Chaos Control and Synchronization in Fractional-Order Lorenz-Like System

Abstract

The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006). The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback.

1. Introduction

A variety of problems in engineering and natural sciences are modeled using chaotic dynamical systems.A chaotic system is a nonlinear deterministic system possessing complex dynamical behaviors such as being extremely sensitive to tiny variations of initial conditions, unpredictability, and having bounded trajectories in the phase space [1]. Controlling the chaotic behavior in the dynamical systems using some form of control mechanism has recently been the focus of much attention. So many approaches are proposed for chaos control namely, OGY method [2], backstepping design method [3], differential geometric method [4], inverse optimal control [5], sampled-data feedback control [6], adaptive control [7], and so on. One simple approach is the linear feedback control [8]. Linear feedback controllers are easy to implement, they can perform the job automatically, and stabilize the overall control system efficiently [9].

The controllers can also be used to synchronize two identical or distinct chaotic systems [10–13]. Synchronization of chaos refers to a process wherein two chaotic systems adjust a given property of their motion to a common behavior due to a coupling. Synchronization has many applications in secure communications of analog and digital signals [14] and for developing safe and reliable cryptographic systems [15].

Fractional calculus deals with derivatives and integration of arbitrary order [16–18] and has deep and natural connections with many fields of applied mathematics, engineering, and physics. Fractional calculus has wide range of applications in control theory [19], viscoelasticity [20], diffusion [21–25], turbulence, electromagnetism, signal processing [26, 27], and bioengineering [28]. Study of chaos in fractional order dynamical systems and related phenomena is receiving growing attention [29, 30]. I. Grigorenko and E. Grigorenko investigated fractional ordered Lorenz system and observed that below a threshold order the chaos disappears [31]. Further, many systems such as Li and Peng [32], Lu [33], Li and Chen [34], Daftardar-Gejji and Bhalekar [35], and unified system [36] were investigated in this regard. Effect of delay on chaotic solutions in fractional order dynamical system is investigated by the present author [37]. It is demonstrated that the chaotic systems can be transformed into limit cycles or stable orbits with appropriate choice of delay parameter. Synchronization of fractional order chaotic systems was also studied by many researchers [38–41].

In this paper, we propose fractional version of the Lorenz-like chaotic dynamical system [42]. We investigate minimum effective dimension of the system for chaos to exist. Then we control the chaos using simple linear feedback control. Further, we synchronize the proposed fractional order system using feedback control.

2. Preliminaries

2.1. Fractional Calculus

Few definitions of fractional derivatives are known [16–18]. Probably the best known is the Riemann-Liouville formulation.

The Riemann-Liouville integral of order 𝜇,𝜇>0 is given by 𝐼^𝜇𝑓(𝑡)=1Γ𝜇^𝑡₀(𝑡−𝜏)^𝜇−1𝑓(𝜏)𝑑𝜏,𝑡>0. (2.1) An alternative definition was introduced by Caputo. Caputo's derivative is defined as 𝐷^𝜇𝑓(𝑡)=𝑑^𝑚𝑑𝑡^𝑚𝑓(𝑡),𝜇=𝑚=𝐼^𝑚−𝜇𝑑^𝑚𝑓(𝑡)𝑑𝑡^𝑚,𝑚−1<𝜇<𝑚, (2.2) where 𝑚∈𝑁. The main advantage of the Caputo's formulation is that the Caputo derivative of a constant is equal to zero, that is not the case for the Riemann-Liouville derivative. Note that for 𝑚−1<𝜇≤𝑚,𝑚∈𝑁, 𝐼^𝜇𝐷^𝜇𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑘=0𝑑^𝑘𝑓𝑑𝑡^𝑘(0)𝑡^𝑘,𝐼𝑘!_𝜇𝑡^𝜈=Γ(𝜈+1)Γ𝑡𝜇+𝜈+1^𝜇+𝜈. (2.3)

2.2. Numerical Method for Solving Fractional Differential Equations

Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). A modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm et al. in [43–45] to solve FDEs.

