Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl. 36 (00) 5 74 Content lit available at ScienceDirect Journal of Mathematical Analyi and Application Chao control and chao ynchronization for multi-croll chaotic attractor generated uing hyperbolic function F. Xu, Pei Yu Department of Applied Mathematic, The Univerity of Wetern Ontario, London, Ontario, N6A 5B7, Canada article info abtract Article hitory: Received 6 September 008 Available online 8 March 009 Submitted by Goong Chen Keyword: Multi-croll chaotic attractor Hyperbolic tangent function Feedback control Lyapunov function Global and exponential tabilization Global and exponential ynchronization In thi paper, we deign a erie of chaotic ytem that can generate one-directional, two-directional and three-directional multi-croll chaotic attractor. Then, baed upon the propertie of thee chaotic ytem, we contruct appropriate Lyapunov function and deign imple linear feedback control to globally exponentially tabilize and ynchronize thee chaotic ytem. Numerical imulation reult are alo preented to how the applicability of the propoed control law. 009 Publihed by Elevier Inc.. Introduction The tudy on multi-croll chaotic attractor ha received increaing attention over the pat few decade. The firt multicroll chaotic attractor can be traced back to Chua circuit [, which diplay a double-croll chaotic attractor. Later, one-directional n-croll chaotic attractor were developed [ 9. Recently the reearch on multi-croll chaotic ytem wa extended to the tudy of two-directional and three-directional chaotic attractor [0. Meanwhile, piecewie linear function played an important part in the deigning of multi-croll chaotic attractor. Chua circuit ued piecewie linear function a the non-linear term [. Recent reearch how that imple piecewie linear function could be extended to the multilevel piecewie linear function to generate multi-directional, multi-croll chaotic attractor [. Chao control and chao ynchronization play a very important role in the tudy of chaotic ytem and have great ignificance in the application of chao. Since chao i very enitive to it initial condition, chao control and chao ynchronization were once believed to be impoible. However, two important dicoverie completely changed thi ituation. One of them i the OGY method [3 developed in the 990. The other i the concept of chao ynchronization propoed by Pecora and Garrol [4 in 990. Up to now, the tudy of chao control and chao ynchronization wa motly focued on claical chaotic ytem uch a the Lorenz ytem, Chua circuit, Chen ytem, Lü ytem etc. (e.g., ee [ 7). Very few reult have been obtained for chao control and chao ynchronization of more complicated multi-directional multi-croll chaotic attractor. In thi paper, other than applying the traditionally widely ued piecewie linear function, we ue hyperbolic function erie in a imple linear ytem to generate a erie of one-directional, two-directional and three-directional multi-croll chaotic attractor. The hyperbolic function, a a continuouly differentiable function, i eaier to analyze chao control and chao ynchronization. On the other hand, ince hyperbolic tangent function i frequently ued to characterize the behavior * Correponding author. Fax: addre: pyu@pyu.apmath.uwo.ca (P. Yu) X/$ ee front matter 009 Publihed by Elevier Inc. doi:0.06/j.jmaa

2 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig.. (a) Sigmoidal hyperbolic function f (x) = tanh(x); (b) piecewie linear tair function; (c) hyperbolic tangent function erie f (x) = j= tanh(x + jτ ) with τ = 0; and (d) hyperbolic tangent function erie f (x) = j= tanh(x + jτ ) with τ = 0, howing the envelop tangent line to the graph, and the number of interection point of the line y = τ x with the graph being =. of neural network, uch a tudy may help undertand complex dynamical behavior exiting in neural network. Then, baed on the propertie of the chaotic ytem, we deign imple feedback control to globally exponentially tabilize and ynchronize the chaotic ytem. Correponding Lyapunov function were preented and ued to prove the propoed control law. Then numerical imulation example were howed to verify the applicability of the control law. The ret of the paper i organized a follow. In Section, the hyperbolic tangent function and hyperbolic tangent function erie are defined. Then -D-n-croll, -D-m n-grid-croll, 3-D-m n l-grid-croll chaotic attractor are obtained in Section 3, 4 and 5, repectively. Chao control and chao ynchronization for -D-n-croll, -D-m n-grid-croll and 3-D-m n l-grid-croll chaotic attractor are tudied in Section 6, 7 and 8, repectively. Concluion i given Section 9.. Definition of hyperbolic tangent function erie The hyperbolic tangent function i decribed by the following equation: f (x) = tanh x = inh x coh x = ex e x e x + e x () which, a hown in Fig. (a), i monotone increaing for < x < and bounded, < tanh(x) <. Thi i a typical characteritic function in decribing many neural network [5, which can be replaced by piecewie linear function (ee Fig. (b)) particularly ued in circuit deign. Recently, multilevel piecewie linear function have been ued in CNN [5. Thi multilevel piecewie linear function may be replaced by an infinitely differentiable function, decribed by the following o-called hyperbolic tangent function erie: F (x) = tanh(x + jτ ), j= r ()

