Startseite Combination synchronization of fractional order n-chaotic systems using active backstepping design
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Combination synchronization of fractional order n-chaotic systems using active backstepping design

  • Vijay K. Yadav und S. Das EMAIL logo
Veröffentlicht/Copyright: 9. April 2019
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 8 Heft 1

Abstract

In this article, a scheme using active backstepping design method is proposed to achieve combination synchronization of n number of fractional order chaotic systems. In the proposed method the controllers are designed with the help of a new lemma and Lyapunov function in a systematic way. Synchronization among three/four fractional order systems have been shown as examples of synchronization of n-chaotic systems. Numerical simulation and graphical results clearly exhibit that the method of this new procedure is easy to implement and reliable for synchronization of fractional order chaotic systems.

Keywords: Backstepping method; Lyapunov stability theory; Synchronization; Fractional order chaotic systems

1 Introduction

The theory of fractional calculus deals with derivatives and integrals of arbitrary order and has applications in various scientific field and engineering including viscoelasticity, fluid mechanics, material science, colored noise, dielectric polarization, electromagnetic wave, bioengineering, biological model, electromechanical system, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The fractional differential equations are the generalization of classical differential equations. The advantage of fractional order system is it allows greater flexibility in the model. The fractional order differential operator is non local but integer order differential operator is a local operator in the sense that fractional order differential operator takes into account the fact that the future state not only depends upon the present state but also upon all of the histories of its previous states. For this realistic property, fractional calculus which was in earlier stage considered as mathematical curiosity now becomes the object of extensive development of fractional order partial differential equations for its the purpose of engineering applications.

Chaos synchronization is an interesting phenomenon of nonlinear dynamical systems and it may occur when two or more chaotic systems are coupled or one chaotic system drives the other. The synchronization of chaotic systems was first given by Pecora and Carroll [12] in 1960, and after which it has been intensively studied due to its potential applications in various fields viz., ecological system, physical system, chemical system, secure communications etc [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In recent years various types of synchronization have been investigated such as complete synchronization, anti-synchronization, lag synchronization, adaptive synchronization, projective synchronization, function projective synchronization etc [23, 24, 25, 26] and also different schemes have been successfully applied to chaos synchronization viz., linear and nonlinear feedback control method, active control method, adaptive control method, sliding mode control method, backstepping method etc [27, 28, 29, 30, 31, 32, 33, 34].

The method here to use is backstepping design, which has been employed by many researchers for controlling and synchronizing chaotic systems as well as hyperchaotic systems. It consists in a recursive procedure that links the choice of a Lyapunov function with the design of a controller. Backstepping design is recognised as powerful design method for chaos synchronization. The design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems [35, 36, 37]. To stabilize and track chaotic systems, the method had been successfully used by Mascolo and Grassi [38] in the year 1999. In 2006, the backstepping control was used by Bin et al. [39] to synchronize two coupled chaotic neurons in external electrical stimulation. Wang and Ge [40] proposed the Adaptive synchronization of uncertain chaotic systems via backstepping design. Backstepping design was successfully applied by Tan et al. [41] during synchronization of the chaotic systems and also by Yu and Zhand [42] to control the uncertain behavior of chaotic systems. Recently, Park [43], Wu et al. [44] have shown that the backstepping method is very simple, reliable and powerful for controlling the chaotic behavior and synchronization of chaotic systems. But to the best of authors’ knowledge the synchronization of fractional order systems using backstepping control has not yet been studied by any researcher. The theme of the present study is to investigate the synchronization procedure for a number of fractional order chaotic systems using this simple and reliable backstepping method. In our earlier article [45], active control method and backstepping approach are used to synchronize the fractional order chaotic systems. The fractional order Chen and Qi systems are taken to synchronize using both the methods. It was shown that the backstepping method takes less time to synchronize as the systems pair approaches from standard order to fractional order. This has motivated the authors to find the required time of synchronization among three/four fractional order chaotic systems.