Consider for 𝛼∈(𝑚−1,𝑚] the initial value problem (IVP) 𝐷^𝛼𝑦(𝑡)(𝑡𝑦=𝑓𝑡,𝑦),0≤𝑡≤𝑇,_(𝑘)(0)=𝑦₀^(𝑘),𝑘=0,1,…,𝑚−1. (2.4) The IVP (2.4) is equivalent to the Volterra integral equation 𝑦(𝑡)=^𝑚−1_𝑘=0𝑦₀^(𝑘)𝑡^𝑘+1𝑘!Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓𝜏,𝑦(𝜏)𝑑𝜏. (2.5) Consider the uniform grid {𝑡_𝑛=𝑛ℎ/𝑛=0,1,…,𝑁} for some integer 𝑁 and ℎ∶=𝑇/𝑁. Let 𝑦_ℎ(𝑡_𝑛) be approximation to 𝑦(𝑡_𝑛). Assume that we have already calculated approximations 𝑦_ℎ(𝑡_𝑗),𝑗=1,2,…,𝑛, and we want to obtain 𝑦_ℎ(𝑡_𝑛+1) by means of the equation 𝑦_ℎ𝑡_𝑛+1=^𝑚−1_𝑘=0𝑡^𝑘_𝑛+1𝑦𝑘!^0(𝑘)+ℎ^𝛼Γ()𝑓𝑡𝛼+2^𝑛+1,𝑦^𝑃_ℎ𝑡_𝑛+1+ℎ^𝛼Γ(𝛼+2)^𝑛_𝑗=0𝑎_𝑗,𝑛+1𝑓𝑡_𝑗,𝑦_𝑛𝑡_𝑗, (2.6) where 𝑎_𝑗,𝑛+1=𝑛^𝛼+1−(𝑛−𝛼)(𝑛+1)^𝛼if𝑗=0,𝑛−𝑗+2^𝛼+1+𝑛−𝑗^𝛼+1−2𝑛−𝑗+1^𝛼+1if1≤𝑗≤𝑛,1if𝑗=𝑛+1. (2.7) The preliminary approximation 𝑦^𝑃_ℎ(𝑡_𝑛+1) is called predictor and is given by 𝑦^𝑃_ℎ𝑡_𝑛+1=^𝑚−1_𝑘=0𝑡^𝑘_𝑛+1𝑦𝑘!^0(𝑘)+1Γ(𝛼)^𝑛_𝑗=0𝑏_𝑗,𝑛+1𝑓𝑡_𝑗,𝑦_𝑛𝑡_𝑗, (2.8) where 𝑏_𝑗,𝑛+1=ℎ^𝛼𝛼𝑛+1−𝑗^𝛼−𝑛−𝑗^𝛼. (2.9) Error in this method is max_{𝑗=0,1,…,𝑁}𝑦𝑡_𝑗−𝑦_ℎ𝑡_𝑗=𝑂ℎ^𝑝, (2.10) where 𝑝=min(2,1+𝛼).

2.3. Asymptotic Stability of the Fractional-Ordered System

Consider the following fractional-ordered dynamical system: 𝐷^𝛼𝑥_𝑖=𝑓_𝑖𝑥₁,𝑥₂,𝑥₃,1≤𝑖≤3. (2.11) Let 𝑝≡(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃) be an equilibrium point of the system (2.11) that is, 𝑓_𝑖(𝑝)=0,1≤𝑖≤3 and 𝜉_𝑖=𝑥_𝑖−𝑥^∗_𝑖 a small disturbance from a fixed point. Then 𝐷^𝛼𝜉_𝑖=𝐷^𝛼𝑥_𝑖=𝑓_𝑖𝑥₁,𝑥₂,𝑥₃=𝑓_𝑖𝜉₁+𝑥^∗₁,𝜉₂+𝑥^∗₂,𝜉₃+𝑥^∗₃=𝑓_𝑖𝑥^∗₁,𝑥^∗₂,𝑥^∗₃+𝜉₁𝜕𝑓_𝑖𝑝𝜕𝑥₁+𝜉₂𝜕𝑓_𝑖𝑝𝜕𝑥₂+𝜉₃𝜕𝑓_𝑖𝑝𝜕𝑥₃+higher-orderedterms≈𝜉₁𝜕𝑓_𝑖𝑝𝜕𝑥₁+𝜉₂𝜕𝑓_𝑖𝑝𝜕𝑥₂+𝜉₃𝜕𝑓_𝑖𝑝𝜕𝑥₃. (2.12) System (2.12) can be written as 𝐷^𝛼𝜉=𝐽𝜉, (2.13) where 𝜉=(𝜉₁,𝜉₂,𝜉₃)^𝑡 and 𝜕𝐽=¹𝑓₁𝑝𝜕₂𝑓₁𝑝𝜕₃𝑓₁𝑝𝜕₁𝑓₂𝑝𝜕₂𝑓₂𝑝𝜕₃𝑓₂𝑝𝜕₁𝑓₃𝑝𝜕₂𝑓₃𝑝𝜕₃𝑓₃𝑝. (2.14) Consider the linear autonomous system 𝐷^𝛼𝜉=𝐽𝜉,𝜉(0)=𝜉₀, (2.15) where 𝐽 is 𝑛×𝑛 matrix and 0<𝛼<1 is asymptotically stable if and only if |arg(𝜆)|>𝛼𝜋/2 for all eigenvalues 𝜆 of 𝐽. In this case, each component of solution 𝜉(𝑡) decays towards 0 like 𝑡^−𝛼 [29, 46].