3 54 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 where the parameter τ i a poitive real value, while the parameter r and are non-negative integer. The function F (x) i depicted in Fig. (c), indicating that thi function mooth the multilevel piecewie linear function (hown in Fig. (b)). The baic idea in generating multi-croll chaotic attractor i to add the above hyperbolic tangent function erie to a linear ytem, given by [ ẋ = y, ẏ = z, ż = ax by cz, (3) where a, b and c are contant parameter. It i eay to ee that the unique equilibrium point of the linear ytem (3) i the origin (x, y, z) = (0, 0, 0). A linear analyi how that when a > 0, c > 0, bc > a, the equilibrium point i aymptotically table. 3. Deign of -D-n-croll chaotic attractor Baed on (3), our propoed model for -D chao generator i given a follow: ẋ = y, ẏ = z, ż = ax by cz + τ [ a (r ) + tanh(x + jτ ), (4) j= r where a, b, c and τ are poitive real value, while and r take non-negative integer, determining the maximum number of chaotic croll. For example, when a = b = c = 0.65, τ = 00, and = r = 0, a double-croll chaotic attractor can be obtained, a hown in Fig. (a) and (b). The time hitory for the x component i depicted in Fig. (c). The double-croll chaotic ytem (4) ha three equilibrium point, S 0 = (0, 0, 0) and S, = (±50, 0, 0). The eigenvalue aociated with the equilibrium point S 0 are λ =.9036 and λ,3 =.7768 ±.7949i, indicating that S 0 i a addle point of index one ince the real part of the conjugate eigenvalue i negative. The eigenvalue aociated with the equilibrium point S, are λ = 0.87 and λ,3 = ± i, implying that S, are addle point of index two ince the real part of the conjugate eigenvalue i poitive. It i known that only equilibria of addle point of index two can generate croll [6,7. Therefore, S,, correponding to the two equilibria of addle point of index two, are reponible for the generation of the double croll, while S 0,onthe other hand, i reponible for the connection of the two chaotic croll. Another approach to determine whether a ytem i chaotic or not i to compute it Lyapunov exponent. If a ytem ha at leat one poitive Lyapunov exponent, and all the trajectorie are ultimately bounded, then the ytem i chaotic [8. A chao i called hyperchao if the ytem ha two or more poitive Lyapunov exponent. For ytem (4) with the given parameter value, numerical imulation reult how that trajectorie are bounded for different initial point. Further, a numerical method [9 ha been employed to obtain the following three Lyapunov exponent: L = 0.069, L = 0.005, L 3 = Hence, L > 0 implie that the ytem i chaotic. The imulation reult of the Lyapunov exponent i hown in Fig. (d). When proper coefficient are choen, ytem (4) can have (r + + ) addle point of index one (denoted by Γ x ) and (r + + ) addle point of index two (denoted by Γ x ): Γ x { τ,( )τ,..., rτ }, { Γ x ( + ) τ },( )τ,..., (r + )τ, (5) where the ubcript x denote the non-zero coordinate of the addle point. Equilibrium point in et Γ x, which have correponding eigenvalue λ < 0 and λ,3 = ξ ± ηi with ξ>0 and η 0, are reponible for the generation of the r + + croll. Equilibrium point in et Γ x, which have correponding eigenvalue λ > 0 and λ,3 = ξ ± ηi with ξ<0 and η 0, are reponible for the connection of the r + + croll. Thu, ytem (4) can generate a many a r + + chaotic croll. For example, when a = b = c = 0.65, τ = 00, = r = 3, the correponding hyperbolic tangent function erie can produce 8-croll chaotic attractor, ee Fig. 3(a) and (b), and the time hitory for the x component i hown in Fig. 3(c). The equilibrium point and their correponding eigenvalue are lited in Table. It i een from the table that the equilibrium point of type two, S 7,8, S 9,0, S,, S 3,4, are reponible for the generation of the 8 croll, while the equilibrium

4 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig.. Simulated reult for a double-croll chaotic attractor of ytem (4) with a = b = c = 0.65, τ = 00, r = = 0: (a) the phae portrait in the x y z plane; (b) the phae portrait projected on the x y plane; (c) the time hitory x(t); and (d) the Lyapunov exponent. point of type one, S 0, S,, S 3,4, S 5,6, are reponible for the connection of the 8 croll chaotic attractor. The Lyapunov exponent are found to be L = 0.337, L = 0.004, L 3 = , howing that the ytem i chaotic. The numerical computation reult of thee Lyapunov exponent i hown in Fig. 3(d). 4. Deign of -D-m n-grid-croll chaotic attractor The -D multi-croll chaotic attractor can be obtained by adding two hyperbolic tangent function erie to ytem (3) to obtain the following ytem: ẋ = y τ [ (r ) + tanh(y + jτ ), j= r ẏ = z, [ [ ż = ax by cz + a τ (r ) + j= r tanh(x + jτ ) + b τ (r ) + j= r tanh(y + jτ ), (6) where a, b, c, τ and τ are poitive real value, while r, r,, are non-negative integer. m and n are determined by m = r + +, n = r + +. (7)

5 56 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 Fig. 3. Simulated reult for -D 8-croll chaotic attractor of ytem (4) with a = b = c = 0.65, τ = 00, r = = 3: (a) the phae portrait in the x y z plane; (b) the phae portrait projected on the x y plane; (c) the time hitory x(t); and (d) the Lyapunov exponent. Table Equilibrium point and their correponding eigenvalue for the cae of 8 chaotic croll (Fig. 3). Equilibrium point Correponding eigenvalue Type of equilibrium point S 0 (0, 0, 0).9037,.7768 ±.7949i I S, (±00, 0, 0).9037,.7768 ±.7949i I S 3,4 (±00, 0, 0).9037,.7768 ±.7949i I S 5,6 (±300, 0, 0).9037,.7768 ±.7949i I S 7,8 (±50, 0, 0) 0.87, ± i II S 9,0 (±50, 0, 0) 0.87, ± i II S, (±50, 0, 0) 0.87, ± i II S 3,4 (±350, 0, 0) 0.87, ± i II Define Γ x { } τ,( )τ,..., r τ, { Γ x ( + ) τ,( ) τ,..., (r + ) τ }, Γ y { } τ,( )τ,..., rτ, { Γ y ( + ) τ,( ) τ,..., (r + ) τ }, (8) where the ubcript x and y indicate the non-zero coordinate of the addle point. Then all equilibrium point of ytem (6) can be claified into the following four different et:

6 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig. 4. Simulated reult for -D 4 4-croll chaotic attractor of ytem (6) with a = b = c = 0.7, τ = τ = 00, r = r = = = : (a) the phae portrait in the x y z plane; (b) the phae portrait projected on the x y plane; (c) the time hitory x(t); and (d) the Lyapunov exponent. Γ = { (x, y) x Γ x, y Γ } y, Γ = { (x, y) } x Γ x, y Γ y, Γ 3 = { (x, y) x Γx, y Γ } y, Γ 4 = { (x, y) } x Γx, y Γ y. A direct calculation how that only the equilibrium point in et Γ are addle point of index two with eigenvalue atifying λ < 0 and λ,3 = ξ ± ηi (ξ >0 and η 0). Thee equilibrium point are reponible for the generation of (r + + ) (r + + ) chaotic attractor. For example, etting a = b = c = 0.7, τ = τ = 00, and r = r = = = in ytem (6), we get a 4 4-grid-croll chaotic attractor, a hown in Fig. 4(a) and (b). The three Lyapunov exponent for thi ytem are L = 0.384, L = 0, L 3 = , implying that the ytem i chaotic. Note that the divergence of ytem (6) i V = ẋ x + ẏ y + ż = c < 0. z (9) Hence, the dynamical ytem decribed by (6) i diipative, and an exponential contraction of the ytem (6) i given by dv dt = e ct. Therefore, in dynamical ytem (6), a volume element V 0 i apparently contracted by the flow into a volume element V 0 e ct in time t. It mean that each volume containing the trajectory of thi dynamical ytem hrink to zero a t at an (0)

7 58 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 exponential rate c. So, all the trajectorie of thi ytem are eventually confined by a pecific ubet that ha zero volume, and the aymptotic motion ettle onto an attractor of ytem (6). 5. Deign of 3-D-m n l-grid-croll chaotic attractor Similarly, we may generate 3-D-m n l-grid-croll chaotic attractor by adding three hyperbolic tangent function erie to ytem (3) to obtain the following ytem: ẋ = y τ [ (r ) + tanh(y + jτ ), j= r ẏ = z τ [ 3 3 (r 3 3 ) + tanh(z + jτ 3 ), j= r 3 ż = ax by cz + a τ [ (r ) + tanh(x + jτ ) + b τ [ (r ) + tanh(y + jτ ) j= r j= r + c τ [ 3 3 (r 3 3 ) + tanh(z + jτ 3 ), () j= r 3 where a, b, c, τ, τ and τ 3 are poitive real value, wherea r, r, r 3,,, 3 are non-negative integer. m, n and l are determined by m = r + +, n = r + +, l = r () Similarly, define Γ x { } τ,( )τ,..., r τ, { Γ x ( + ) τ,( ) τ,..., (r + ) τ }, Γ y { } τ,( )τ,..., r τ, { Γ y ( + ) τ,( ) τ,..., (r + ) τ }, Γ z { } 3 τ 3,( 3 )τ 3,..., r 3 τ 3, { Γ z ( 3 + ) τ 3,( 3 ) τ 3,..., (r 3 + ) τ } 3. (3) Then all equilibrium point of ytem () can be claified into the following eight different et: Γ = { (x, y, z) x Γ x, y Γ y, z Γ } z, Γ = { (x, y, z) x Γ x, y Γ y, z Γ } z, Γ 3 = { (x, y, z) x Γx, y Γ y, z Γ } z, Γ 4 = { (x, y, z) x Γx, y Γ y, z Γ } z, Γ 5 = { (x, y, z) x Γ x, y Γ y, z Γ z}, Γ 6 = { (x, y, z) } x Γ x, y Γ y, z Γ z, Γ 7 = { (x, y, z) x Γx, y Γ y, z Γ z}, Γ 8 = { (x, y, z) } x Γx, y Γ y, z Γ z. Calculation how that only the equilibrium point in et Γ,Γ 4 and Γ 6 are addle point of index two with eigenvalue atifying λ < 0 and λ,3 = ξ + ηi (ξ >0 and η 0). Thee equilibrium point are reponible for the generation of (r + + ) (r + + ) (r ) chaotic attractor. For example, chooing a = b = c = 0.8, τ = 60, τ = 00, τ 3 = 80, and r = r = r 3 = = = 3 = inytem(),weobtaina6 6 6-grid-croll chaotic attractor, a depicted in Fig. 5. The time hitory for the x component i hown in Fig. 6(a).

8 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig. 5. Simulated reult for 3-D croll chaotic attractor of ytem () with a = b = c = 0.8, τ = 60, τ = 00, τ 3 = 80, and r = t = r 3 = = = 3 = : (a) the phae portrait in the x y z plane; (b) the phae portrait projected on the x y plane; (c) the phae portrait projected on the x z plane; (d) and the phae portrait projected on the y z plane. Fig. 6. Simulated reult for 3-D croll chaotic attractor of ytem () with a = b = c = 0.8, τ = 60, τ = 00, τ 3 = 80, and r = r = r 3 = = = 3 = : (a) the time hitory x(t); and (b) the Lyapunov exponent. Numerical computation give the following three Lyapunov exponent for the ytem: L = 0.56, L = , L 3 = 0.696, implying that the ytem generate a hyperchao. The computation reult of thee Lyapunov exponent i hown in Fig. 6(b).

9 60 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 In ytem (), letting r =, r = and r 3 = 3 yield ẋ = y τ ẏ = z τ 3 j= r tanh(y + jτ ), 3 j= r 3 tanh(z + jτ 3 ), ż = ax by cz + a τ j= r tanh(x + jτ ) + b τ j= r tanh(y + jτ ) + c τ 3 3 j= r 3 tanh(z + jτ 3 ). (4) Apparently, ytem (4) i ymmetric about the origin, ince the ytem i invariant under the tranformation (x, y, z) ( x, y, z). Therefore, ytem (4) with the given parameter value i a pecial cae of ytem (), a hown in Fig Synchronization and tabilization of -D-n-croll chaotic attractor To further tudy the global and exponential ynchronization of two -D-n-croll chaotic attractor, conider ytem (4) a a drive ytem ẋ d = y d, ẏ d = z d, ż d = ax d by d cz d + a τ [ (r ) + tanh(x d + jτ ), (5) j= r where the ubcript d indicate the drive. Then the correponding driven (or receiving) ytem i ẋ r = y r + u (x d x r, y d y r, z d z r ), ẏ r = z r + u (x d x r, y d y r, z d z r ), ż r = ax r by r cz r + τ [ a (r ) + tanh(x r + jτ ) + u 3 (x d x r, y d y r, z d z r ), (6) j= r where the ubcript r indicate the receive. Here, u i are the continuou, linear function of it variable, atifying u i (0, 0, 0) = 0, i =,, 3. Let e x = x d x r, e y = y d y r, e z = z d z r. Then the error ytem i given by ė x = e y u (e x, e y, e z ), ė y = e z u (e x, e y, e z ), ė z = ae x be y ce z + τ a j= r f (ξ)e x u 3 (e x, e y, e z ), (7) where by the intermediate value theorem, f (ξ)e x = tanh(x d + jτ ) tanh(x r + jτ ) with min(x r, x d )<ξ<max(x r, x d ).We will ue the property (tanh x) = tanh x = ech x < + to tudy the globally exponential ynchronization and globally exponential tability. Let X = (x, y, z ) denote any equilibrium point of ytem (4). Then, define X = X X = (x x, y y, z z ).Thu ytem (7) can be rewritten a x = ȳ u (x, y, z), ȳ = z u (x, y, z), z = ax by cz + a τ f (η) x u 3 (x, y, z), (8) j= r where η i a real value between x and x, and f (η) x = tanh(x + jτ ) tanh(x + jτ ). Next, we will how that poibly the implet feedback control law u, u and u 3 can be choen uch that the zero olution of the error ytem (7) or 8) i globally exponentially tabilized. Thu, ytem (5) and (6) are globally exponentially ynchronized, or the equilibrium point X = X i globally exponentially tabilized.