Initially the prediction of a system had been confined through finding the analytical solution of the formal modelling of the systems via mathematical modelling with a set of parameters and initial/boundary conditions. But after the advent of modern computers and related software packages, the simulation has become a useful technique of modelling of many streams of science and engineering as well as computational sociology. Nowadays it is used in technology to optimize the performance, safety engineering, also during modelling of natural and human systems. Simulation is described as the limitation of operation of a real world system over time. Thus before performing simulation, it requires to develop a model which will represent key features of the selected physical or abstract systems. Thus simulation basically represents the operation of the system over time. During synchronization of identical or non-identical chaotic systems the simulation is used to find requirement of minimum time after which the states of slave system behave similar to the master system.

The synchronization of three chaotic dynamical systems in integer order are first studied by the Runzi et al. [46] in 2011. In 2013 Zhang et al. [47] studied the combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters. In 2016 the combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters is studied by Wang et al. [48]. It is seen from literature survey that the synchronization between three and more chaotic systems are few in numbers. Runzi et al. [46] had stated the cause of investigation between two drive systems and one response system through a physical application in secure communication as transmitted signals can be split into several parts, each part loaded in different drive systems which shows that transmitted signals have stronger anti-attack ability and anti-translated capability than that transmitted by the usual transmission model. This has motivated the authors to study the generalization of synchronization between chaotic systems, when the systems have memory effects. The main contribution of the present scientific contribution is introduction of a new type of synchronization scheme known as combination synchronization which used to synchronize a number of fractional order chaotic systems. The backstepping method is applied during synchronization of fractional order chaotic systems using Lyapunov stability theory and a new lemma for Caputo derivative. The combination synchronization of three and four fractional order chaotic systems are found through numerical simulation which is presented graphically to show the effectiveness and feasibility of the proposed scheme and method. The salient feature of the present study is the pictorial presentation of requirement of less time during synchronization of four systems compared to three systems.

The article is organized as follows: In section 2, the definition and Lemma for Caputo derivative in fractional order case are introduced. Section 3 includes the combination synchronization scheme of fractional order n-chaotic systems are presented. In section 4, the fractional order chaotic systems are introduced. Sections 5 and 6 contain the combination synchronization of three and four chaotic systems using backstepping design respectively. Section 5.1 and 6.1 provides numerical simulation results and followed by a conclusion of the overall research work given in Section 7.

2 Basic definitions of Fractional order derivative and Lemma

The definitions of fractional order derivative given by B Riemann and J. Liouville and also by M. Caputo are as follows.

2.1

Definition 1

The Riemann–Liouville (R-L) fractional derivative operator of order q > 0 of a function f(t) is defined by [49]

Dxqft=dndtnJxn−qft,n−1<q≤n,n∈N,

where the fractional integral operator of order q > 0 of a function f(t) is given by

Jtqft=1Γq∫0tt−ξq−1fξdξ,

with Jt0ft=ft..

2.2

Definition 2

The Caputo order fractional derivative of a function f(t) is [50, 51]

Dtqft=1Γn−q∫0tt−ξn−q−1fnξdξ,n−1<q<n,n∈N,
Dtqft=dnftdxn,q=n,

where Dtqf(t) satisfies the following basic property

JtqDtqft=ft−∑k=0n−1fk0+tkk!,t≥0,n−1<q<n,n∈Nandf∈Cμn,μ≥−1.

Between these two, the Caputo derivative is commonly used by the researchers. The main difference of considering the Caputo derivative is that its derivative of a constant is zero whereas in the case of finite value of lower terminal ′a′ the R-L derivative of a constant is aDtqc=ct−qΓ(1−q). However for the case a → −∞ this becomes zero. Thus for the steady state dynamical process fractional order derivative for both Caputo and R-L provide the same results.

Another reason of choosing Caputo derivative is that it holds for homogeneous and non-homogeneous initial conditions whereas R-L derivative require homogeneous initial condition during the solution of initial value problems.

2.3

Lemma 1

[52] Letx(t) ∈ R be a continuous and derivable function. Then for any time instant t ≥ t0,

12t0cDtqx2(t)≤x(t)t0cDtqx(t),∀q∈(0,1).

3 The scheme of combination synchronization of fractional order n-chaotic systems

In this scheme, (n-1) drive systems and one response system are assumed to be in fractional order system.