This shows that if |arg(𝜆)|>𝛼𝜋/2 for all eigenvalues 𝜆 of 𝐽 then the solution 𝜉_𝑖(𝑡) of the system (2.13) tends to 0 as 𝑡→∞. Thus, the equilibrium point 𝑝 of the system is asymptotically stable if |arg(𝜆)|>𝛼𝜋/2, for all eigenvalues 𝜆 of 𝐽, that is, if min_𝑖arg𝜆_𝑖>𝛼𝜋2. (2.16)

3. Fractional Lorenz-Like System

In [42], Chongxin et al. proposed novel Lorenz-like chaotic system ,̇𝑥̇=𝑎𝑦−𝑥𝑦̇=𝑏𝑥−𝑙𝑥𝑧,𝑧=−𝑐𝑧+ℎ𝑥²+𝑚𝑦², (3.1) where 𝑎=10,𝑏=40,𝑐=2.5,𝑚=ℎ=2,𝑙=1 and initial conditions ((2.2),(2.3), and (28)). In this paper, we study the corresponding fractional order system 𝐷^𝛼,𝐷𝑥=𝑎𝑦−𝑥_𝛼𝐷𝑦=𝑏𝑥−𝑙𝑥𝑧,_𝛼𝑧=−𝑐𝑧+ℎ𝑥²+𝑚𝑦², (3.2) where 𝛼∈(0,1). The equilibrium points of the system (3.1) and the eigenvalues of corresponding Jacobian matrix 𝐽𝑥,𝑦,𝑧=−𝑎𝑎0𝑏−𝑙𝑧0−𝑙𝑥2ℎ𝑥2𝑚𝑦−𝑐 (3.3) are given in Table 1. An equilibrium point 𝑝 of the system (3.1) is called as saddle point if the Jacobian matrix at 𝑝 has at least one eigenvalue with negative real part (stable) and one eigenvalue with nonnegative real part (unstable). A saddle point is said to have index one (/two) if there is exactly one (/two) unstable eigenvalue/s. It is established in the literature [47–51] that scrolls are generated only around the saddle points of index two. Saddle points of index one are responsible only for connecting scrolls.

Table 1 

Equilibrium points and corresponding eigenvalues.

It is clear from Table 1 that the equilibrium points 𝐸₁ and 𝐸₂ are saddle points of index two; hence, there exists a two-scroll attractor [47] in the system (3.2).

The system (3.2) shows regular behavior if it satisfies (2.16), that is, the system is stable if 2𝛼<𝜋min_𝑖arg𝜆_𝑖≈0.96345. (3.4) Thus, the system does not show chaotic behavior for 𝛼<0.96345. This result is supported by numerical experiments. Figure 1 shows phase portrait in 𝑥𝑦-plane for 𝛼=0.96. It is observed that the system shows chaotic behavior for 𝛼≥0.97. For 𝛼=0.97  𝑥𝑧-phase portrait is shown in Figure 2. Figures 3 and 4 show 𝑥𝑦- and 𝑦𝑧-phase portraits, respectively, for 𝛼=0.98. The phase portraits in 𝑥𝑦- and 𝑥𝑧-plane are drawn for 𝛼=0.99 in Figures 5 and 6 respectively. Thus, the minimum effective dimension of the system is 0.97×3=2.91.