10 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Definition. With properly choen feedback control law u i, (x d (0), y d (0), z d (0)) R 3 and correponding (x r (0), y r (0), z r (0)) R 3, if the zero olution of (7) i globally exponentially tabilized (globally aymptotically tabilized), then ytem (5) and (6) are aid to be globally exponentially ynchronized (globally ynchronized). Definition. With properly choen feedback control law u i, (x d (0), y d (0), z d (0)) R 3 and correponding (x r (0), y r (0), z r (0)) R 3, if the zero olution of (8) i globally exponentially tabilized (globally aymptotically tabilized), then the equilibrium point X = X of ytem (4) i aid to be globally exponentially tabilized (globally tabilized). For convenience, we define two imple linear feedback control law a follow: u = δ x e x, u = δ y e y, u 3 = δ z e z, (9) and u = δ x x, u = δ y ȳ, u 3 = δ z z, (0) where δ a, δ y and δ z are called control gain coefficient, to be pecified in the following theorem and corollarie. Theorem. For ytem (7), under the control law (9) if one of the following condition hold: () δ x >, δ y >, δ z > a + b c + a τ ( + r + ); () δ x > a + a τ ( + r + ), δ y > b +, δ z > c; then the zero olution of (7) i globally exponentially tabilized, and thu ytem (5) and (6) are globally exponentially ynchronized. Proof. Contruct poitive definite, radially unbounded vector Lyapunov function for ytem (7): V = ( e x, e y, e z ) T. Along the olution of ytem (7), evaluating the Dini derivative of V yield D + e x δ x 0 e x D + e y 0 δ y D + e z a + a τ e y. () (7) ( + r + ) b c δ z e z Conider the comparion equation of ytem (): η x δ x 0 η x η x η y = 0 δ y η y := η z a + a τ A η y, () ( + r + ) b c δ z η z η z which ha the olution: η x (t) η x (t 0 ) η y (t) = e A (t t 0 ) η y (t 0 ). η z (t) η z (t 0 ) If one of the condition in Theorem i atified, A i a Hurwitz matrix [0. Thu, there exit contant M, α > 0, atifying e A (t t 0 ) M e α (t t 0 ). By the comparion principal, we have e x (t) η e y (t) e z (t) x (t) η η y (t) η z (t) M e α x (t 0 ) (t t 0 ) η y (t 0 ) η z (t 0 ). (3) Eq. (3) how that the zero olution of ytem (7) i globally exponentially tabilized, and thu by Definition we know that ytem (5) and (6) are globally exponentially ynchronized. Remark. One of the criteria in the control deign i that the control hould not alter the baic tructure of the ytem. An extreme example of uch control i the o-called cancellation control, which cancel term on the right-hand ide of the error ytem (7), or that of the controlled ytem (6).

11 6 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 For the control propoed in Theorem, ubtituting the feedback control (9) into Eq. (6) yield the following controlled ytem: ẋ r = y r + δ x (x d x r ), ẏ r = z r + δ y (y d y r ), ż r = ax r by r cz r + δ z (z d z r ) + τ [ a (r ) + tanh(x r + jτ ). (4) j= r It i obviou that the original term y r and z r in the firt two equation of (4) are not canceled by the control. For the third equation, ince the control gain δ z i either great than a + b c + a τ ( + r + ) or greater than c, the control term δ z z r cannot cancel the original term z r. Next, we will preent an example of cancellation control. Take the following linear control law: u = e y + e x, u = e z + e y, u 3 = (ae x + be y + ce z ) + e z, (5) then the linear term on the right-hand ide of the error ytem (7) are all canceled and the reulting ytem become ė x = e x, ė y = e y, ė x = e z + τ a f (ξ)e x, j= r (6) which i globally aymptotically table. However, ubtituting thi control law into the correponding ytem (7) cancel all the linear term on the right-hand ide of the ytem. Such control deign (5) i not deirable from the view point of control theory. Corollary. For any given equilibrium point X = X of ytem (8), chooe the linear control law (0). Then if one of the following condition i atified: () δ x >, δ y >, δ z > a + b c + a τ ( + r + ); () δ x > a + a τ ( + r + ), δ y > b +, δ z > c; X = (x, y, z ) can be globally exponentially tabilized. Remark. Sytem (4) may have multiple equilibrium point. However, when the feedback control law (0) i applied, which i related to the equilibrium point X = (x, y, z ), all the other equilibrium point diappear. The only remaining one become the equilibrium point to be globally exponentially tabilized. Theorem. Chooe the linear feedback control law (9) for ytem (7). Then under the condition: δ x > + a b + aτ 4b ( + r + ), δ y >, δ z > c + aτ 4 ( + r + ) + a, the zero olution of the error ytem (7) i globally exponentially tabilized, and thu ytem (5) and (6) are globally exponentially ynchronized. Proof. Contruct poitive definite, radially unbounded Lyapunov function for ytem (7): T V = ( ) e x e x e x + e y + e z = e y P e y. b e z e z Let λ m (P) and λ M (P) denote the minimum and maximum eigenvalue of 0 0 P = 0 0, 0 0 b repectively. Then we have