The drive systems are considered as

Dtqx1=f1(x1),(1)
Dtqx2=f2(x2),(2)
………………..………………..Dtqxn−1=fn−1(xn−1),(3)

and the response system is taken as

Dtqxn=fn(xn)+U(x1,x2,......,xn),(4)

where x1=(x11,x21,........,xn1),x2=(x12,x22,........,xn2),………………,xn−1=(x1n−1,x2n−1,........,xnn−1) and xn=(x1n,x2n,........,xnn) with x1, x2,   .........., xn − 1, xn ∈ Rn are the state vectors of the n-chaotic systems. f1, f2, ......., fn − 1, fn : Rn → Rn are the n-continuous vector functions and U(x1, x2, ......, xn) : Rn×Rn×.......×Rn→Rnntimes is a controller which will be designed latter.

Definition

The fractional order (n-1) drive systems and one response system follow combination synchronization among n-chaotic systems if there exists n constant matrixes called scaling matrixes A1, A2, ....., An ∈ Rn with An ≠ 0 such that

limn→+∞A1x1+A2x2+.......−Anxn=0, where ∥ . ∥ represent the matrix norm.

It is noted that if A1 ≠ 0, A2 = A3 = ......... = An − 1 = 0, An = I then this problem is reduced to the projective synchronization, where I is an n × n identity matrix. If the scaling matrix A1 is considered as a function, then synchronization problem is reduced into function projective synchronization problem. Again if A1 = A2 = ......... = An − 1 = 0, then the problem becomes a chaos control problem.

4 Systems’ descriptions

4.1 Fractional order Newton-Leipnik system

The fractional order Newton-Leipnik system [53] was first studied in the year 2008, which is given by

dqx1dtq=−a1x1+x2+10x2x3
dqx2dtq=−x1−0.4x2+5x1x3(5)
dqx3dtq=a2x3−5x1x2,0<q<1,

where a1 and a2 are the variable parameters’ and a2 ∈ (0, 0.8). The system is ill-behaved when a2 takes the values outside of this interval. If a2 becomes close to zero, the system shows uninteresting dynamic and if a2 ≥ 0.8, the given system becomes explosive i.e., the solution of this system diverges to infinity for any initial condition other than the critical points.

For the parameters’ values a1 = 0.4, a2 = 0.175 and the initial condition (0.19, 0, −0.18), the Newton-Leipnik system shows chaotic behaviour at q = 0.95 which is depicted through Fig. 1(a).

Fig. 1 Phase portraits of fractional order (a) Newton-Leipnik system, (b) Liu system, (c) Lotka-Voltra system, (d) Chen system for the order of derivative q = 0.95.
Fig. 1

Phase portraits of fractional order (a) Newton-Leipnik system, (b) Liu system, (c) Lotka-Voltra system, (d) Chen system for the order of derivative q = 0.95.

4.2 Fractional order Liu system

The Liu system [54] was studied in the year 2009, which was later extended to fractional order Liu system by Gejji and Bhalekar [55] in 2010 as

dqy1dtq=−b1y1−b4y22
dqy2dtq=b2y2−b5y1y3(6)
dqy3dtq=−b3y3+b6y1y2.

The phase portrait of the system is described through Fig. 1(b), which shows that the system exhibits chaos at the lowest fractional order q = 0.92 for the values of parameters b1 = 1, b2 = 2.5, b3 = 5, b4 = 1, b5 = 4, b6 = 4 and initial condition (0.2, 0, 0.5). The chaos control of Liu system is given in [56]. The trajectories of the system are controlled at its all equilibrium points for order of the derivative 0 < q ≤ 1.

4.3 Fractional order Lotka-Voltra system

The fractional order Lotka-Voltra system [57] is given as

dqz1dtq=c1z1−c2z1z2+c5z12−c6z3z12
dqz2dtq=−c3z2+c4z1z2(7)
dqz3dtq=−c7z3+c6z3z12.

The chaotic attractor of the system is described through Fig. 1(c) at q = 0.95 for the values of the parameters c1 = c2 = c3 = c4 = 1, c5 = 2, c6 = 2.7, c7 = 3 and initial condition (1, 1.4, 1).

