly chosen nonlinear vector function. Moreover system (6) is sai to be a global observer of system (4) if lim t + [y(t) H(x(t τ))] = for any initial conitions x() an y(). That is to say, the error ynamical system ė(t) = ẏ(t) Ḣ(x(t τ)) = ẏ(t) DH(x(t τ))ẋ(t τ) = Ay(t) + F (y(t)) + C + g(s(h(x(t τ))) s(y(t)) DH(x(t τ))[ax(t τ) + F (x(t τ)) + C], (7) amits a globally asymptotically stable equilibrium point for e =, where DH(x(t τ)) is the Jacobian matrix of H(x(t τ)) H 1 (x(t τ)) H 1 (x(t τ)) H 1 (x(t τ)) x 1 (t τ) x 2 (t τ) x n (t τ) H 2 (x(t τ)) H 2 (x(t τ)) H 2 (x(t τ)) x DH(x(t τ)) = 1 (t τ) x 2 (t τ) x n (t τ), (8)... H n (x(t τ)) H n(x(t τ)) H n (x(t τ)) x 1 (t τ) x 2(t τ) x n (t τ) Proposition 1 Let s(h(x(t τ))) = (D t A + )H(x(t τ)), s(y(t)) = (D t A + )y(t), (9) g(s(h(x(t τ))) s(y(t))) = s(h(x(t τ))) s(y(t), (1) where D t = iag(/t, /t,..., /t), = δ ij R n n. Then the generalize (lag (τ > ), anticipate (τ < ), an complete (τ = )) synchronization occurs between systems (4) an (6) if δ ii > a ii, δ ij = a ij (i > j) or δ ii > a ii, δ ij = a ij (i < j). Proof The substitution of Eq. (9) an (1) into the generalize-type synchronization error system (7) yiels ė = ẏ D t H(x(t τ)) = Ay(t) + f(y(t)) + C + (D t A + )H(x(t τ)) (D t A + )y(t) D t H(x(t τ)) = (A )e(t), When δ ii > a ii, δ ij = a ij (i > j) or δ ii > a ii, δ ij = a ij (i < j), we have that all eigenvalues of the error ynamical system (11) are all the negative reals, i.e., lim t + e(t) = lim t + y(t) H(x(t τ)) =. Therefore the generalize (lag (τ > ), anticipate (τ < ) an complete (τ = )) synchronization occurs between systems (6) an (4) occurs. This completes the proof of the proposition. Remark 2 There also exist other types of solutions for δ ij such that the error system (11) is globally asymptotically stable. Remark 3 The propose scheme is more powerful than the scheme use in Refs. [13], [14], an [17]. 3 Generalize-type Synchronization of 4D Hyperchaotic Oscillators The simple 4D oscillators can be written as (11)
74 YAN Zhen-Ya Vol. 44 ẋ 1 ẋ 2 ẋ 3 ẋ 4 = A x 1 x 2 x 3 x 4 3(x 4 1)Q(x 4 1), A =.7 1 1 1 3 3 3, (12) System (12) contains an opamp, two LC circuits, an a ioe. Its hyperchaotic behaviors an synchronization have been investigate. [14,15] The (x[1], x[2], x[4]) an (x[2], x[3], x[4]) projects of hpyerchaotic attractors of system (12) are shown in Figs. 1 an 2. Fig. 1 (x[1], x[2], x[4]) projects of hpyerchaos. Fig. 2 (x[2], x[3], x[4]) projects of hpyerchaos. The erive system is chosen as ẏ 1.7 1 1 ẏ 2 ẏ 3 = 1 3 3 3 ẏ 4 y 1 y 2 y 3 y 4 where the vector transmitte signal s(h(x(t τ))) is 3(y 4 1)Q(y 4 1) + s(h(x(t τ))) s(y(t)), (13) s(h(x(t τ))) = (D t A + )H(x(t τ)) t.7 + δ 11 1 + δ 12 1 + δ 13 δ 14 1 + δ 21 t + δ 22 δ 23 δ 24 H 1 (x(t τ)) H 2 (x(t τ)) = 3 + δ 31 δ 32 t + δ H 33 3 + δ 34 3 (x(t τ)), (14) H 4 (x(t τ)) δ 41 δ 42 3 + δ 43 t + δ 44 where = (δ ij ) n n is a parameter matrix to be etermine later. Let e(t) = y(t) H(x(t τ)). Then the error ynamical systems between (12) an (13) is ė 1.7 δ 11 1 δ 12 1 δ 13 δ 14 e 1 ė 2 ė 3 = 1 δ 21 δ 22 δ 23 δ 24 e 2 3 δ 31 δ 32 δ 33 3 δ 34 e 3. (15) ė 4 δ 41 δ 42 3 δ 43 δ 44 If δ 11 >.7, δ 22 >, δ 33 >, δ 44 >, δ 21 = 1, δ 31 = 3, δ 32 = δ 41 = δ 42 =, δ 43 = 3, (16) then system (15) has four negative eigenvalues.7 δ 11, δ 22, δ 33, δ 44. Therefore the error system is globally asymptotically stable, i.e., lim t + e(t) = lim t + y(t) H(x(t τ)) =. In what follows we woul like to use the numerical simulations to verify the effectiveness of the above-esigne scheme. The initial values of system (12) an (13) as x 1 () = 1, x 2 () =.5, x 3 () = 2, x 4 () = 5, an y 1 () = 2, e 4
No. 1 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal 75 y 2 () = 1, y 3 () = 3, y 4 () = 6. Here we only consier the special case that the smooth vector function H(x(t τ)) is the hyperbolic function H(x(t τ)) = [sech(x 1 (x τ)), sech(x 2 (x τ)), tanh(x 3 (x τ)), tanh(x 4 (x τ))] T. (17) Case 1 Generalize lag synchronization In the case τ >, without loss of generality, we set τ = 1. Let δ 11 = 2, δ 22 = 3, δ 33 = 4, δ 44 = 6, δ 12 = 1, δ 13 = 2, δ 14 = 5, δ 23 = 2, δ 24 = 3, δ 34 = 3, which satisfies Eq. (15). Then the initial values of the error ynamical system (15) is e 1 () = y 1 () sech(x 1 ( 1)) = 2.419 613 335, e 2 () = y 2 () sech(x 2 ( 1)) =.63 74 917 4, e 3 () = y 3 () tanh(x 3 ( 1)) = 2. 21 61, e 4 () = y 4 () tanh(x 4 ( 1)) = 7. The ynamical of generalize lag synchronization errors for the rive system (12) an the response system (13) are shown in Figs. 3(a) 3(). Fig. 3 Generalize lag synchronization errors: (a) e 1(t) = y 1(t) sech(x 1(t 1)), (b) e 2(t) = y 2(t) sech(x 2(t 1)), (c) e 3(t) = y 3(t) tanh(x 3(t 1)), () e 4(t) = y 4(t) tanh(x 4(t 1)). Case 2 Generalize anticipate synchronization In the case τ <, without less of generality, we set τ = 2. Let δ 11 = 2, δ 22 = 3, δ 33 = 4, δ 44 = 1, δ 12 = 1, δ 13 = 4, δ 14 = 5, δ 23 = 2, δ 24 = 3, δ 34 = 11, which satisfies Eq. (16). The initial values of the error ynamical system (15) is e 1 () = y 1 () sech(x 1 (2)) = 2.8 44 881, e 2 () = y 2 () sech(x 2 (2)) =.966 58 31, e 3 () = y 3 () tanh(x 3 (2)) = 2. 18 592, e 4 () = y 4 () tanh(x 4 (2)) = 5.14 73 3. The ynamical of generalize lag
76 YAN Zhen-Ya Vol. 44 synchronization errors for the rive system (12) an the response system (13) are shown in Figs. 4(a) 4(). Fig. 4 Generalize anticipate synchronization errors: (a) e 1(t) = y 1(t) sech(x 1(t+2)), (b) e 2(t) = y 2(t) sech(x 2(t+ 2)), (c) e 3(t) = y 3(t) tanh(x 3(t + 2)), () e 4(t) = y 4(t) tanh(x 4(t + 2)). Case 3 Generalize synchronization In the case τ =, let δ 11 = 2, δ 22 = 5, δ 33 = 1, δ 44 = 8, δ 12 = 1, δ 13 = 4, δ 14 = 15, δ 23 = 2, δ 24 = 2, δ 34 = 8, which satisfies Eq. (16). The initial values of the error ynamical system (15) is e 1 () = y 1 () sech(x 1 ()) = 2.648 54 274, e 2 () = y 2 () sech(x 2 ()) =.113 181 116, e 3 () = y 3 () tanh(x 3 ()) = 2.35 972 42, e 4 () = y 4 () tanh(x 4 ()) = 5. 9 796. The ynamical of generalize lag synchronization errors for the rive system (12) an the response system (13) are shown in Figs. 5(a) 5().
