Chaos Control and Synchronization of a Fractional-order Autonomous System

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1 Chaos Control and Snchronization of a Fractional-order Autonomous Sstem WANG HONGWU Tianjin Universit, School of Management Weijin Street 9, 37 Tianjin Tianjin Universit of Science and Technolog College of Science TEDA, Street 3, 37 Tianjin CHINA lzwhw@6.com MA JUNHAI Tianjin Universit School of Management Weijin Street 9, 37 Tianjin CHINA mjhtju@ahoo.com.cn Abstract: Liu, Liu, and Liu, in the paper A novel three-dimensional autonomous chaos sstem, Chaos, Solitons and Fractals. 39 (9) , introduce a novel three-dimensional autonomous chaotic sstem. In this paper, the fractional-order case is considered. The lowest order for the sstem to remain chaotic is found via numerical simulation. Stabilit analsis of the fractional-order sstem is studied using the fractional Routh-Hurwitz criteria. Furthermore, the fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order sstem to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractionalorder autonomous sstem can be controlled to its equilibrium points. In addition, the snchronization of the fractional-order sstem and the fractional-order Liu sstem is studied using active control technique. Numerical results show the effectiveness of the theoretical analsis. Ke Words: Fractional-order; Chaos; Predictor-correctors scheme; The fractional Routh-Hurwitz criteria; Feedback control; Snchronization; Active control Introduction As a 3-ear-old mathematical topic, fractional calculus has alwas attracted the interest of man famous ancient mathematicians, including L Hospital, Leibniz, Liouville, Riemann, Grunwald, and Letnikov []. Although it has a long histor, it was not applied in our real life because it seems to be more difficult and fewer theories have been established than for classical differential e- quations. In recent decades, fractional-order differential equations have been found to be a powerful tool in more and more fields, such as phsics, chemistr, biolog, economics, and other comple sstems [, 3]. The interest in the stud of fractional-order nonlinear sstems lies in the fact that fractional derivatives provide an ecellent tool for the description of memor and hereditar properties, which are not taken into account in the classical integer-order models. Studing dnamics in fractional-order nonlinear sstems has become an interesting topic and the fractional calculus is plaing a more and more important role for analsis of the nonlinear dnamical sstems. Some work has been done in the field, and the chaos and control in fractional-order sstems have been studied, including Lorenz sstem [4], Chua sstem [5], Chen sstem [6], Rossler sstem [7], Newton-Leipnic sstem [8], Liu sstem [9] and so on. In 9, Liu et al. gave a novel threedimensional autonomous chaotic sstem [], which is not different from Liu sstem in Ref. []. In this paper, the dnamical behaviors and feedback control to stead state of the fractionalorder autonomous sstem are studied. The lowest order for the sstem to remain chaotic is found via numerical simulation. Furthermore, we are going to use the fractional Routh-Hurwitz conditions given in [, 3, 4] to stud the stabilit conditions for the fractional-order sstem, and the conditions for linear feedback control are obtained as well. Moreover, the snchronization of the fractional-order autonomous sstem and the fractional-order Liu sstem is also studied using the active control method in Ref. [5]. This paper is organized as follows: Section gives the integer-order autonomous sstem. Section 3 presents the fractional derivatives and the lowest order for the fractional-order autonomous sstem E-ISSN: Issue 8, Volume, August