4.4 Fractional order Chen system

The fractional order Chen system [58] is considered as

dqw1dtq=d1(w2−w1)
dqw2dtq=(d3−d1)w1−w1w3+d3w2(8)
dqw3dtq=w1w2−d2w3.

Fig. 1(d) shows the chaotic attractors of the system at the fractional order q = 0.95 for the parameters’ values d1 = 35, d2 = 3, d3 = 28 and the initial condition (1, 1.4, 1). The chaos control of Chen system is given in [58]. The system is controlled at order q = 0.9, q = 0.8 at its equilibrium points. The trajectories of the system is stable at all the equilibrium points of the system and it will also remain be controlled for order 0 < q ≤ 1.

5 Synchronization of fractional order Newton-Leipnik, Lotka-Voltra and Liu systems

For the study of synchronization among three fractional order chaotic systems, two systems Newton-Leipnik (5) and Lotka-Voltra (7) are considered as drive system-I and drive system-II and third system Liu system is considered as response system. The response system with the control functions u1, u2, u3 is defined as

dqy1dtq=−b1y1−b4y22+u1
dqy2dtq=b2y2−b5y1y3+u2(9)
dqy3dtq=−b3y3+b6y1y2+u3.

Defining the error functions as ei = yi − zi − xi, i = 1, 2, 3, we obtain the error system as

dqe1dtq=−b1e1−b4e2+φ1+u1
dqe2dtq=b2e2−b5e1e3−b5e3(z1+x1)−b5e1(z3+x3)+φ2+u2(10)
dqe3dtq=−b3e3+b6e1e2+b6e2(z1+x1)+b6e1(z2+x2)+φ3+u3,

where φ1 = −b1(z1 + x1) − b4(z2 + x2) − c1z1 + c2z1z2 − c5z12+c6z3z12+a1x1−x2−10x2x3

φ2=−b5(z1+x1)(z3+x3)+b2(z2+x2)+c3z2−c4z1z2+x1+0.4x2−5x1x3φ3=b6(z1+x1)(z2+x2)−b3(z3+x3)+c7z3−c6z3z12−a2x3+5x1x2.

Now the control functions would be designed using backstepping approach for combination synchronization of three fractional order chaotic systems.

Theorem 1

If the control functions are chosen as

u1=−φ1,u2=b5v1(z3+x3)−b2v2+b4v1−v2−φ2u3=−b6v1(z2+x2)+(b5−b6)v2(z1+x1)+(b5−b6)v1v2−φ3,

where v1 = e1, v2 = e2, v3 = e3, then the drive system I & II will be combination synchronized with response system.

Proof

To achieve the results, let us use the active backstepping procedure through following three steps.

Step-I: Defining v1 = e1, we get

dqv1dtq=dqe1dtq=−b1v1−b4e2+φ1+u1,(11)

where e2 = α1(v1) is regarded as a virtual controller. For designing α1(v1) to stabilizev1 - subsystem, choosing the Lyapunov function V1 as

V1=12v12.

The-th order fractional derivative of V1 w. r. to t is

dqV1dtq=12dqv12dtq≤v1dqv1dtq(using Lemma-1)i.e.≤v1[−b1v1−b4α1(v1)+φ1+u1].

If we take α1(v1) = 0 and u1 = φ1, then dqV1dtq≤−b1v12 < 0, which implies that subsystem (11) is asymptotically stable. Since virtual control function α1 (v1) is an estimate function, defining the error variable v2 between e2 and α1 (v1) as

v2=e2−α2(v1),

we obtain the following (v1, v2) -subsystem as

dqv1dtq=−b1v1−b4v2dqv2dtq=b2v2−b5v1e3−b5e3(z1+x1)−b5v1(z3+x3)+φ2+u2,(12)

where e3 = α2 (v1, v2) is regarded as an virtual controller.

Step II: To stabilize (v1, v2) - subsystem (12), choose Lyapunov function as

V2=V1+12v22=12v12+12v22.