12 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) dv dt = e x ė x + e y ė y + (7) b e zė z T e x a δ x b + aτ (r + + ) e x 4b e y δ y 0 e y a e z b + aτ 4b (r + + ) 0 c+δ z b e z T e x e x := e y A e y e z e z λ M (A ) ( e x + e y + e z), where λ M (A ) i the maximum eigenvalue of A. It follow from the condition in Theorem that λ M (A )<0. Hence, dv dt λ M(A ) λ M(P) ( e λ M (P) x + e y + λ M (A ) z) e λ M (P) V, which implie that ( V X(t) ) ( V X(t 0 ) ) λ M (A ) e λ M (P) (t t 0). Hence, e x (t) + e y (t) + V (X(t)) e z (t) λ m (P) V (X(t λm (A) 0)) λ m (P) e λ M (P) (t t 0). (7) Eq. (7) how that the zero olution of (7) i globally exponentially tabilized, and thu ytem (5) and (6) are globally exponentially ynchronized. Corollary. For any given equilibrium point X = X of ytem (4), chooe the feedback control law (0). If δ x > + a b + aτ 4b ( + r + ), δ y >, δ z > c + aτ 4 ( + r + ) + a, then the equilibrium point of ytem (4), X= X, i globally exponentially tabilized. The proof of Corollary i imilar to that of Theorem. Theorem 3. Chooe the linear feedback control law (9) for ytem (7). When δ x > a + a τ (r + + ), δ y = + b, δ z = c, the zero olution of (7) i globally exponentially tabilized, i.e., the two ytem (5) and (6) are globally exponentially ynchronized. Proof. Contruct poitive definite, radially unbounded Lyapunov function for Eq. (7): V = e x + e y + e z. Then we have D + V (7) = ė x ign e x + ė y ign e y + ė z ign e z e y + e z δ x e x δ y e y δ z e z c e z +a τ (r + + ) e x +a e x +b e y ( δ x + a + a τ ) (r + + ) e x +( δ y + b + ) e y +( δ z c + ) e z = ( δ x + a + a τ ) (r + + ) e x, (8) which implie that the zero olution of (7) i globally exponentially tabilized with repect to the partial variable e x. Next, we conider the coefficient matrix of the linear part of ytem (7) with the feedback control law given in Theorem 3: A 3 = δ x 0 0 δ y a b

13 64 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 from which we obtain the characteritic polynomial: λ + δ x 0 det(λi 3 A 3 ) = det 0 λ + + b a b λ + = λ 3 + (b + + δ x )λ + [ δ x (b + ) + b + λ + δ x (b + ) + a := λ 3 + pλ + qλ + r. It i well known that the ufficient and neceary condition for A 3 to be a Hurwitz matrix i p > 0, pq > r > 0. Uing a = b, wehave pq = (b + + δ x ) [ δ x (b + ) + b + = (b + )(b + ) + δ x (b + ) + b(b + ) + δ x (b + ) + δx (b + ) + bδ x >δ x (b + ) + b = r > 0, which implie that A 3 i a Hurwitz matrix. On the other hand, the olution of (7) can be written a e x (t) e x (t 0 ) t 0 e y (t) = e A 3(t t 0 ) e y (t 0 ) A + e 3 (t τ ) 0 dτ. e z (t) e z (t 0 ) a τ j= r f (ξ)e x (τ ) t 0 Since A 3 i a Hurwitz matrix, there exit contant M and α > 0 atifying e A 3 (t t 0 ) M e α (t t 0 ). (9) Thu, e x (t) e e y (t) e z (t) x (t 0 ) t M e y (t 0 ) e z (t 0 ) e α (t t 0 ) + t 0 M e α (t τ ) τ a ( + r + ) ex (τ ) dτ. With the property lim t e x (t) = 0, we can prove that ε > 0, σ > 0, when e x (t 0 ), e y (t 0 ), e z (t 0 ) σ, e x (t 0 ) M e y (t 0 ) e z (t 0 ) e α (t t 0 ) < ε 3. Moreover, for any t > t 0, when e x (t 0 ), e y (t 0 ), e z (t 0 ) σ, we have and t t 0 t t M e α (t τ ) τ a ( + r + ) ex (τ ) ε dτ < 3, M e α (t τ ) τ a ( + r + ) ex (τ ) ε dτ < 3. Therefore, e x (t) e e y (t) e z (t) x (t 0 ) t M e y (t 0 ) e z (t 0 ) e α (t t 0 ) + M e α (t τ ) τ a ( + r + ) ex (τ ) dτ t + t M e α (t τ ) τ a ( + r + ) ex (τ ) dτ < ε 3 + ε 3 + ε 3 = ε. t 0