2 to remain chaotic. Section 4 gives stabilit analsis of the fractional-order chaotic autonomous sstem. Section 5 presents chaos control of the fractional-order autonomous sstem, and the numerical simulation is also given. In Section 6, the active control technique is applied to snchronize the novel fractional-order autonomous sstem and the fractional-order Liu sstem [9]. Finall, in Section 7, concluding comments are given. The integer-order chaotic autonomous sstem Ref. [] reported a three-dimensional autonomous sstem which relies on two multipliers and one quadratic term to introduce the nonlinearit necessar for folding trajectories. The chaotic attractor obtained from the new sstem according to the detailed numerical as well as theoretical analsis is also the butterfl shaped attractor, ehibiting the abundant and comple chaotic dnamics. This chaotic sstem is a new attractor which is similar to Lorenz chaotic attractor. The chaotic sstem is described b the following nonlinear integer-order differential e- quations, called sstem (): d dt d dt dz = a e = b + kz () dt = cz m Liu et al. had studied and analzed its forming mechanism. The compound structure of the butterfl attractor obtained b merging together two simple attractors after performing one mirror operation was eplored. When a =, b =.5, c = 5, e =, k = 4, m =, the sstem () has three real equilibria E (,, ), E (.5,.8,.559) and E 3 (.5,.8,.559). The chaotic attractor and the equilibria are show in Fig., and the initial value of the sstem is selected as (.,,.5),as in Ref. []. At equilibrium E (,, ), the Jacobian matri of sstem () is given b J = = a e kz b k m m c.5 5 () These eigenvalues of the Jacobian matri computed at equilibrium E (,, ) are given b λ =, λ =.5, λ 3 = 5 Here λ is a positive real number, λ and λ 3 are two negative real numbers. Therefore, the equilibrium E (,, ) is a saddle point. So, this equilibrium point is unstable. At equilibrium E (.5,.8,.559), the Jacobian matri of sstem () is equal to J = (3) These eigenvalues of the Jacobian matri computed at equilibrium E (.5,.8,.559) are given b λ = , λ = j λ 3 = j Here λ is a negative real number, λ and λ 3 are a pair of comple conjugate eigenvalues with positive real parts. Therefore, the equilibrium E (.5,.8,.559) is a saddle-focus point. So, this equilibrium point E (.5,.8,.559) is unstable. At equilibrium E 3 (.5,.8,.559), the Jacobian matri of sstem () is equal to J 3 = (4) These eigenvalues of the Jacobian matri computed at equilibrium E 3 (.5,.8,.559) are given b λ = , λ = j λ 3 = j Here λ is a negative real number, λ and λ 3 are a pair of comple conjugate eigenvalues with positive real parts. Therefore, the equilibrium E 3 (.5,.8,.559) is also a saddle-focus point, and the equilibrium point E 3 (.5,.8,.559) is also unstable. For dnamical sstem (), we can obtain V = (d dt )+ (d dt )+ z (dz dt ) = a+b c = p (5) E-ISSN: Issue 8, Volume, August

3 Figure : The chaotic attractor and the equilibria of sstem (). With p = a + b c = 3.5, here p is a negative constant, dnamical sstem described b () is one dissipative sstem, and an eponential contraction of the sstem () is dv dt = ept = e 3.5t (6) In dnamical sstem (), a volume element V is apparentl contracted b the flow into a volume element V e pt = V e 3.5t in time t. It means that each volume containing the trajector of this dnamical sstem shrinks to zero as t + at an eponential rate p. So, all this dnamical sstem orbits are eventuall confined to a specific subset that have zero volume, the asmptotic motion settles onto an attractor of the sstem () [6]. 3 Fractional derivatives and chaos in the fractional-order chaotic autonomous sstem 3. Fractional derivatives and the fractional-order autonomous sstem There are several definitions for the fractionalorder differential operator, but the following definition is most used: D α () = J m α (m) (), α > (7) Where m = α, i.e., m is the first integer which is not less than α, (m) is the general m-order derivative, J β is the β-order Riemann-Liouville integral operator [7], which is epressed as follows: J β z() = ( t) β z(t)dt, β > (8) Γ(β) a The operator D α is generall called α-order Caputo differential operator. If the initial value a =, D α is denoted b D α. The fractional-order autonomous sstem is described b the following nonlinear fractionalorder differential equations, called sstem (9): d q dt q d q dt q d q 3 z = a e = b + kz (9) dt q 3 = cz m where the fractional differential operator is the Caputo differential operator; i.e., d q i /dt q i = D q i, i =,, 3, < q, q, q 3, and its order is denoted b q = (q, q, q 3 ), D q = (d q /dt q, d q /dt q,d q 3 /dt q 3 ) here. For comparing with the integer-order sstem (), we also let a =, b =.5, c = 5, e =, k = 4 and m =, with initial state (.,,.5). E-ISSN: Issue 8, Volume, August