The order fractional derivative of V2 w. r. to t is

dqV2dtq=12dqv12dtq+12dqv22dtq

≤v1dqv1dtq+v2dqv2dtq, (from Lemma 1)

i.e. ≤ − b1v12 − b4v1v2 + v2[b2v2 − b5v1α2(v1, v2) − b5α2(v1, v2)(z1 + x1) − b5v1(z3 + x3) + φ2 + u2]

If α2(v1, v2) = 0 andu2 = b5v1(z3 + x3) − b2v2 + b4v1 − v2 − φ2, then dqV2dtq≤−b1v12−v22<0, which implies that (v1, v2)-subsystem (12) is asymptotically stable.

Again defining the error variable as

v3=e3−α2(v1,v2),

the (v1, v2, v3) - subsystem becomes

dqv1dtq=−b1v1−b4v2dqv2dtq=−v2+b4v1−b5v1v3−b5v3(z1+x1)dqv3dtq=−b3v3+b6v1v2+b6v2(z1x1)+b6v1(z2+x2)+φ3+u3(13)

Step III: To stabilize the (v1, v2, v3) - subsystem (13), choosing the Lyapunov function V3 as

V3=V2+12v32=12v12+12v22+12v32.

The fractional derivative of V3 is

dqV3dtq=12dqv12dtq+12dqv22dtq+12dqv32dtq

≤v1dqv1dtq+v2dqv2dtq+v3dqv3dtq, (from Lemma 1)

i.e. ≤ − b1v12−v22 − b5v1v2v3 − b5v2v3 (z1 + x1) + v3[−b3v3 +b6v1v2 +b6v2 (z1 +x1) +b6v1 (z2 +x2) +φ3 + u3]

Taking u3 = −b6v1 (z2 +x2) +(b5 − b6) v2 (z1 +x1) +(b5 − b6) v1v2 − φ3, we obtain dqV3dtq≤−b1v12−v22−b3v32<0. Thus the system is asymptotically stable. Thus for v1 = e1, v2 = e2 − α1(v1) = e2 and v3 = e3 − α2 (v1, v2) = e3, the error systems ei → 0, i = 1, 2, 3, which helps to obtain combination synchronization among the three considered fractional order systems.

5.1 Numerical simulation and results

In the numerical simulation the parameters’ values of the fractional order Newton-Leipnik, Lotka-Voltra and Liu systems are taken as a1 = 0.4, a2 = 0.175, and b1 = 1, b2 = 2.5, b3 = 5, b4 = 1, b5 = 4, b6 = 4 respectively. The initial conditions of two drive systems and response system are taken as (0.19, 0, −0.18), and (0.2, 0, 0.5) respectively. Fig. 2 shows the synchronization among three fractional order chaotic systems are achieved through backstepping approach at q = 0.95. Figs. 2(a), 2(b) and 2(c) depict the time response of the state trajectories xi(t) + zi(t) and yi(t), where i = 1, 2, 3 represent the drive systems (5), (7) and response system (9) respectively. The error states are displayed through Fig. 2(d).

Fig. 2 Combination synchronization among three fractional order chaotic systems (5), (6) and (7) for fractional order q = 0.95: (a) between x1(t) +z1(t) and y1(t), (b) between x2(t) + z2(t) and y2(t), (c) between x3(t) + z3(t) and y3(t), (d) the evaluation of error function e1(t), e2(t) sand e3(t).
Fig. 2

Combination synchronization among three fractional order chaotic systems (5), (6) and (7) for fractional order q = 0.95: (a) between x1(t) +z1(t) and y1(t), (b) between x2(t) + z2(t) and y2(t), (c) between x3(t) + z3(t) and y3(t), (d) the evaluation of error function e1(t), e2(t) sand e3(t).

