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. Preliminaries2.1. 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
(2.1)Iμf(t)=1Γ(μ)∫0t(t-τ)μ-1f(τ)dτ,t>0.
An alternative definition was introduced by Caputo. Caputo's derivative is defined as
(2.2)Dμf(t)=dmdtmf(t),μ=m=Im-μdmf(t)dtm,m-1<μ<m,
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,
(2.3)IμDμf(t)=f(t)-∑k=0m-1dkfdtk(0)tkk!,Iμtν=Γ(ν+1)Γ(μ+ν+1)tμ+ν.
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)
(2.4)Dαy(t)=f(t,y(t)),0≤t≤T,y(k)(0)=y0(k),k=0,1,…,m-1.
The IVP (2.4) is equivalent to the Volterra integral equation
(2.5)y(t)=∑k=0m-1y0(k)tkk!+1Γ(α)∫0t(t-τ)α-1f(τ,y(τ))dτ.
Consider the uniform grid {tn=nh/n=0,1,…,N} for some integer N and h:=T/N. Let yh(tn) be approximation to y(tn). Assume that we have already calculated approximations yh(tj),j=1,2,…,n, and we want to obtain yh(tn+1) by means of the equation
(2.6)yh(tn+1)=∑k=0m-1tn+1kk!y0(k)+hαΓ(α+2)f(tn+1,yhP(tn+1))+hαΓ(α+2)∑j=0naj,n+1f(tj,yn(tj)),
where
(2.7)aj,n+1={nα+1-(n-α)(n+1)αifj=0,(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1if1≤j≤n,1ifj=n+1.
The preliminary approximation yhP(tn+1) is called predictor and is given by
(2.8)yhP(tn+1)=∑k=0m-1tn+1kk!y0(k)+1Γ(α)∑j=0nbj,n+1f(tj,yn(tj)),
where
(2.9)bj,n+1=hαα((n+1-j)α-(n-j)α).
Error in this method is
(2.10)maxj=0,1,…,N|y(tj)-yh(tj)|=O(hp),
where p=min(2,1+α).
2.3. Asymptotic Stability of the Fractional-Ordered System
Consider the following fractional-ordered dynamical system:
(2.11)Dαxi=fi(x1,x2,x3),1≤i≤3.
Let p≡(x1*,x2*,x3*) be an equilibrium point of the system (2.11) that is, fi(p)=0,1≤i≤3 and ξi=xi-xi* a small disturbance from a fixed point. Then
(2.12)Dαξi=Dαxi=fi(x1,x2,x3)=fi(ξ1+x1*,ξ2+x2*,ξ3+x3*)=fi(x1*,x2*,x3*)+ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3=+higher-orderedterms≈ξ1∂fi(p)∂x1+ξ2∂fi(p)∂x2+ξ3∂fi(p)∂x3.
System (2.12) can be written as
(2.13)Dαξ=Jξ,
where ξ=(ξ1,ξ2,ξ3)t and
(2.14)J=(∂1f1(p)∂2f1(p)∂3f1(p)∂1f2(p)∂2f2(p)∂3f2(p)∂1f3(p)∂2f3(p)∂3f3(p)).
Consider the linear autonomous system
(2.15)Dαξ=Jξ,ξ(0)=ξ0,
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 ξi(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
(2.16)mini|arg(λi)|>απ2.
3. Fractional Lorenz-Like System
In [42], Chongxin et al. proposed novel Lorenz-like chaotic system
(3.1)x˙=a(y-x),y˙=bx-lxz,z˙=-cz+hx2+my2,
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
(3.2)Dαx=a(y-x),Dαy=bx-lxz,Dαz=-cz+hx2+my2,
where α∈(0,1). The equilibrium points of the system (3.1) and the eigenvalues of corresponding Jacobian matrix
(3.3)J(x,y,z)=(-aa0b-lz0-lx2hx2my-c)
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.
Equilibrium points and corresponding eigenvalues.
Equilibrium point
Eigenvalues
O(0,0,0)
-25.6155,15.6155,-2.5
E1(5,5,40)
-13.8776,0.688787±11.9851i
E2(-5,-5,40)
-13.8776,0.688787±11.9851i
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
(3.4)α<2πmini|arg(λi)|≈0.96345.