4 3. Chaos in the fractional-order autonomous sstem Consider the following fractional-order differential equation: D q () = f(t, (t)), t T (k) () = (k) k =,,, m, m = q () It is equivalent to the Volterra integral equation (t) = m k= (k) t k k! + t (t s) q f(s, (s))ds Γ(q) () Diethelm et al. have given a predictorcorrectors scheme [8, 9], based on the Adams- Bashforth-Moulton algorithm to integrate E- q.(). B appling this scheme to the fractionalorder autonomous sstem, and setting h = T/N, t n = nh, n =,,, N Z +, sstem (9) can be discredited as follows: n+ = + + hq Γ(q + ) n+ = + + hq Γ(q + ) z n+ = z + where + hq 3 Γ(q 3 + ) p n+ = + Γ(q ) p n+ = + Γ(q ) z p n+ = z + Γ(q 3 ) h q Γ(q + ) ( ap n+ ep n+ p n+ ) n α,j,n+ ( a j e j j ) j= h q Γ(q + ) (bp n+ + kp n+ zp n+ ) n α,j,n+ (b j + k j z j ) j= h q 3 Γ(q 3 + ) ( czp n+ mp n+ p n+ ) n α 3,j,n+ ( cz j m j j ) j= n β,j,n+ ( a j e j j ) j= n β,j,n+ (b j + k j z j ) j= n β 3,j,n+ ( cz j m j j ) j= α i,j,n+ = n q i+ (n n q i )(n + ) q i, j = (n j + ) q i+ +(n j) q i+ (n j + ) q i+, j n, j = n + β i,j,n+ = hq i ((n j + ) q i + (n j) q i ), j n. q i The error estimate of the above scheme is ma j=,,,n { (t j ) h (t j ) } = O(h p ), in which p = min(, + q i ) and q i >, i =,, 3. Using predictor-correctors scheme, when q = q = q 3 = α, the simulation results demonstrate that the lowest order for the sstem (9) to remain chaotic is α =.9. When α =.89, α =.9 and α =.9, the phase portrait is show in Fig. to Fig.4. 4 Stabilit analsis of the fractional-order chaotic autonomous sstem 4. Fractional-order Routh-Hurwitz conditions Consider a three-dimensional fractional-order sstem d (t) dt q = f(,, z) d q (t) dt = g(,, z) d q 3 z(t) dt q 3 = h(,, z) () Where q, q, q 3 (, ). The Jacobian matri of the sstem () at the equilibrium points is f/ f/ f/ z g/ g/ g/ z h/ h/ h/ z (3) The eigenvalues equation of the equilibrium point is given b the following polnomial: P (λ) = λ 3 + a λ + a λ + a 3 (4) a, a and a 3 is the coefficients of the polnomial and its discriminant D(P ) is given as: D(P ) = 8a a a 3 + (a a ) 4a 3 a 3 4a 3 7a 3 (5) Lemma If the eigenvalues of the Jacobian matri (3) satisf arg(λ) πα/, α = ma(q, q, q 3 ) (6) E-ISSN: Issue 8, Volume, August

5 Figure : The phase portrait of sstem (9) when α =.89. i.e. all the roots of the polnomial Eq.(4) satisf Eq.(6), then the sstem is asmptoticall stable at the equilibrium points. The stabilit region of the fractional-order sstem is illustrated in Fig.5, in which σ, ω refer to the real and imaginar parts of the eigenvalues, respectivel. It is eas to show that the stabilit region of the fractional-order case is greater than the stabilit region of the integer-order case. Using the results of Ref. [3, 9], we have the following fractional Routh-Hurwitz conditions: (i) If D(P ) >, then the necessar and sufficient condition for the equilibrium point to be locall asmptoticall stable, is a >, a 3 >, a a a 3 >. (ii) If D(P ) <, a, a, a 3 >, then the equilibrium point is locall asmptoticall stable for α < /3. However, if D(P ) <, a <, a <, α > /3, then all roots of Eq.(4) satisf the condition. (iii) If D(P ) <, a >, a >, a a a 3 =, then the equilibrium point is locall asmptoticall stable for all α [, ). (iv) The necessar condition for the equilibrium point, to be locall asmptoticall stable, is a 3 >. The proof for above conditions can be seen in detail from Ref. [, 3]. 4. The stabilit of equilibrium points We assume q = q = q 3 = α, and the fractional sstem (9) has the same equilibria E (,, ), E (.5,.8,.559) and E 3 (.5,.8,.559) as integer sstem (). When α.9sstem (9) ehibits chaotic behavior, and the attractor and the equilibria are shown in Fig.6, when α =.9. If the equilibrium point is E(,, z ), the Jacobian matri of sstem (9) is J = a e kz b k m m c The characteristic equation of the Jacobian matri J is λ 3 +(a b+c)λ +( ab bc+ac+km +ek z )λ +( abc+akm ekm +cek z ) = (7) Theorem For the parameters a =, b =.5, c = 5, e =, k = 4 and m =, the equilibrium point E of sstem.(9) is unstable for an α (, ). Proof: When a =, b =.5, c = 5, e =, k = 4 and m =, substituting coordinate of E into Eq.(7), we can get the characteristic polnomial as follows λ λ λ.5 = (8) The eigenvalues of Eq.(8) is λ = 5, λ =, λ 3 =.5. Here λ 3 is a positive real number. Therefore, according to Lemma the equilibrium point E is unstable for an α (, ). E-ISSN: Issue 8, Volume, August

