A new fractional order chaotic dynamical system and its synchronization using optimal control
AA new fractional order chaotic dynamical system and itssynchronization using optimal control
Madhuri Patil , Sachin Bhalekar , Department of Mathematics, Shivaji University, Kolhapur - 416004, India, School of Mathematics andStatistics, University of Hyderabad, Hyderabad, India, Email:[email protected] (Madhuri Patil),[email protected], sbb [email protected] (Sachin Bhalekar)
Abstract
In this work, we introduce a new three-dimensional chaotic differential dynamical system. We find equilibriumpoints of this system and provide the stability conditions for various fractional orders. Numerical simulations willbe used to investigate the chaos in the proposed system. A simple linear control will be used to control the chaoticoscillations. Further, we propose an optimal control which is based on the fractional order of the system and useit to synchronize new chaotic system.
Keywords : Fractional derivative, Chaos, Synchronization, Optimal control.
The differential equation is a prime tool used by Scientists in modeling various natural phenomena. To makeit more realistic, the generalized operator viz. fractional derivative [1, 2] is introduced by the researchers. Theorder in fractional derivative can be any real or complex number, a function of time or may be distributed oversome interval. This flexible order makes fractional derivative more suitable to model the intermediate processesand the memory properties. The fractional differential equations (FDE) have applications in Bioengineering [3],Viscoelasticity [4], Control theory [5], and so on. The analysis of FDEs is presented in [6, 7, 8, 9, 10, 11, 12]. Thedifficult task of numerical solutions of FDEs is handled in [13, 14].The signals generated by a higher-order deterministic nonlinear system which are aperiodic for all the timeand depend sensitively on initial conditions are termed as “chaotic” [15, 16]. The chaos can occur in a nonlinearautonomous differential dynamical system of order three or more, a delay differential equation and a discrete map.The most celebrated examples of chaos are the Lorenz system [17] and the logistic map [18]. Few other examplesinclude the systems viz. Chen [19], Chua [20], Lu [21], Rossler [22], Bhalekar-Gejji(BG) [23], Modified BG [24],Proto BG [25], Pehlivan [26], and so on.The fractional order counterparts of these classical systems also produce chaotic signals for certain values. Fora commensurate order case, it is observed that the system remains chaotic up to some threshold value of fractionalorder and then becomes stable. Few examples of fractional order chaotic systems are described in [27, 28, 29, 30,31, 32, 33].Chaos in fractional ordered delayed systems is analyzed in [34, 35, 36, 37, 38].The paper is organized as follows:The basics are described in Section 2. We propose a new chaotic system and present the stability, bifurcation andchaos in Section 3. Sections 4 and 5 deal with the chaos control and synchronization, respectively. The commentson the incommensurate order case are presented in Section 6. The conclusions are summarized in Section 7.
This section deals with basic definitions and results given in the literature [1, 2, 8, 39]. Throughout this section, wetake n ∈ N . 1 a r X i v : . [ n li n . C D ] J u l efinition 2.1. Let α ≥ ( α ∈ R ). Then Riemann-Liouville (RL) fractional integral of function f ∈ C [0 , b ] , t > of order ‘ α ’ is defined as, I αt f ( t ) = 1Γ( α ) (cid:90) t ( t − τ ) α − f ( τ ) d τ. (1) Definition 2.2.
The Riemann-Liouville (RL) fractional derivative of order α > of function f ∈ C [0 , b ] , t > isdefined as, RL D αt f ( t ) = (cid:40) n − α ) d n dt n (cid:82) t ( t − τ ) n − α − f ( τ ) d τ, if n − < α < n d n dt n f ( t ) , if α = n. (2) Definition 2.3.
The Caputo fractional derivative of order α > , n − < α ≤ n is defined for f ∈ C n [0 , b ] , t > as, C D αt f ( t ) = (cid:40) n − α ) (cid:82) t ( t − τ ) n − α − f ( n ) ( τ ) d τ, if n − < α < n d n dt n f ( t ) , if α = n. (3) Definition 2.4.