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=0.97xz-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.
xy-phase portrait for α=0.96.
xz-phase portrait for α=0.97.
xy-phase portrait for α=0.98.
yz-phase portrait for α=0.98.
xy-phase portrait for α=0.99.
xz-phase portrait for α=0.99.
4. Control of Chaos
In this section, we control the chaos in system (3.2). Consider
(4.1)Dαx=10(y-x),Dαy=40x-xz,Dαz=-2.5z+2x2+2y2+u,
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
(4.2)J1(x,y,z)=(-1010040-z0-xk+4x4y-2.5).
One eigenvalue e0 of the matrix J1 at point P1 is
(4.3)-4.1667-(0.419974(218.75+0.375k2+0.375k1600+k2))/(-43843.8-19.6875k2-19.6875k1600+k2+((-43843.8-19.6875k2-19.6875k1600+k2)2+4(218.75+0.375k2+0.375k1600+k2)3))1/3-(0.132283±0.229122i)(-43843.8-19.6875k2-19.6875k1600+k2∓+(0.132283±0.229122i)+((-43843.8-19.6875k2-19.6875k1600+k2)2∓+(0.132283±0.229122i)+++4(218.75+0.375k2+0.375k1600+k2)3))1/3.
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
(4.4)-4.1667-((0.209987±0.363708i)(218.75+0.375k2+0.375k1600+k2))/(-43843.8-19.6875k2-19.6875k1600+k2+((-43843.8-19.6875k2-19.6875k1600+k2)2+4(218.75+0.375k2+0.375k1600+k2)3))1/3-(0.132283±0.229122i)(-43843.8-19.6875k2-19.6875k1600+k2∓+(0.132283±0.229122i)+((-43843.8-19.6875k2-19.6875k1600+k2)2∓+(0.132283±0.229122i)+++4(218.75+0.375k2+0.375k1600+k2)3))1/3.
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.
Estimate for the control k.
Controlled signal α=1(k=22).
Controlled signal α=0.99(k=20).
Controlled signal α=0.98(k=15).
Controlled signal α=0.97(k=7).
5. Synchronization
Present section deals with synchronization of proposed fractional-order system. Consider the master system
(5.1)Dαx1=a(y1-x1),Dαy1=bx1-lx1z1,Dαz1=-cz1+hx12+my12,
and the slave system
(5.2)Dαx2=a(y2-x2)+u1,Dαy2=bx2-lx2z2+u2,Dαz2=-cz2+hx22+my22+u3.
The unknown terms u1,u2,u3 in (5.2) are control functions to be determined. Define the error functions as
(5.3)e1=x1-x2,e2=y1-y2,e3=z1-z2.
Equation (5.3) together with (5.1) and (5.2) yields the error system
(5.4)Dαe1=a(e2-e1)-u1,Dαe2=be1+lx2z2-lx1z1-u2,Dαe3=-ce3+h(x12-x22)+m(y12-y22)-u3.
The control terms ui are chosen so that the system (5.4) becomes stable. There is not a unique choice for such functions. We choose
(5.5)u1=ae2,u2=lx2z2-lx1z1+e2,u3=h(x12-x22)+m(y12-y22).
With the choice of ui given by (5.5), the error system (5.4) becomes
(5.6)Dαe1=-ae1=-10e1,Dαe2=be1-e2=40e1-e2,Dαe3=-ce3=-2.5e3.
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 ei(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.
Synchronized signals x1;x2(α=0.99).
Synchronized signals y1;y2(α=0.99).
Synchronized signals z1;z2(α=0.99).
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.