6 Figure 3: The phase portrait of sstem (9) when α =.9. Theorem 3 When the parameters a =, b =.5, c = 5, e =, k = 4 and m =, if α <.9, then equilibrium points E and E 3 of sstem (9) are stable. Proof: When a =, b =.5, c = 5, e =, k = 4 and m =, substituting coordinate of E or E 3 into Eq.(7), we can get the characteristic polnomial as follows λ λ λ = (9) The eigenvalues of Eq.(8) is λ = 4.388, λ = j, λ 3 = j. λ = is a negative real number, and the arguments of λ = j, λ 3 = j are and.4389, respectivel. arg(λ,3 ) =.4389 > πα/ =.494. Therefore, according to Lemma, if α <.9, then sstem (9) is stable at equilibrium points E and E 3, although the real part of the eigenvalues is positive. 5 Chaos control of the fractional-order autonomous sstem 5. Chaos control of the fractionalorder autonomous sstem The controlled fractional-order autonomous sstem is given b: d q = a e k ( ) dt q d q dt q d q 3 z dt q 3 = b + kz k ( ) = cz m k 3(z z ) () where E(,, z ) represents an arbitrar equilibrium point of sstem (9). The parameters k, k, k 3 are feedback control gains which can make the eigenvalues of the linearized equation of the controlled sstem () satisf one of the above-mentioned Routh-Hurwitz conditions, then the trajector of the controlled sstem () asmptoticall approaches the unstable equilibrium point E(,, z ). Substituting the coordinate of E(,, z ) into sstem (), we get the Jacobian matri refers to E(,, z ) as follows J = k a e kz k + b k m m k 3 c () and the corresponding characteristic equation is P (λ) = λ 3 + (a b + c + k + k + k 3 )λ +( ab bc + ac bk + ck + ak + ck + k k +ak 3 bk 3 + k k 3 + k k 3 + km +ek z )λ +( abc bck + ack + ck k abk 3 bk k 3 +ak k 3 + k k k 3 + akm +kk m ekm +cek z +ekk 3 z ) = () and its discriminant is D(P ) = 8a a a 3 + (a a ) 4a 3 a 3 4a 3 7a 3 E-ISSN: Issue 8, Volume, August

7 Figure 4: The phase portrait of sstem (9) when α =.9. where a = a b + c + k + k + k 3 a = ab bc + ac bk + ck + ak + ck + k k +ak 3 bk 3 + k k 3 + k k 3 + km +ek z a 3 = abc bck + ack + ck k abk 3 bk k 3 +ak k 3 + k k k 3 + akm +kk m ekm +cek z +ekk 3 z (3) 5. Some conditions for stabilizing the equilibrium point When a =, b =.5, c = 5, e =, k = 4 and m =, because of the compleit of the condition, we onl consider the situation k = k = k 3 = β. When β >.5, we have D(P ) >, a >, a 3 >, a a > a 3 at the equilibrium point E, and all the eigenvalues are real and located in the left half of the s-plane. So Routh-Hurwitz conditions are the necessar and sufficient conditions for the fulfillment of Lemma. When a =, b =.5, c = 5, e =, k = 4, m = and k = k = k 3 = β R, we can obtain D(P ) <, a >, a >, a 3 >, then the equilibrium point E and E 3 is locall asmptoticall stable for an α < / Simulation results The parameters are chosen as α =.9, to ensure the eistence of chaos in the absence of control. The initial state is taken as (.,,.5), the time step is.(s), and the control is active when t 8(s) in order to make a comparison between the behavior before activation of control and after it. We simulate the process of which sstem () stabilizes to the equilibrium point E (,, ). When k = k = k 3 = β = 3, we have D(P ) >, a >, a 3 > and a a > a 3, and the three roots are λ = 8, λ = 4, λ 3 =.5, so sstem () is asmptoticall stable at E (,, ), Fig.7 displas the simulation result. We have also simulated the process of which sstem () stabilizes to equilibrium points E (.5,.8,.559) and E 3 (.5,.8,.559), using the feedback control method. When k = k = k 3 = β =., α =.6, we have D(P ) <, a, a, a 3 > and the roots of P (λ) = are λ = 4.488, λ = j, λ 3 = j. In this case, the integer-order sstem is unstable because the real part of λ and λ 3 is positive, however, the arguments of the right half-plane eigenvalues are 84., indicating that the eigenvalues are located in the stable region of the fractional-order autonomous sstem as shown in Fig.5, and the sstem s cone makes an angle 54. Hence, E-ISSN: Issue 8, Volume, August