The one-parameter Mittag-Leffler function is defined as, E α ( z ) = ∞ (cid:88) k =0 z k Γ( αk + 1) , z ∈ C , ( α > . (4) The two-parameter Mittag-Leffler function is defined as, E α,β ( z ) = ∞ (cid:88) k =0 z k Γ( αk + β ) , z ∈ C , ( α > , β > . (5) Properties (i) Let n − < α ≤ n and β ≥ C D αt t β = (cid:40) Γ( β +1)Γ( − α + β +1) t β − α , if β > n − , β ∈ R , if β ∈ { , , , . . . , n − } . (ii) C D αt I βt f ( t ) = I β − αt f ( t ) , if β > αf ( t ) , if β = α C D α − βt f ( t ) , if α > β. (iii) C D αt c = 0 , where c is a constant.(iv) RL D αt c = c t − α Γ(1 − α ) , where c is a constant. Theorem 2.1. [40] Solution of homogeneous fractional order differential equation C D αt x ( t ) + λx ( t ) = 0 , < α < (6) is given by, x ( t ) = x (0) E α ( − λt α ) . (7) Stability Analysis
Consider the following fractional order system, C D α t x = f ( x , x , . . . , x n ) , C D α t x = f ( x , x , . . . , x n ) , ... C D α n t x n = f n ( x , x , . . . , x n ) (8)where < α i < are fractional orders. If α = α = · · · = α n , then the system (8) is called commensurate ordersystem, otherwise incommensurate order system. 2 point E = ( x ∗ , x ∗ , . . . , x ∗ n ) is called an equilibrium point of the system (8) if f i ( E ) = f i ( x ∗ , x ∗ , . . . , x ∗ n ) = 0 , for each i = 1 , , . . . , n. (a) Commensurate order system:Theorem 2.2. [41, 42] Consider α = α = α = · · · = α n in (8). An equilibrium point E of the system (8) islocally asymptotically stable if all the eigenvalues of the Jacobian matrix J = ∂f ∂x ∂f ∂x · · · ∂f ∂x n ∂f ∂x ∂f ∂x · · · ∂f ∂x n ... ... ... ... ∂f n ∂x ∂f n ∂x · · · ∂f n ∂x n (9) evaluated at E = ( x ∗ , x ∗ , . . . , x ∗ n ) satisfy the following condition | arg(Eig( J | E )) | > απ . (b) Incommensurate order system:Theorem 2.3. [7] Consider the incommensurate fractional ordered dynamical system given by (8). Let α i = v i u i ,gcd ( u i , v i ) = 1 , u i , v i be positive integers. Define M to be the least common multiple of u i ’s.Define, ∆( λ ) = diag (cid:0)(cid:2) λ Mα , λ Mα , . . . , λ Mα n (cid:3)(cid:1) − J (10)where, J is the Jacobian matrix as defined in (9) evaluated at point E . If all the roots λ ’s of det(∆( λ )) = 0 satisfy | arg( λ ) | > απ , then E is locally asymptotically stable. This condition is equivalent to the followinginequality π M − min i | arg( λ i ) | < . (11)The term π M − min i | arg( λ i ) | is called as the instability measure for equilibrium points in fractional order systems(IMFOS). Hence, a necessary condition [7] for fractional order system (8) to exhibit chaotic attractor is IMFOS ≥ . (12)Note that, the condition (12) is not sufficient [7, 29] for chaos to exist. We propose the following chaotic dynamical system, ˙ x = ax − y , ˙ y = by − z + dxz, ˙ z = − gz + 4 xy − hx , (13)where a , b , d , g , h ∈ R are parameters. When a = − , b = 2 . , d = − , g = 5 . and h = − . , system (13)shows chaotic behavior.If X ∗ = ( x ∗ , y ∗ , z ∗ ) is an equilibrium point of (13), then the Jacobian J ( X ∗ ) of this system at X ∗ is given by J ( X ∗ ) = a − y ∗ dz ∗ b − dx ∗ − hx ∗ + 4 y ∗ x ∗ − g . .1 Bifurcation Analysis Out of five parameters a , b , d , g , h of the system (13), we hold any four parameters fixed and vary the remain-ing one to present the