6 Synchronization of fractional order Newton-Leipnik, Liu, Lotka-Voltra systems and Chen system

In this section to synchronize four fractional order chaotic systems, we consider fractional order Newton-Leipnik system (5), fractional order Liu system (6) and fractional order Lotka-Voltra system (7) as the drive systems I, II and III respectively. The fractional order Chen system (8) is taken as response system with control function u1′,u2′,u3′ as

dqw1dtq=d1(w2−w1)+u1′dqw2dtq=(d3−d1)w1−w1w3+d3w2+u2′dqw3dtq=w1w2−d2w3+u3′.(14)

Defining error functions as ei = wi −zi − yi −xi, i = 1, 2, 3, we obtain the error system as

dqe1dtq=d1(e2−e1)+ψ1+u1′dqe2dtq=(d3−d1)e1−e1e3−e3(z1+y1+x1)−e1(z3+y3+x3)+d3e2+ψ2+u2′dqe3dtq=e1e2+e2(z1+y1+x1)+e1(z2+y2+x2)−d2e3+ψ3+u3′,(15)

where

ψ1=d1z2+(d1b4y2)y2+(d1−1)x2−(d1+c1)z1−(d1−b1)y1−(d1−a1)x1+c2z1z2−c5z12+c6z3z12−10x2x3
ψ2=(d3−d1)(z1+y1+x1)−(z1+y1+x1)(z3+y3+x3)+d3(z2+y2+x2)+c3z2−c4z1z2−b2y2+b5y1y3+x1+0.4x2−5x1x3ψ3=(z1y1x1)(z2+y2+x2)−d2(z3+y3+x3)+c7z3−c6z3z12+b3y3−b6y1y2−a2x3+5x1x2

Next the control functions u1′,u2′ and u3′ would be designed using backstepping approach for combination synchronization of four fractional order chaotic systems.

Theorem: 2

If the control functions are chosen as

u1′=−ψ1,u2′=−d3v2+v1(z3+y3+x3)−d2v1−v2−ψ2,u3′=−v1(z2+y2+x2)−ψ3

where v1 = e1, v2 = e2, v3 = e3, then the drive systems (5), (6) and (7) will be combination synchronized with response system (14).

Proof

For synchronization, backstepping procedure is used through following steps.

Step-I: Considering v1 = e1,

dqv1dtq=dqe1dtq=d1(e2−e1)+ψ1+u1′,(16)

where e2 = α1 (v1) is regarded as a virtual controller. To stabilize v1 -subsystem, let us define the Lyapunov function V1 as

V1=12v12,

whose fractional derivative is

dqV1dtq=12dqv12dtq≤v1dqv1dtqi.e.,≤v1[d1(α1(v1)−v1)+ψ1+u1′]

Taking α1(v1) = 0 and u1′ = − ψ1, we get dqV1dtq≤−d1v12 < 0, which implies that v1-subsystem (16) is asymptotically stable. For the virtual control function α1(v1), a variable v2 between e2 and α1(v1) is defined as

v2=e2−α1(v1).

Then (v1, v2)-subsystem is obtained as

dqv1dtq=d1(v2−v1)dqv2dtq=(d3−d1)v1−v1e3−e3(z1+y1+x1)−v1(z3+y3+x3)+d3v2+ψ2u2′(17)

Let us consider v3 = α2 (v1, v2) is a virtual controller.

Step II: In this step to stabilize (v1, v2)-subsystem (17), let us define the Lyapunov function V2 as

V2=V1+12v22=12v12+12v22.

Now

dqV3dtq=12dqv12dtq+12dqv22dtq≤v1dqv1dtq+v2dqv2dtqi.e.,≤d1v1v2−d1v12+v2[(d3−d1)v1−v1α2(v1,v2)−α2(v1,v2)(z1+y1+x1)−v1(z3+y3+x3)+d3v2+ψ2+u2′]

Taking α2 (v1, v2) = 0 and u2′ = −d3v2 + v1 (z3 +y3 +x3) − d2v1 − v2 − ψ2, we get dqV2dtq≤−d1v12−v22<0, which makes subsystem (17) asymptotically stable.