14 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) For any (e x (t 0 ), e y (t 0 ), e z (t 0 )) R 3, it i eay to how that e x (t 0 ) t lim M t e y (t 0 ) e z (t 0 ) e α (t t 0 ) + lim M e α (t τ ) τ a t ( + r + ) ex (τ ) dτ = 0. (30) t 0 Thu, the zero olution of (7) i globally aymptotically tabilized, and o ytem (5) and (6) are globally exponentially ynchronized. Corollary 3. For any equilibrium point X = X of ytem (4), chooe the linear feedback control law (0). When δ x > a + a τ (r + + ), δ y = + b, δ z = c, X can be globally exponentially tabilized. 7. Synchronization and tabilization of -D-m n-grid-croll chaotic attractor In thi ection, we conider ynchronization and tabilization of the -D-m n-grid-croll chaotic attractor. Aume that the drive ytem i ẋ d = y d τ [ (r ) + tanh(y d + jτ ), j= r ẏ d = z d, ż d = ax d by d cz d + aτ [ (r ) + tanh(x d + jτ ) + bτ [ (r ) + tanh(y d + jτ ), (3) j= r j= r while, the correponding driven ytem i given by ẋ r = y r τ [ (r ) + tanh(y r + jτ ) + u (e x, e y, e z ), j= r ẏ r = z r + u (e x, e y, e z ), ż r = ax r by r cz r + aτ [ (r ) + tanh(x r + jτ ) + bτ [ (r ) + tanh(y r + jτ ) j= r j= r + u 3 (e x, e y, e z ). The error ytem i then obtained a ė x = e y τ f (ξ)e y u (e x, e y, e z ), j= r ė y = e z u (e x, e y, e z ), ė z = ae x be y ce z + aτ f (ξ)e x + bτ j= r (3) j= r f (ξ)e y u 3 (e x, e y, e z ). (33) Theorem 4. In ytem (33), chooe the linear control law (9). If one of the following condition i atified: () δ x > + τ ( + r + ), δ y >, δ z > a + aτ ( + r + ) + b + bτ ( + r + ) c; () δ x > a + aτ ( + r + ), δ y > + τ ( + r + ) + b + bτ ( + r + ), δ z > c; then the zero olution of (33) i globally exponentially tabilized, and thu ytem (3) and (3) are globally exponentially ynchronized. Proof. We contruct poitive definite, radially unbounded vector Lyapunov function: V = ( e x, e y, e z ) T, and then we have

15 66 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 D + e x D + V (33) = D + e y D + e z (33) Conider the comparion equation of (34), given by η x η y = η z δ x + τ ( + r + ) 0 0 δ y a + aτ ( + r + ) b + bτ ( + r + ) δ z c δ x + τ ( + r + ) 0 0 δ y a + aτ ( + r + ) b + bτ ( + r + ) δ z c The olution of Eq. (35) can be written a η x (t) η x (t 0 ) η y (t) = e A 4(t t 0 ) η y (t 0 ). η z (t) η z (t 0 ) e x e x e y := A 4 e y. (34) e z e z η x η x := A 4. (35) From the condition of the Theorem 4, we know that A 4 i a Hurwitz matrix. Hence, there exit M 4 andα 4 > 0uch that e A 4 (t t 0 ) M4 e α 4(t t 0 ). Further, from the comparion principle we know that ( ex (t), e y (t), ez (t) ) T ( ηx (t), η y (t), η z (t) ) T e A 4 (t t 0 ) ( η x (t 0 ), η y (t 0 ), η z (t 0 ) ) T η y η z M 4 e α 4(t t 0 ) ( η x (t 0 ), η y (t 0 ), η z (t 0 ) ) T, (36) η y η z which indicate that the concluion of Theorem 4 i true. Corollary 4. For any given equilibrium point X = X of ytem (6), if chooe the linear feedback control law (0), where the control gain coefficient δ x, δ y and δ z atify the condition in Theorem 4, the zero olution of the following error ytem x = ȳ τ f (ξ)y u (x, y, z), j= r ẏ = z u (x, y, z), ż = ax b y cz + aτ f (ξ)x + bτ j= r j= r f (ξ)y u 3 (x, y, z) (37) i globally exponentially tabilized, i.e., the equilibrium point X = (x, y, z ) i globally exponentially tabilized. Theorem 5. In ytem (33), again chooe the linear control law (9). If and δ x > + τ 4 ( + r + ) + τ 4 ( + r + ), δ y > + τ 4 ( + r + ) + bτ 4a ( + r + ) + b a, δ z > a + b + aτ 4 (r + + ) + bτ 4 (r + + ) c, then the zero olution of (33) i globally exponentially tabilized. Thu, ytem (3) and (3) are globally exponentially ynchronized. Proof. Contruct the poitive definite, radially unbounded Lyapunov function: V = ( Then we have e x + e y + e z a ).

16 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) dv dt δ x e x + e xe y + τ (33) ( + r + ) e x e y +e y e z δ y e y e xe z b a e ye z c a e z δ ze z a + τ ( + r + ) e x e z + bτ a ( + r + ) e y e z δ x + τ 4 ( + r + ) T e x e y e z + τ 4 ( + r + ) e x := e y e z T A 5 + τ 4 ( + r + ) δ y + b a + bτ e x e y. e z + b c a a δ z a + bτ 4a (r + + ) + τ 4 ( + r + ) e x 4a ( + r + ) e y e z From the propoed condition we know that A 5 i negative definite. Similar to the proof of Theorem, it i eay to how that e x (t) + e y (t) + V (X(t)) e z (t) λ m (P) V (X(t λm (A5) 0)) λ m (P) e λ M (P) (t t 0), (38) indicating that the concluion of Theorem 5 i true. Corollary 5. For any given equilibrium point X = X of ytem (3), under the linear control law (0), when the condition in Theorem 5 hold, the zero olution of ytem (37) i globally exponentially table. Thu the equilibrium point X = X of (3) i globally exponentially tabilized. 8. Synchronization and tabilization of 3-D-m n l-grid-croll chaotic attractor In thi ection, we tudy ynchronization and tabilization of the 3-D-m n l-grid-croll chaotic attractor. For thi cae, the drive ytem i ẋ d = y d τ [ (r ) + tanh(y d + jτ ), j= r [ ẏ d = z d τ 3 ż d = ax d by d cz d + aτ + cτ 3 3 (r 3 3 ) + tanh(z d + jτ 3 ), j= r 3 [ (r ) + tanh(x d + jτ ) + bτ [ (r ) + tanh(y d + jτ ) j= r j= r [ (r 3 3 ) + tanh(z d + jτ 3 ), (39) j= r while the correponding driven ytem with control i decribed by ẋ r = y r τ [ (r ) + tanh(y r + jτ ) + u (e x, e y, e z ), j= r ẏ r = z r τ [ 3 3 (r 3 3 ) + tanh(z r + jτ 3 ) + u (e x, e y, e z ), j= r 3 ż r = ax r by r cz r + aτ [ (r ) + tanh(x r + jτ ) + bτ [ (r ) + tanh(y r + jτ ) j= r j= r + cτ [ 3 3 (r 3 3 ) + tanh(z r + jτ 3 ) + u 3 (e x, e y, e z ). (40) j= r 3 The error ytem can then be obtained a