bifurcation analysis. The bifurcation diagrams and the trajectories/phase portraits of thecorresponding cases are presented in Figure 1- Figure 4.Fixed parameters Changing parameter Behavior of trajectories Corresponding figures b = 2 . a = − Stable orbit Figure 1(a) d = − a = − . Limit cycle Figure 1(b) g = 5 . a = − . Limit cycle Figure 1(c) h = − . a = − . Limit cycle Figure 1(d) a = − Chaos Figure 1(e) a = − b = 1 . Stable orbit Figure 2(a) d = − b = 1 . Chaos Figure 2(b) g = 5 . b = 2 . Chaos Figure 2(c) h = − . b = 4 . Limit cycle Figure 2(d) b = 5 . Limit cycle Figure 2(e) b = 2 . g = 2 . Limit cycle Figure 3(b) d = − g = 3 . Limit cycle Figure 3(c) h = − . g = 3 . Limit cycle Figure 3(d) a = − g = 4 . Chaos Figure 3(a) g = 5 . Chaos Figure 3(e) a = − h = − . Stable orbit Figure 4(a) b = 2 . h = − Limit cycle Figure 4(b) d = − h = − . Limit cycle Figure 4(c) g = 5 . h = 0 . Limit cycle Figure 4(d) h = 0 . Chaos Figure 4(e)Table 1: Observations for different values of parameters
The equilibrium point of three dimensional dynamical system is called a saddle point of index (respectively, index ) if it generates one (respectively, two) unstable eigenvalue(s). The scrolls of chaotic attractors are usually aroundthe saddle points of index , whereas the saddles of index connect these scrolls [43, 44].In Table 2, we classify equilibrium points of the proposed system (13) with parameter values a = − , b = 2 . , d = − , g = 5 . and h = − . . It is observed that the chaotic attractor in this system is in the neighborhood of theequilibrium points O , E and E .Equilibrium points X ∗ Eigenvalues of J ( X ∗ ) Nature of X ∗ O (0 , , − . , . , − saddle point of index E ( − . , − . , . − . , . ± . i saddle point of index E ( − . , . , − . − . , . ± . i saddle point of index E ( − . , − . , . − . , − . , . saddle point of index Table 2: Equilibrium points of (13), eigenvalues of Jacobian at these points and the nature of equilibriums4 - - xyz (a) a = − - - - - - (b) a = − . - - - - - - (c) a = − . - - - - - (d) a = − . - - - - - (e) a = − - - - - - - (f) Bifurcation with parameter a Figure 1: Bifurcation analysis for parameter a - - - - (a) b = 1 . - - - - (b) b = 1 . - - - - - (c) b = 2 . - - - - - (d) b = 4 . - - - - - - (e) b = 5 . - - - (f) Bifurcation with parameter b Figure 2: Bifurcation analysis for parameter b - - - - - (a) g = 2 . - - - - - (b) g = 3 . - - - - - - (c) g = 3 . - - - - - - - - - (d) g = 4 . - - - - - (e) g = 5 . - - (f) Bifurcation with parameter g Figure 3: Bifurcation analysis for parameter g - - xyz (a) h = − . - - - - - - - - - - - (b) h = − - - - - - - (c) h = − . - - - - - - (d) h = 0 . - - - - (e) h = 0 . - - - - - (f) Bifurcation with parameter h Figure 4: Bifurcation analysis for parameter h .3 Chaos Consider the chaotic system, ˙ x = − x − y ˙ y = 2 . y − z − xz ˙ z = − . z + 4 xy − . x . (14)We know that, extreme sensitivity to initial conditions is one of the significant characteristic of chaos. In theFigure 5, we have plotted the trajectories x ( t ) with initial conditions (0 . , . , . and (0 . , . , . withsmall variation of . . We can see that, there is huge amount of difference between these two trajectories. Thishigh sensitivity to initial conditions indicates chaos in (14).