8 Figure 5: Stabilit region of the fractional-order sstem. 3.5 E3.5 E3 E z E E.5 E z.5 E E3 z E3 E.5 E E Figure 6: The chaotic attractor and the equilibria of sstem (9). E-ISSN: Issue 8, Volume, August

9 3 z,,z t(s) Figure 7: Stabilizing the equilibrium point E. the controlled fractional-order sstem () is asmptoticall stable at E and E 3, as shown in Fig.8. and Fig.9. 6 Snchronization between two different fractional-order sstems The snchronization of chaotic fractional-order sstems have attracted increasing attention, because of its application in secure communication and control processing [, ]. In this section, we focus on investigating two different sstems: the novel fractional-order autonomous sstem and the fractional-order Liu sstem, using the active control technique in Ref. [?]. Let the novel fractional-order autonomous sstem (9) be the drive sstem: d q dt q = a e d q dt q = b + k z (4) d q 3 z dt q 3 = cz m and the response sstem is the fractional-order Liu sstem, which given as d q dt q = α( ) + u d q dt q = β θ z + u d q 3 z dt q 3 = δz + h + u 3 (5) Where U = (u, u, u 3 ) T is the active control function. Here, the aim is to determinate the controller U which is required for the drive sstem (4) to snchronize with the response sstem (5). For this purpose, we define the error states between (4) and (5) as e =, e = and e 3 = z z. The error dnamic sstem can be epressed b d q e dt q = α(e e ) + (a α) + α + e + u d q dt q = βe + β b θ z k z + u d q 3 z dt q 3 = δe 3 + (c δ)z + h + m + u 3 (6) Define the active control function U = (u, u, u 3 ) T as follows: u = (a α) α e + V (t) u = β + b + θ z + k z + V (t) u 3 = (c δ)z h m + V 3 (t) (7) and V (t), V (t) and V 3 (t) are the control inputs. Substituting (7) into (6) gives d q e dt q d q dt q d q 3 z dt q 3 = α(e e ) + V (t) = βe + V (t) (8) = δe 3 + V 3 (t) The error sstem (8) is a linear sstem with the control input function [V, V, V 3 ] T = H[e, e, e 3 ] T, where H is a 3 3 matri. Let J denote the Jacobi matri of sstem (8) at e- quilibrium point e i = (i =,, 3). α α J = H + β δ E-ISSN: Issue 8, Volume, August

10 3 z,,z t(s) Figure 8: Stabilizing the equilibrium point E. 3.5 z.5,,z t(s) Figure 9: Stabilizing the equilibrium point E 3. E-ISSN: Issue 8, Volume, August