Considering v3 = e3 − α2 (v1, v2), the (v1, v2, v3) - subsystem is obtained as

dqv1dtq=d1(v2−v1)dqv2dtq=−v2−d1v1−v1v3−v3(z1+y1+x1)dqv3dtq=v1v2+v2(z1+y1x1)+v1(z2+y2+x2)−d2v3+ψ3+u3′.(18)

Step III: In order to stabilize (v1, v2, v3) - subsystem (18), choosing the Lyapunov function as

V3=V2+12v32=12v12+12v22+12v32,

we get

dqV3dtq=12dqv12dtq+12dqv22dtq+12dqv32dtq≤v1dqv1dtq+v2dqv2dtq+v3dqv3dtq,i.e.,≤v1[d1(v2−v1)]+v2[−v2−d1v1−v1v3−v3(z1+y1+x1)]+v3[v1v2+v2(z1+y1+x1)+v1(z2+y2+x2)−d2v3+ψ3+u3′]=−d1v12−v22+v3[v1(z2+y2+x2)−d2v3+ψ3+u3′]

If u3′=−v1(z2+y2+x2)−ψ3,dqV3dtq≤−d1v12−v22−d2v32<0, negative definite. In view of v1 = e1, v2 = e2 − α1 (v1) = e2, v3 = e2 − α2 (v1, v2) = e3, the errors states ei → 0, i = 1, 2, 3 will converge to zero after a finite period of time, and thus the combination synchronization among four fractional order chaotic systems will be achieved.

6.1 Numerical simulation and results

During synchronization the earlier values of the parameters and initial conditions for drive systems I, II, III and response system are considered. The time step size is taken as 0.005. The synchronization among four fractional order chaotic systems are achieved through Fig. 3 using the same method at q = 0.95. Figs. 3(a), 3(b) and 3(c) show the time response of the states xi(t) + yi(t) +i(t) and wi(t), where i = 1, 2, 3 represent the drive systems (5), (6), (7) and the response system (8). The error states for this case are described through Fig. 3(d). It is noticed that it takes less time for synchronization among four systems (Fig. 3(d)) compared to that of three systems (Fig. 2(d)) for the considered systems in both fractional order as well as integer order case (Fig. 4).

Fig. 3 Combination synchronization among four fractional order chaotic systems (5), (6), (7) and (8) for fractional order q = 0.95: (a) between x1(t) + y1(t) + z1 (t) and w1(t), (b) between x2 (t) + y2 (t) + z2 (t) and w2(t), (c) between x3 (t) + y3 (t) +z3(t) and w3(t), (d) the evaluation of error functions e1 (t), e2 (t) and e3(t).
Fig. 3

Combination synchronization among four fractional order chaotic systems (5), (6), (7) and (8) for fractional order q = 0.95: (a) between x1(t) + y1(t) + z1 (t) and w1(t), (b) between x2 (t) + y2 (t) + z2 (t) and w2(t), (c) between x3 (t) + y3 (t) +z3(t) and w3(t), (d) the evaluation of error functions e1 (t), e2 (t) and e3(t).

Fig. 4 The evaluation of error functions e1(t), e2(t) and e3(t) at q = 1: (a) for three systems; (b) for four systems.
Fig. 4

The evaluation of error functions e1(t), e2(t) and e3(t) at q = 1: (a) for three systems; (b) for four systems.

7 Conclusion

In the present study, the combination synchronization among a number of fractional order drive and response systems is successfully demonstrated using backstepping method. For validation, the combination synchronization of three and four systems are considered separately taking two systems and three systems as drive system respectively, while one system as response system, which clearly exhibit that the applied method is effective and convenient to achieve global synchronization of a number of non-identical fractional order chaotic systems. The exhibition of requirement of less time during synchronization of four systems compared to three systems is one of the major contributions of the present study. It is worth mentioning that this scientific contribution of combination synchronization among the fractional order chaotic systems will be significant to the research community involved in the area of modelling of fractional order dynamical systems.

Acknowledgement

The authors are extending their heartfelt thanks to the revered reviewers for their valuable comments to upgrade the present manuscript.

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Received: 2017-06-05
Revised: 2018-05-12
Accepted: 2018-06-26
Published Online: 2019-04-09
Published in Print: 2019-01-28

© 2019 Vijay K. Yadav and S. Das, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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https://doi.org/10.1515/nleng-2017-0073
Schlagwörter für diesen Artikel
Backstepping method; Lyapunov stability theory; Synchronization; Fractional order chaotic systems
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Artikel in diesem Heft

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
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  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
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  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
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Heruntergeladen am 8.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/nleng-2017-0073/html
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