20 40 60 80 100 t - - - ( t ) ( x ( ) ,y ( ) ,z ( ))=( )( x ( ) ,y ( ) ,z ( ))=( ) Figure 5: Solution trajectory x ( t ) of the system (14) with slightly different initial conditions.Chaotic attractor and wave forms are shown in Figure 6. Consider the fractional order generalization of (14) as C D αt x = − x − y C D βt y = 2 . y − z − xz C D γt z = − . z + 4 xy − . x . (15)Equilibrium points of fractional-order system (15) and the eigenvalues of the Jacobian evaluated at these equi-librium points are same as their classical counterparts, described in Table 2. Consider the fractional order system (15) with α = β = γ .Define α ∗ ( E ) = 2 π (cid:20) min λ E | arg( λ E ) | (cid:21) as stability bound for the equilibrium point E . Therefore, E is stable if < α < α ∗ ( E ) and unstable for α ∗ ( E ) <α ≤ . Note that, instability of all the equilibrium points is the necessary condition [45] to exists chaos in thecommensurate order system (15).In this case, α ∗ ( O ) = 0 ,α ∗ ( E ) = 0 . α ∗ ( E ) = 0 . . t - - - - ( t ) (a) z ( t ) - - - x - - y (b) xy − plane Figure 6: Chaos in system (14)We define ¯ α = max E α ∗ ( E ) . (16)In this case, ¯ α = 0 . . If α < ¯ α , then at least one of the equilibrium points O , E or E becomes stable andthe system cannot be chaotic.We further define the “threshold value” α t such that the system is chaotic for α > α t and achaotic for α < α t .For the commensurate order system (15) α t = 0 . (cf. Figures 7 (a-d)).Note that, α t is the value obtained using numerical observations and α t ≥ ¯ α .Based on the observations of various fractional order commensurate order chaotic systems we propose thefollowing conjecture. Conjecture 1.
Suppose that the system C D αt X = f ( X ) (17) is chaotic with the threshold value α t . If ¯ α defined in (16) corresponds to the equilibrium points associated withthe chaotic attractor then α t = ¯ α . We consider the system (15) with two cases viz. α ≥ . , β = γ = 1 and α = β = 1 , γ ≥ . . We show thatIMFOS ≥ is not sufficient condition to exist chaos in the system. Numerical simulations:Case 1:
Let α = = 0 . and β = γ = 1 .In this case M = LCM (50 , ,
1) = 50 , ∆ = diag( λ , λ , λ ) − J ( O )Det(∆ ) = (1 + λ )( − . λ )(5 . λ ) IMFOS ( O ) = π − . > Similarly, we can find that, IMFOS ( E ) = π − . . > andIMFOS ( E ) = π − . . > So, the system is unstable but there does not exist chaos (cf. Figure 8(a)).10 t - - - z (a) α = 0 . , Chaotic trajectory - - - - - - x - - - z (b) α = 0 . , Chaoticattractor
150 200 250 300 t - - - z (c) α = 0 . , stable
120 140 160 180 200 t - - - - x (d) α = 0 . , stable Figure 7: Stability of commensurate fractional order system (15)11 ase 2:
Let α = = 0 . and β = γ = 1 .In this case M = LCM (10 , ,
1) = 10 , ∆ = diag( λ , λ , λ ) − J ( O )Det(∆ ) = (1 + λ )( − . λ )(5 . λ ) IMFOS ( O ) = π − . > Similarly, we can find that, IMFOS ( E ) = π − . . > andIMFOS ( E ) = π − . . > The system is chaotic (cf. Figure 8(b)).
Case 3:
Let γ = = and α = β = 1 .In this case M = LCM (20 , ,
1) = 20 , ∆ = diag( λ , λ , λ ) − J ( O )Det(∆ ) = (5 . λ )( − . λ )(1 + λ ) IMFOS ( O ) = π − . > Similarly, we can find that, IMFOS ( E ) = π − . . > andIMFOS ( E ) = π − . . > The system is chaotic (cf. Figure 8(c)). 12
20 40 60 80 100 t z (a) α = 0 . , β = γ = 1 - - - x - - y -
10 1 z (b) α = 0 . , β = γ = 1 - - y - - - z (c) γ = 0 . , α = β = 1 Figure 8: Stability of incommensurate fractional order system13
Chaos control using linear control
Let us consider the controlled system, C D αt x = − x − y C D αt y = 2 . y − z − xz + u C D αt z = − . z + 4 xy − . x , (18)where u is the linear feedback control term. We set u = ky , where k is a parameter to be determined so that thesystem (18) is stable. Clearly O = (0 , , is one of the equilibrium points of the system (18).The Jacobian of system (18) evaluated at O is, − . k −
10 0 − . . Eigenvalues of this Jacobian matrix are − , . k and − . . Therefore, if we choose k < − . then theequilibrium point O becomes stable and the system (18) loose the chaotic nature. In Figure 9 we observe stablebehavior of system (18) with α = 0 . and k = − .