17 68 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 ė x = e y τ ė y = e z τ 3 j= r f (ξ)e y u (e x, e y, e z ), 3 j= r 3 f (ξ)e z u (e x, e y, e z ), ė z = ae x be y ce z + aτ f (ξ)e x + bτ j= r f (ξ)e y + cτ 3 j= r 3 j= r 3 f (ξ)e z u 3 (e x, e y, e z ). (4) Theorem 6. In ytem (4), chooe the linear feedback control law (9). Ifδ x 0, δ y 0 and δ z 0 are properly choen uch that δ x + τ ( + r + ) 0 A 6 = 0 δ y + τ 3 ( 3 + r 3 + ) aτ ( bτ + r + ) + a ( + r + ) + b δ z c + cτ 3 ( 3 + r 3 + ) i a Hurwitz matrix,then the zero olution of (4) i globally exponentially table. Thu, ytem (39) and (40) are globally exponentially ynchronized. Proof. Contruct poitive definite, radially unbounded vector Lyapunov function: V = ( e x, e y, e z ) T. Then we have D + e x D + V (4) = D + e y D + e z δ x + τ ( + r + ) 0 e x 0 δ y + τ 3 ( 3 + r 3 + ) e y aτ ( bτ + r + ) + a ( + r + ) + b e x δ z c + cτ 3 ( 3 + r 3 + ) e z := A 6 e y. e z Conider the comparion equation of (4): η x δ x + τ ( + r + ) 0 η y = 0 δ y + τ 3 ( 3 + r 3 + ) η aτ z ( bτ + r + ) + a ( + r + ) + b δ z c + cτ 3 ( 3 + r 3 + ) Since A 6 i a Hurwitz matrix, there exit M 6 and α 6 > 0 uch that ( e x, e y, e z ) T (η x, η y, η z ) T M 6 e α 6(t t 0 ) ( η x (t 0 ), η y (t 0 ), η z (t 0 ) ) T, (4) η x η x := A 6. (43) η y η z η y η z indicating that the concluion of Theorem 6 i true. Theorem 7. In ytem (4), chooe the ame linear control law a that for the proof of Theorem 6. If δ x + τ 4 ( + r + ) + τ 4 ( + r + ) + τ a+b 4 ( + r + ) δ y a + τ 3 4 ( 3 + r 3 + ) A 7 = bτ 4a ( + r + ) + τ a+b 4 ( + r + ) a + τ 3 4 ( 3 + r 3 + ) c a + cτ 3 a ( 3 + r 3 + ) δ z a bτ 4a ( + r + ) i a Hurwitz matrix, then the zero olution of ytem (4) i globally exponentially table. Thu, ytem (39) and (40) are globally exponentially ynchronized. Proof. We ue the ame Lyapunov function a that in the proof of Theorem 5: V = ( ) e x + e y + e z a

18 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) and then we obtain dv dt e x e y + τ (4) ( + r + ) e x e y δ x e x + e ye z + τ 3 ( 3 + r 3 + ) e y e z δ y e y + e xe z + b a e ye z c a e z implying + τ ( + r + ) e x e z + bτ a ( + r + ) e y e z + cτ 3 a ( 3 + r 3 + )e z a δ ze z T e x e x = e y A 7 e y, e z e z ( e x (t) + e y (t) + e z (t)) V (X(t)) λ m (P) V (X(t 0)) λ m (P) e λm (A7) λ M (P) (t t 0), where the definition of λ M (P), λ m (P) and λ M (A 7 ) are imilar to thoe ued in Theorem. Thu, the concluion i true. Corollary 6. If the condition in Theorem 6 and 7 hold, the zero olution of the following ytem x = δ x x + ȳ τ ẏ = δ y y τ 3 3 j= r f (η)y, j= r 3 f (η)z, ż = (δ z + c)z ax b y + aτ f (η)x + bτ j= r f (η)y + cτ 3 j= r i globally exponentially table. Thu the equilibrium point X = X i globally exponentially tabilized. 9. Numerical imulation example 3 (44) j= r 3 f (η)z (45) In thi ection, we preent ome numerical example to demontrate the applicability of control law propoed in the previou ection. The analytical prediction for the control law are verified by numerical imulation. For the ynchronization problem, we how three example for -D, -D and 3-D multi-croll chaotic ytem uing the control law given in Theorem 3, 4 and 6, repectively, ince the remaining cae are imilar. The initial condition choen repectively for -D, -D and 3-D ytem are: x d (0) = 4, y d (0) =, z d (0) =, (46) and x r (0) =, y r (0) = 5, z r (0) = 3 (47) for the -D ytem; x d (0) = 0, y d (0) = 5, z d (0) = 5, (48) and x r (0) = 5, y r (0) = 8, z r (0) = (49) for the -D ytem; and and x d (0) = 0, y d (0) =, z d (0) =, (50) x r (0) = 8, y r (0) = 5, z r (0) = 4 (5) for the 3-D ytem. Obviouly, the initial condition choen here for the drive ytem and driven ytem are quite different. Under the control law given in Theorem 3, 4 and 6, time hitorie for the error ignal, e x (t), e y (t) and e z (t), obtained for the three cae are diplayed in Fig. 7, 8 and 9, repectively. All the three cae how the exponential convergence of the error to zero, a expected.