Figure 1 

𝑥𝑦-phase portrait for 𝛼=0.96.

Figure 2 

𝑥𝑧-phase portrait for 𝛼=0.97.

Figure 3 

𝑥𝑦-phase portrait for 𝛼=0.98.

Figure 4 

𝑦𝑧-phase portrait for 𝛼=0.98.

Figure 5 

𝑥𝑦-phase portrait for 𝛼=0.99.

Figure 6 

𝑥𝑧-phase portrait for 𝛼=0.99.

4. Control of Chaos

In this section, we control the chaos in system (3.2). Consider 𝐷^𝛼,𝐷𝑥=10𝑦−𝑥_𝛼𝐷𝑦=40𝑥−𝑥𝑧,_𝛼𝑧=−2.5𝑧+2𝑥²+2𝑦²+𝑢, (4.1) where 𝑢 is the linear feedback control term. We set 𝑢=𝑘𝑥, where 𝑘 is a parameter to be determined so that the system (4.1) is stable. Equilibrium points of system (4.1) are 𝑂=(0,0,0), 𝑃₁=(−0.125(𝑘+√1600+𝑘²),−0.125(𝑘+√1600+𝑘²),40) and 𝑃₂=(−0.125(𝑘−√1600+𝑘²),−0.125(𝑘−√1600+𝑘²),40). The points 𝑃₁ and 𝑃₂ decide. The stability of the system. The Jacobian matrix is given by 𝐽₁𝑥,𝑦,𝑧=−1010040−𝑧0−𝑥𝑘+4𝑥4y−2.5. (4.2) One eigenvalue 𝑒₀ of the matrix 𝐽₁ at point 𝑃₁ is −4.1667−0.419974218.75+0.375𝑘²+0.375𝑘√1600+𝑘²/−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²+√−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²²+4218.75+0.375𝑘²+0.375𝑘√1600+𝑘^231/3−(0.132283±0.229122𝑖)−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²+√−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²²+4218.75+0.375𝑘²+0.375𝑘√1600+𝑘^231/3. (4.3) The eigenvalue 𝑒₀ is having negative real part for all 𝑘≥0 and hence stable for all 0≤𝛼≤1. Other two complex-conjugate eigenvalues 𝜆_+,− are given by −4.1667−(0.209987±0.363708𝑖)218.75+0.375𝑘²+0.375𝑘√1600+𝑘²/−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²+√−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²²+4218.75+0.375𝑘²+0.375𝑘√1600+𝑘^231/3−(0.132283±0.229122𝑖)−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²+√−43843.8−19.6875𝑘²−19.6875𝑘√1600+𝑘²²+4218.75+0.375𝑘²+0.375𝑘√1600+𝑘^231/3. (4.4) The stability of eigenvalues 𝜆_+,− depends on 𝑘. We have plotted the curves |arg(𝜆)| and 𝛼𝜋/2 for 𝛼=0.97,0.98,0.99,1 in Figure 7. The intersection points of the curve |arg(𝜆)| with the curves 0.97𝜋/2, 0.98𝜋/2, 0.99𝜋/2 and 𝜋/2 are 𝑘≈3.5, 𝑘≈8.5, 𝑘≈14 and 𝑘≈21, respectively. Following stability condition (2.16), it is clear that the chaos in the system can be controlled if we take the value of 𝑘 greater than the corresponding intersection point. Figure 8 shows controlled time series for 𝛼=1 and 𝑘=22. Similarly, Figures 9, 10, and 11 show controlled time series for 𝛼=0.99,0.98,0.97, and 𝑘=20,15,7, respectively.

Figure 7 

Estimate for the control 𝑘.

Figure 8 

Controlled signal 𝛼=1(𝑘=22).

Figure 9 

Controlled signal 𝛼=0.99(𝑘=20).

Figure 10 

Controlled signal 𝛼=0.98(𝑘=15).

Figure 11 

Controlled signal 𝛼=0.97(𝑘=7).