20 40 60 80 100 t - - xyz Figure 9: α = 0 . , k = − , X (0) = ( − . , . , − . T We say that the chaotic systems get synchronized if the difference between their states tends to zero with increasingtime. Synchronization between many pairs of fractional ordered chaotic systems are studied in the literature. Mostof the fractional ordered chaotic systems are synchronized by using active control [46, 47], adaptive control [48, 49],sliding mode control [50, 51], impulsive control [52, 53], projective control [54, 55], etc. In this section, we proposea synchronization technique based on optimal control which depends on the fractional order of the system.Here we have taken the system (15) as both drive and response system.Let us consider a drive system as C D αt x = − x − y C D αt y = 2 . y − z − x z C D αt z = − . z + 4 x y − . x (19)14nd response system as C D αt x = − x − y + u C D αt y = 2 . y − z − x z + u C D αt z = − . z + 4 x y − . x + u . (20)We define the error functions as, e = x − x , e = y − y , e = z − z (21)The error system can be given by using equations (19), (20) and (21) as C D αt e = − e + y − y − u C D αt e = 2 . e − e − x z + 5 x z − u C D αt e = − . e + 4 x y − x y − . x − . x − u . (22)Let us choose the control terms u i ( t ) in system (20) as, u = − e + y − y − cos( απ (cid:15) ) e − sin( απ (cid:15) ) e u = 2 . e − e − x z + 5 x z + sin( απ (cid:15) ) e − cos( απ (cid:15) ) e u = 4 x y − x y − . x − . x , (23)where (cid:15) is any small positive real number.Note that, the control terms u i depend on fractional order α . Hence, for the given fractional order α the optimalstrength of the controls is utilized in contrast with the conventional controls.Now the error system (22) becomes, C D αt e = cos( απ (cid:15) ) e + sin( απ (cid:15) ) e C D αt e = − sin( απ (cid:15) ) e + cos( απ (cid:15) ) e C D αt e = − . e . (24)The eigenvalues of the coefficient matrix of linear system (24) are, λ ± = e ± i ( απ + (cid:15) ) and λ = − . . (25)Clearly, all the eigenvalues satisfy the condition, | arg ( λ ) | > απ .So, the error system (24) is stable and hence the systems (19) and (20) get synchronized.Figures 10(a)-(c) show the synchronization where the response system is shown by the dashed line. Here we havetaken α = 0 . and the initial conditions as x (0) = − . , y (0) = 0 . , z (0) = − . , x (0) = − . , y (0) = − . and z (0) = 0 . . In Figure 10(d), the error terms e i ( t ) of drive and response systems are plotted.15 t - - - - - x , x (a) Signals x , x
10 20 30 40 t - - y , y (b) Signals y , y
10 20 30 40 t - - - z , z (c) Signals z , z
10 20 30 40 50 60 t - - - - ( e , e , e ) (d) error system Figure 10: Synchronization of chaos in the proposed system
The behavior of the commensurate fractional order systems is relatively simpler than that of incommensurate order.Analyzing stability is a difficult task when the system is incommensurate.If the commensurate order system is stable for some value α ∗ then it remains stable for all the orders α ∈ (0 , α ∗ ) . We cannot expect such nice behavior from the incommensurate order systems. We illustrate this with thefollowing example.Consider the system (15) with α = γ = 1 . The system is chaotic for α = 0 . and is stable for α = 0 . .However, it does not remain stable for all α ∈ (0 , . e.g. α = 0 . produces unstable oscillations (cf. Fig. 11).Order of derivative IMFOS evaluated at equilibrium points Stability Corresponding figures O E E β = 0 . , α = γ = 1 0 . > . > . > Unstable Figure 11(a) β = 0 . , α = γ = 1 0 . > − . < . > Stable Figure 11(b) β = 0 . , α = γ = 1 0 . > . > . > Chaotic Figure 11(c)Table 3: Weired behavior of incommensurate order system16 t - - - - x (a) β = 0 . , α = γ = 1 , system is unstable
10 20 30 40 t - - - - - y (b) β = 0 . , α = γ = 1 , system is stable
20 40 60 80 100 t - - xyz (c) β = 0 . , α = γ = 1 , system is chaotic Figure 11
This article presents a new example of chaotic differential dynamical system. The bifurcation analysis with respectto various parameters is presented in the details. The stability analysis of fractional order generalization of theproposed system is described. Chaos in the commensurate as well as incommensurate order cases is investigated.The chaos in the system is controlled using a simple linear controller. The synchronization in the system is achievedusing a nonlinear feedback controller. The peculiarity of this control is that it depends on fractional order. Unlikein other control strategies, the coupling strength is adjustable with the fractional order.