AlligoodK. T.SauerT. D.YorkeJ. A.OttE.GrebogiC.YorkeJ. A.Controlling chaosLüJ.ZhangS.Controlling Chen's chaotic attractor using backstepping design based on parameters identificationFuhC. C.TungP. C.Controlling chaos using differential geometric methodSanchezE. N.PerezJ. P.MartinezM.ChenG.Chaos stabilization: an inverse optimal control approachLuJ. A.XieJ.LuJ. H.ChenS. H.Control chaos in transition system using sampled-data feedbackCaoY. J.A nonlinear adaptive approach to controlling chaotic oscillatorsChenG.DongX.On feedback control of chaotic continuous-time systemsZongminQ.JiaxingC.A novel linear feedback control approach of Lorenz chaotic systemProceedings of the International Conference on Computational Intelligence for Modelling, Control and Automation (CIMCA '06), Jointly with International Conference on Intelligent Agents Web Technologies and International Commerce (IAWTIC '06)December 2006672-s2.0-3884915946810.1109/CIMCA.2006.23PecoraL. M.CarrollT. L.Synchronization in chaotic systemsPecoraL. M.CarrollT. L.Driving systems with chaotic signalsKocarevL.ParlitzU.General approach for chaotic synchronization with applications to communicationCarrollT. L.HeagyJ. F.PecoraL. M.Transforming signals with chaotic synchronizationHilferR.HeR.VaidyaP. G.Implementation of chaotic cryptography with chaotic synchronizationPodlubnyI.SamkoS. G.KilbasA. A.MarichevO. I.KilbasA. A.SrivastavaH. M.TrujilloJ. J.SabatierJ.PoullainS.LatteuxP.ThomasJ. L.OustaloupA.Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test benchCaputoM.MainardiF.A new dissipation model based on memory mechanismMainardiF.LuchkoY.PagniniG.The fundamental solution of the spacetime fractional diffusion equationAgrawalO. P.Solution for a fractional diffusion-wave equation defined in a bounded domainSunH.ChenW.ChenY.Variable-order fractional differential operators in anomalous diffusion modelingJesusI. S.Tenreiro MacHadoJ. A.Fractional control of heat diffusion systemsJesusI. S.MachadoJ. A. T.BarbosaR. S.Control of a heat diffusion system through a fractional order nonlinear algorithmAnastasioT. J.The fractional-order dynamics of brainstem vestibulo-oculomotor neuronsOrtigueiraM. D.MachadoJ. A. T.Fractional calculus applications in signals and systemsMaginR. L.MatignonD.Stability results for fractional differential equations with applications to control processing2Proceedings of the Computational Engineering in Systems and Application MulticonferenceJuly 1996Lille, France963968Kiani-BA.FallahiK.ParizN.LeungH.A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filterGrigorenkoI.GrigorenkoE.Chaotic dynamics of the fractional Lorenz systemLiC.PengG.Chaos in Chen's system with a fractional orderLuJ. G.Chaotic dynamics of the fractional-order Lü system and its synchronizationLiC. G.ChenG.Chaos and hyperchaos in the fractional order Rossler equationsDaftardar-GejjiV.BhalekarS.Chaos in fractional ordered Liu systemWuX.LiJ.ChenG.Chaos in the fractional order unified system and its synchronizationBhalekarS.Daftardar-GejjiV.Fractional ordered Liu system with time-delayDengW. H.LiC. P.Chaos synchronization of the fractional Lü systemZhouT.LiC.Synchronization in fractional-order differential systemsWangJ.XiongX.ZhangY.Extending synchronization scheme to chaotic fractional-order Chen systemsBhalekarS.Daftardar-GejjiV.Synchronization of different fractional order chaotic systems using active controlChongxinL.LingL.TaoL.PengL.A new butterfly-shaped attractor of Lorenz-like systemDiethelmK.FordN. J.FreedA. D.A predictor-corrector approach for the numerical solution of fractional differential equationsDiethelmK.An algorithm for the numerical solution of differential equations of fractional orderDiethelmK.FordN. J.Analysis of fractional differential equationsTavazoeiM. S.HaeriM.Regular oscillations or chaos in a fractional order system with any effective dimensionTavazoeiM. S.HaeriM.A necessary condition for double scroll attractor existence in fractional-order systemsChuaL. O.KomuroM.MatsumotoT.The double-scroll familySilvaC. P.Shil'nikov's theorem-a tutorialCafagnaD.GrassiG.New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ringLüJ.ChenG.YuX.LeungH.Design and analysis of multiscroll chaotic attractors from saturated function series