11 We can choose the suitable matri H, and let the eigenvalues of J satisf the Lemma : arg(λ) πα/,α = ma(q, q, q 3 ). Thus snchronization between the fractional-order drive sstem (4) and the fractional-order response sstem (5) is achieved. For eample, we can choose H = α α β δ. In the following numerical simulation, the parameters are chosen as a =, b =.5, c = 5, e =, k = 4, m =, α =, β = 4, δ =.5, θ = and h = 4. The fractional order is set as q = q = q 3 =.98, and the initial states of the drive sstem (4) and the response sstem (5) are taken as (.,,.5) and (.,.4, 38) respectivel. Simulation results of the snchronization between the fractional sstems (4) and (5) are displaed in Fig..From Fig., we can find the novel autonomous sstem and Liu sstem are globall snchronized asmptoticall for t > Conclusion In this paper, chaos control of a fractional-order autonomous chaotic sstem is studied. Using predictor-correctors scheme the lowest order for the fractional-order autonomous sstem to remain chaotic is found via numerical simulation. Furthermore, we have studied the local stabilit of the equilibria using the fractional Routh-Hurwitz conditions. Analtical conditions for linear feedback control have been implemented, showing the effect of the fractional order on controlling chaos in this sstem. The snchronization of the fractional-order autonomous sstem and the fractional-order Liu sstem is also investigated using the active control method. Simulation results have illustrated the effectiveness of the proposed controlled method. Acknowledgements: The authors thank the reviewers for their careful reading and providing some pertinent suggestions. The research was supported b the National Natural Science Foundation of China (No: 97). References: [] P. Butzer, U. Westphal, An introduction to fractional calculus, Applications of Fractional Calculus in Phsics, (), pp [] P. Arena, R. Caponetto, Nonlinear noninteger order circuits and sstems: an introduction, World Scientific Pub Co Inc,. [3] R. Hilfer, others, Applications of fractional calculus in phsics, World Scientific Singapore,. [4] I. Grigorenko, E. Grigorenko, Chaotic dnamics of the fractional Lorenz sstem, Phsical Review Letters, 9, (3), 34(4 pages) [5] T.T. Hartle, C.F. Lorenzo, H. Killor Qammer, Chaos in a fractional order Chua s sstem, Circuits and Sstems I: IEEE Transactions on Fundamental Theor and Applications, 4, (995),pp [6] C. Li, G. Peng, Chaos in Chen s sstem with a fractional order, Chaos, Solitons and Fractals,, (4), pp [7] C. Li, G. Chen, Chaos and hperchaos in the fractional-order R?ssler equations, Phsica A: Statistical Mechanics and its Applications, 34, (4), pp [8] L. J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, et al., Chaos in the Newton-Leipnik sstem with fractional order, Chaos, Solitons and Fractals, 36, (8), pp [9] A. E. Matouk, Dnamical behaviors, linear feedback control and snchronization of the fractional order Liu sstem, Journal of Nonlinear Sstems and Applications, 3, (), pp [] C. Liu, L. Liu, T. Liu, A novel threedimensional autonomous chaos sstem, Chaos, Solitons and Fractals, 39, (9),pp [] C. Liu, T. Liu, L. Liu, K. Liu, A new chaotic attractor, Chaos, Solitons and Fractals,, (4), pp [] X. Wang, Y. He, M. Wang, Chaos control of a fractional order modified coupled dnamos sstem, Nonlinear Analsis: Theor, Methods and Applications, 7, (9), pp [3] Ahmed, E. et al., On some Routh Hurwitz conditions for fractional order differential e- quations and their applications in Lorenz, Rossler, Chua and Chen sstems, Phs. Lett. A, 358, (6), pp. 4 [4] K. Zhang, H. Wang, H. Fang, Feedback control and hbrid projective snchronization of a fractional-order Newton-Leipnik sstem, Communications in Nonlinear Science and Numerical Simulation, 7, (), pp E-ISSN: Issue 8, Volume, August

12 4 35 e e e3 3 5 e,e,e t(s) Figure : Simulation results of snchronization between sstem (4) and (5). [5] E. W. Bai, K. E. Lonngren, Sequential snchronization of two Lorenz sstems using active control, Chaos, Solitons and Fractals,, (), pp [6] J. Lu, G. Chen, S. Zhang, Dnamical analsis of a new chaotic attractor, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering,, (), pp. -5. [7] I. Podlubn, Fractional differential equations:(an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications), Academic press, 999. [8] K. Diethelm, N. J. Ford, Analsis of fractional differential equations, Journal of Mathematical Analsis and Applications, 65, (), [9] E. Ahmed, A. El-Saed, H. El-Saka, Equilibrium points, stabilit and numerical solutions of fractional-order predator-pre and rabies models, Journal of Mathematical Analsis and Applications, 35, (7), [] T. Zhou, C. Li, Snchronization in fractionalorder differential sstems, Phsica D: Nonlinear Phenomena,, (5) pp. -5. [] G. Peng, Y. Jiang, F. Chen, Generalized projective snchronization of fractional order chaotic sstems, Phsica A: Statistical Mechanics and its Applications, 387, (8), p- p E-ISSN: Issue 8, Volume, August