19 70 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 Fig. 7. Time hitory of error ytem (7) for a = b = c = 0.65, τ = 00, r = = 3 uing the control law given in Theorem 3, with the initial condition, x d (0) = 4, y d (0) =, z d (0) = andx r (0) =, y r (0) = 5, z r (0) = 3, when δ x = 50, δ y =.8 andδ z = 0.5. For the tabilization problem, we how four example uing the control law given in Corollarie, 3 and 4, and the remaining three cae are imilar. The initial condition choen for thee four cae are repectively given by x(0) = 80, y(0) =, z(0) = 5, (5) x(0) = 0., y(0) = 5, z(0) = 00, (53) x(0) = 0, y(0) =, z(0) = 5, (54)

20 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig. 8. Time hitory of error ytem (33) for a = b = c = 0.7, τ = τ = 00, r = r = = = uing the control law given in Theorem 4(), with the initial condition, x d (0) = 0, y d (0) = 5, z d (0) = 5andx r (0) = 5, y r (0) = 8, z r (0) =, when δ x = 60, δ y =.05 and δ z = 50. and x(0) = 0.6, y(0) = 0.6, z(0) = 00. (55) When the control law given in Corollary i applied to ytem (8), uing δ x = δ y =.05 and δ z = 33.3, with E = (0, 0, 0) a the deigned equilibrium point, it i hown (ee Fig. 0(a)) that the trajectory converge to E. Similarly, for the other three cae, numerical imulation reult demontrate that the all olution trajectorie converge to the deigned equilibrium point with proper control law applied (ee Fig. 0(b) and ).

21 7 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) 5 74 Fig. 9. Time hitory of error ytem (4) for a = b = c = 0.8, τ = 60, τ = 00, τ 3 = 80, r = r = = = uing the control law given in Theorem 6, with the initial condition, x d (0) = 0, y d (0) =, z d (0) = andx r (0) = 8, y r (0) = 5, z r (0) = 4, when δ x = 55, δ y = 05 and δ z = Concluion In thi paper, we ued hyperbolic tangent function erie to develop a method to generate n-, m n- and m n l- grid-croll chaotic attractor. Baed on the ytem equilibrium point and correponding eigenvalue, imple mathematical analyi i given to identify -D, -D and 3-D multi-grid-croll chaotic attractor. For each chaotic ytem, with the aid of proper choen Lyapunov function we deigned imple linear feedback control law to globally exponentially tabilize the ytem and ynchronize two chaotic ytem with ame tructure. Numerical imulation reult are preented to confirm the analytical prediction.

22 F. Xu, P. Yu / J. Math. Anal. Appl. 36 (00) Fig. 0. (a) Trajectory of a double-croll chaotic attractor for ytem (8) when a = b = c = 0.65, τ = 00, r = = 0 uing the control law given in Corollary () for δ x = δ y =.05 and δ z = 33.3 with the initial condition x(0) = 80, y(0) =, z(0) = 5, convergent to the equilibrium point E: (0, 0, 0); (b) trajectory of -D-8-croll chaotic attractor for ytem (8) when a = b = c = 0.65, τ = 00, r = = 3 uing the control law given in Corollary 3 for δ x = 50, δ y =.8 andδ z = 0.5 with the initial condition x(0) = 0., y(0) = 5, z(0) = 00, convergent to the equilibrium point E: (0, 0, 0). Fig.. (a) Trajectory of a -D-4 4-croll chaotic attractor for ytem (37) when a = b = c = 0.7, τ = τ = 00, r = r = = = uing the control law given in Corollary 4() for δ x = 60, δ y =.05 and δ z = 50 with the initial condition x(0) = 0, y(0) =, z(0) = 5, convergent to the equilibrium point E: (0, 0, 0); (b)trajectoryofa3-d croll chaotic attractor for ytem (45) when a = b = c = 0.8, τ = 60, τ = 00, τ 3 = 80, and r = r = r 3 = = = 3 = uing the control law given in Corollary 6 for δ x = 55, δ y = 05 and δ z = 690 with the initial condition x(0) = 0.6, y(0) = 0.6, z(0) = 00, convergent to the equilibrium point E: (0, 0, 0). Acknowledgment Thi work wa upported by the Natural Science and Engineering Reearch Council of Canada (NSERC, No. R686A0). Reference [ L. Chua, M. Komuro, T. Matumoto, The double croll family, Part I: Rigorou proof of chao, Circuit Syt. 33 (986) [ J. Suyken, J. Vandewalle, Quailinear approach to nonlinear ytem and the deign of n-double croll (n =,, 3, 4,...), IEE Circuit Device Syt. Ser. 38 (5) (99) [3 F.L. Han, X.H. Yu, Y.Y. Wang, Y. Feng, G.R. Chen, n-scroll chaotic ocillator by econd-order ytem and double-hyterei block, Electronic Lett. 39 (3) (003) [4 M.E. Yalcin, J.A.K. Suyken, J. Vandewalle, Experimental confirmation of 3- and 5-croll attractor from a generalized Chua circuit, Circuit Syt. I, Fundam. Theory Appl. 47 (3) (000) [5 S.M. Yu, J.H. Lü, H. Leung, G.R. Chen, N-croll chaotic attractor from a general jerk circuit, in: Circuit Sytem ISCAS, vol., 005, pp [6 S.M. Yu, J.H. Lü, G.R. Chen, Experimental confirmation of n-croll hyperchaotic attractor, in: Circuit Sytem ISCAS, 006, pp. 4. [7 M.E. Yalcm, J.A.K. Suyken, J. Vandewalle, Multi-croll and hypercube attractor from Joephon junction, in: Circuit Sytem ISCAS, 006, pp [8 S. Özoǧuz, A.S. Elwakil, K.N. Salama, n-scroll chao generator uing nonlinear tranconductor, Electronic Lett. 38 (4) (00) [9 K.N. Salama, S. Özoǧuz, A.S. Elwakil, Generation of n-croll chao uing nonlinear tranconductor, in: Circuit Sytem ISCAS, vol. 3, 003, pp [0 W.H. Deng, J.H. Lü, Deign of multi-directional multi-croll chaotic attractor baed on fractional differential ytem, in: Circuit Sytem ISCAS, 007, pp [ J.H. Lü, G.R. Chen, X.H. Yu, H. Leung, Deign and analyi of multi-croll chaotic attractor from aturated function erie, IEEE Tran. Circuit Syt. I Regul. Pap. 5 () (004) [ J.H. Lü, S. Yu, H. Leung, Deign of 3-D multi-croll chaotic attractor via baic circuit, in: Proceeding of the 4th International DCDIS Conference on Engineering Application and Computational Algorithm, 005, pp [3 E. Ott, G. Grebogi, J.A. Yorke, Controlling chao, Phy. Rev. Lett. 64 (990)

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