5. Synchronization

Present section deals with synchronization of proposed fractional-order system. Consider the master system 𝐷^𝛼𝑥₁=𝑎𝑦₁−𝑥₁,𝐷^𝛼𝑦₁=𝑏𝑥₁−𝑙𝑥₁𝑧₁,𝐷^𝛼𝑧₁=−𝑐𝑧₁+ℎ𝑥²₁+𝑚𝑦²₁, (5.1) and the slave system 𝐷^𝛼𝑥₂=𝑎𝑦₂−𝑥₂+𝑢₁,𝐷^𝛼𝑦₂=𝑏𝑥₂−𝑙𝑥₂𝑧₂+𝑢₂,𝐷^𝛼𝑧₂=−𝑐𝑧₂+ℎ𝑥²₂+𝑚𝑦²₂+𝑢₃. (5.2) The unknown terms 𝑢₁,𝑢₂,𝑢₃ in (5.2) are control functions to be determined. Define the error functions as 𝑒₁=𝑥₁−𝑥₂,𝑒₂=𝑦₁−𝑦₂,𝑒₃=𝑧₁−𝑧₂. (5.3) Equation (5.3) together with (5.1) and (5.2) yields the error system 𝐷^𝛼𝑒₁=𝑎𝑒₂−𝑒₁−𝑢₁,𝐷^𝛼𝑒₂=𝑏𝑒₁+𝑙𝑥₂𝑧₂−𝑙𝑥₁𝑧₁−𝑢₂,𝐷^𝛼𝑒₃=−𝑐𝑒₃+ℎ𝑥²₁−𝑥²₂+𝑚𝑦²₁−𝑦²₂−𝑢₃. (5.4) The control terms 𝑢_𝑖 are chosen so that the system (5.4) becomes stable. There is not a unique choice for such functions. We choose 𝑢₁=𝑎𝑒₂,𝑢₂=𝑙𝑥₂𝑧₂−𝑙𝑥₁𝑧₁+𝑒₂,𝑢₃=ℎ𝑥²₁−𝑥²₂+𝑚𝑦²₁−𝑦²₂. (5.5) With the choice of 𝑢_𝑖 given by (5.5), the error system (5.4) becomes 𝐷^𝛼𝑒₁=−𝑎𝑒₁=−10𝑒₁,𝐷^𝛼𝑒₂=𝑏𝑒₁−𝑒₂=40𝑒₁−𝑒₂,𝐷^𝛼𝑒₃=−𝑐𝑒₃=−2.5𝑒₃. (5.6) The eigenvalues of the coefficient matrix of linear system (5.6) are −10,−1, and −2.5. Hence, the stability condition (2.16) is satisfied for 0≤𝛼≤1 and the errors 𝑒_𝑖(𝑡) tend to zero as 𝑡→∞. Thus, we achieve the required synchronization. The simulation results in case 𝛼=0.99 are summarized in Figures 12–15. Synchronization is shown in Figure 12 (signals 𝑥₁,𝑥₂), Figure 13 (signals 𝑦₁,𝑦₂), and Figure 14 (signals 𝑧₁,𝑧₂). Note that the master systems are shown by solid lines whereas slave systems are shown by dashed lines. The errors 𝑒₁(𝑡) (solid line), 𝑒₂(𝑡) (dashed line) and 𝑒₃(𝑡) (dot-dashed line) in the synchronization are shown in Figure 15. We have studied other cases of 𝛼 namely, 0.97, and 0.98 but the results are omitted.

Figure 12 

Synchronized signals 𝑥1;𝑥2(𝛼=0.99).

Figure 13 

Synchronized signals 𝑦1;𝑦2(𝛼=0.99).

Figure 14 

Synchronized signals 𝑧1;𝑧2(𝛼=0.99).

Figure 15 

Synchronization errors.

6. Conclusion

In the present work, we demonstrate the fractional order Lorenz-like system. We have observed that the system is chaotic for the fractional order 𝛼≥0.97, that is, the minimum effective dimension of the system is 2.91. We have used simple linear feedback controller 𝑢=𝑘𝑥, (𝑘>0) and given sufficient condition on 𝑘 to control the chaos in the proposed system. Further, we have synchronized the system using feedback control.