Acknowledgment
S. Bhalekar acknowledges the Science and Engineering Research Board (SERB), New Delhi, India for the ResearchGrant (Ref. MTR/2017/000068) under Mathematical Research Impact Centric Support (MATRICS) Scheme. M.Patil acknowledges Department of Science and Technology (DST), New Delhi, India for INSPIRE Fellowship(Code-IF170439). Authors are grateful to the anonymous reviewers for their insightful comments leading to theimproved manuscript.
References [1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).172] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition UsingDifferential Operators of Caputo Type, Springer Science & Business Media, New York, (2010).[3] R. L. Magin, Fractional calculus in bioengineering, Begell House, Redding, (2006).[4] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Mod-els, World Scientific, Singapore, (2010).[5] D. Baleanu, J. Antonio, T. Machado, A. C. J. Luo, Fractional Dynamics and control, Springer, New York,(2012).[6] D. Matignon, “Stability results for fractional differential equations with applications to control processing,”Computational engineering in Systems and Application multiconference, IMACS, lille, france, , 963–968(1996).[7] M. S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems, Physica D, ,2628–2637 (2008) .[8] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integral and Derivatives: Theory and Applications,Gordon and Breach Science, Yverdon, (1993).[9] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley,(1993).[10] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, Journal ofMathematical Analysis and Applications, (2), 804–812 (2000).[11] V. Daftardar-Gejji, H. Jafari, Analysis of a system of nonautonomous fractional differential equations involv-ing Caputo derivatives, Journal of Mathematical Analysis and Applications, (2), 1026–1033 (2007).[12] V. Daftardar-Gejji, A. Babakhani, Analysis of a system of fractional differential equations, Journal of Mathe-matical Analysis and Applications, (2), 511–522 (2004).[13] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractionaldifferential equations, Nonlinear Dynamics, (1-4), 3–22 (2002).[14] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equa-tions, Applied Mathematics and Computation, , 158–182 (2014).[15] R. Devaney, An introduction to chaotic dynamical systems, CRC Press, (2018).[16] K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos, Springer, New York, (1996).[17] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., , 130 (1963).[18] R. M. May, Simple mathematical models with very complicated dynamics, Nature, , (1976) 459–467.[19] G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifur. Chaos, , 1465–1466 (1999).[20] T. Matsumoto, A chaotic attractor from Chua’s circuit, IEEE Trans. Circuits Syst., (12), 1055–1058 (1984).[21] J. H. Lu, G. R. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos, (3), 659–661 (2002).[22] O. E. Rossler, An equation for continuous chaos, Phys. Lett. A, , 397–398 (1976).[23] S. Bhalekar, V. Daftardar-Gejji, A new chaotic dynamical system and its synchronization, Proceedings of theinternational conference on mathematical sciences in honor of Prof. A. M. Mathai, 3-5 January 2011, Palai,Kerla- 686 574, India.[24] P. P. Singh, B. K. Roy, Comparative performances of synchronisation between different classes of chaoticsystems using three control techniques, Annual Reviews in Control, , 152–165 ( 2018).1825] M. Aqeel, A. Azam, S. Ahmad, The proto Bhalekar-Gejji system, Chinese Journal of Physics, (3), 1220–1231 (2018).[26] V. Sundarapandian, Global chaos synchronization of the Pehlivan systems by sliding mode control, Interna-tional Journal on Computer Science and Engineering, (5), 2163–2169 (2011).[27] I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., , 485–490 (1995).