No. 1 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal 77 Fig. 5 Generalize synchronization errors: (a) e 1(t) = y 1(t) sech(x 1(t)), (b) e 2(t) = y 2(t) sech(x 2(t)), (c) e 3(t) = y 3(t) tanh(x 3(t)), () e 4(t) = y 4(t) tanh(x 4(t)). 4 Hyperchaotic Oscillators with Gyrators Consier the hyperchaotic oscillators with gyrators ẋ 1.55 1 ẋ 2 ẋ 3 = 1 1/.31 1/33 ẋ 4 x 1 x 2 x 3 x 4 4(x 1 x 3 1)Q(x 1 x 3 1) 4(x 1 x 3 1)Q(x 1 x 3 1), (18) where Q(z) is the Heavisie function efine by Eq. (13). System (18) contains a negative impeance converter, two capacitors, two gyrators an a ioe. The hyperchaotic oscillators realise without inuctance coils has been escribe in Ref. [16], an synchronization has been investigate. [14] Accoring to the above-mentione scheme, the erive system is given by ẏ 1 ẏ 2 ẏ 3 ẏ 4 =.55 1 1 1/.31 1/33 y 1 y 2 y 3 y 4 4(y 1 y 3 1)Q(y 1 y 3 1) 4/.31(y 1 y 3 1)Q(y 1 y 3 1) + s(h(x(t τ))) s(y(t)) (19) with the vector transmitte signal is s(h(x(t τ))) = (D t A + )H(x(t τ)) t.55 + δ 11 1 + δ 12 δ 13 δ 14 1 + δ 21 = t + δ H 1 (x(t τ)) 22 δ 23 δ 24 H 2 (x(t τ)) δ 31 δ 32 t + δ 33 1/.31 + δ H 3 (x(t τ)). (2) 34 H 4 (x(t τ)) δ 41 δ 42 1/.33 + δ 43 t + δ 44 Let e(t) = y(t) H(x(t τ)). Then the error ynamical system between the rive system an the response system ė 1.55 δ 11 1 δ 12 δ 13 δ 14 e 1 ė 2 ė 3 = 1 δ 21 δ 22 δ 23 δ 24 e 2 δ 31 δ 32 δ 31 1/.33 δ 34 e 3. (21) ė 4 δ 41 δ 42 1/.33 δ 43 δ 44 e 4
78 YAN Zhen-Ya Vol. 44 If δ 11 >.55, δ 22 >, δ 33 >, δ 44 >, δ 21 = 1, δ 31 = δ 32 = δ 41 = δ 42 =, δ 43 = 1/.33, (22) then system (21) has four negative eigenvalues.55 δ 11, δ 22, δ 33, δ 44. Therefore the error system is global asymptotically stable, i.e., lim t + e(t) = lim t + y(t) H(x(t τ)) =. Similar to Sec. 3, we also use the numerical simulations to verify the effectiveness of the above-esigne controllers. Here we omit them. 5 Conclusions an Discussions In brief, we have presente a systematic scheme to investigate the generalize (lag, anticipate, an complete) synchronization between the rive system an response system base on the active control iea. Moreover we have chosen the 4D hyperchaotic oscillator an the hyperchaotic oscillator with gyrators. Numerical simulations are use to verify the effectiveness of the propose scheme. The scheme was also applie to other chaos systems. [21] Moreover the propose hyperchaotic generalize-type synchronization can be also applie to secure communications. This will be consiere in future. References [1] G. Chen an X. Dong, From Chaos to Orer, Worl Scientific, Singapore (1998). [2] T. Yang, et al., IEEE. Trans. CAS 44 (1997) 469. [3] T. Yang an L.O. Chua, Int. J. Bifurcation an Chaos 7 (1997) 645. [4] M.G. Rosenblum an A.S. Pikovsky, Phys. Rev. Lett. 76 (1995) 184. [5] L.M. Pecora an T.L. Caroll, Phys. Rev. Lett. 64 (199) 821. [6] T.L. Caroll an L.M. Pecora, IEEE Trans. Circ. Syst. I 38 (1991) 453. [7] S. Boccaletti, et al., Phys. Rep. 366 (22) 1. [8] N.F. Rulkov, et al., Phys. Rev. E 51 (1995) 98. [9] L. Kocarev an U. Parlitz, Phys. Rev. Lett. 76 (1996) 1816. [1] M.G. Rosenblum, et al., Phys. Rev. Lett. 78 (1997) 4193. [11] S. Sivaprakasam, et al., Phys. Rev. Lett. 87 (21) 15411. [12] M. Ho, et al., Phys. Lett. A 296 (22) 43. [13] G. Grassi an S. Mascolo, Electronics Lett. 34 (1998) 1844. [14] G. Grassi an S. Mascolo, Electronics Lett. 34 (1998) 424. [15] A. Tamasevicius, et al., Electronics Lett. 32 (1996) 957. [16] A. Tamasevicius, et al., Electronics Lett. 33 (1997) 542. [17] C. Li, et al., Chaos, Solitons an Fractals 23 (25) 183. [18] O.E. Rossler, Phys. Lett. A 71 (1979) 155. [19] T. Matsumoto, et al., IEEE Trans. Circ. Syst. I, 33 (1986) 1143. [2] E. Bai an K.E. Lonngren, Chaos, Solitons an Fractals 8 (1997) 51. [21] Z.Y. Yan, Chaos 15 (25) 1311.