[29] V. Gejji, S. Bhalekar, Chaos in fractional ordered Liu system, Computers and Mathematics with Applications, , 1117–1127 (2010).[30] S. Bhalekar, V. Gejji, D. Baleanu, R. Magine, Transient chaos in fractional Bloch equations, Computers andMathematics with Applications , (2012) 3367–3376.[31] V. K. Yadav, V. K. Shukla, S. Das, A. Y. T. Leung, M. Srivastava, Function projective synchronization offractional order satellite system and its stability analysis for incommensurate case. Chinese journal of physics, (2), 696–707 (2018).[32] M. Faieghi, H. Delavari, Chaos in fractional-order Genesio-Tesi system and its synchronization, Communica-tions in Nonlinear Science and Numerical Simulation, (2), 731–741 (2012).[33] I. Petr´aˇs, Chaos in the fractipona-order Volta’s system, modeling and simulation, Nonlinear Dynamics, (1-2), 157–170 (2012).[34] S. Bhalekar, V. Gejji, Fractional ordered Liu system with time-delay, Commun Nonlinear Sci Numer Simulat., (8), 084306 (2016).[37] S. Bhalekar, Stability analysis of a class of fractional delay differential equations. Pramana, (2), 215–224(2013).[38] S. Bhalekar, Dynamical analysis of fractional order Uar prototype delayed system. Signal, Image and VideoProcessing, (3), 513–519 (2012).[39] S. Das, Functional Fractional Calculus, Springer Science & Business Media, Berlin, (2011).[40] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputoderivatives, Acta Math. Vietnam., , 207–233 (1999).[41] M.S. Tavazoei, M. Haeri, Regular oscillations or chaos in a fractional order system with any effective dimen-sion, Nonlinear Dynamics, (3), 213–222 (2008).[42] D. Matignon, B. d’Andr´ea-Novel, Some results on controllability and observability of finite-dimensional frac-tional differential systems, In Computational engineering in systems applications, IMACS, IEEE-SMC Lille,France, , 952–956 (1996).[43] J. M. T. Thompson, H. B. Stewart, Nonlinear dynamics and chaos, John Wiley & Sons, (2002).[44] D. Cafagna, G. Grassi, New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring, InternationalJournal of Bifurcation and Chaos, (10), 2889–2903 (2003).1945] M. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems,Physics Letters A, , 102–113 (2007).[46] Bai, E.W. and Lonngren, K.E., 1997. Synchronization of two Lorenz systems using active control. Chaos,Solitons & Fractals, (1), pp.51-58.[47] S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using activecontrol. Communications in Nonlinear Science and Numerical Simulation, (11), 3536–3546 (2010).[48] T. L. Liao, Adaptive synchronization of two Lorenz systems, Chaos, Solitons & Fractals, (9), 1555–1561(1998).[49] M. T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system. Applied Mathematicsand Computation, (1), 113–128 (2003).[50] A. Razminia, D. Baleanu, Complete synchronization of commensurate fractional order chaotic systems usingsliding mode control, Mechatronics, (7), 873–879 (2013).[51] P. Muthukumar, P. Balasubramaniam, K. Ratnavelu, Sliding mode control design for synchronization of frac-tional order chaotic systems and its application to a new cryptosystem. International Journal of Dynamics andControl, (1), 115–123 (2017).[52] W. Xing-Yuan, Z. Yong-Lei, L. Da, Z. Na, Impulsive synchronisation of a class of fractional-order hyper-chaotic systems. Chinese Physics B, (3), 030506 (2011).[53] L. Jin-Gui, A novel study on the impulsive synchronization of fractional-order chaotic systems. ChinesePhysics B, (6), 060510 (2013).[54] X. Wang, Y. He, Projective synchronization of fractional order chaotic system based on linear separation,Phys. Lett. A, , 435-441 (2008).[55] G. Peng, Y. Jiang, F. Chen, Generalized projective synchronization of factional order chaotic systems, PhysicaA, , 3738-3746 (2008).[56] A. Jhinga, V. Daftardar-Gejji, A new finite-difference predictor-corrector method for fractional differentialequations, Applied Mathematics and Computation,336