## 1 Introduction

The beginning of the fractional calculus [1] is considered to be the Leibniz’s letter to L’Hospital in 1695, where the notation for differentiation of non-integer order 1/2 is discussed. In addition, Leibniz wrote, “Thus it follows that *d*^{1/2}*x* will be equal to

Though the idea of chaos theory came from observing weather patterns [3], but eventually it has become applicable to a variety of other disciplines. Some areas in mathematics, computer science, microbiology, meteorology, biology, engineering, geology, finance, economics, algorithmic trading, politics, population dynamics, psychology, philosophy and robotics are already benefited by chaos theory.

Chaos synchronization is an important topic in the nonlinear science. Generally speaking, the synchronization phenomenon has the feature that the trajectories of two systems (drive and response systems) are identical, in spite of starting from different initial conditions. However, slight changes in the initial conditions may lead to completely different trajectories. Therefore, how to control two chaotic systems to be synchronized have received a great deal of interest in past two decades to the researchers working in the field of chaos theory.

Various forms of synchronization that have been observed in the literature are phase synchronization, lag synchronization, generalized synchronization and sequential synchronization. Notable among the various methods for achieving this aim include linear feed-back, adaptive synchronization, backstepping control, sliding mode control, active control method, OGY method, projective synchronization, inverse optimal control ([4–12]).

Active control method, proposed by Bai and Lonngren [13] has widely been accepted as an efficient technique for the synchronization of non-identical chaotic systems, a feature for which it has got advantage over other synchronization methods. After giving a generalized design of the method by Ho and Hung [14], the method had been treated as one of the most interesting control strategies for its simplicity. Despite of the fact that active control method is an expensive strategy as its takes comparative more time for synchronization as compared to other exiting methods, it is noticed that the method is effective and convenient to synchronize since Lyapunov exponents are not required mathematical calculation. The method has been well tested to many practical systems such as spatiotemporal dynamical systems, nonlinear bloch equations modeling, nuclear magnetic resonance, electric circuits modeling, complex dynamos, acoustic gravity waves, parametrically excited systems and RCL-shunted Josephson junctions etc. It is seen from literature that Vincent and Laoye [15] used active control based synchronization scheme for controlling directed transport arising from co-existing attractors in non-equilibrium physics.

Backstepping design and active control are both recognised as powerful design methods for chaos synchronization. Backstepping design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems ([16–18]). The method has been employed for controlling and synchronizing many chaotic systems as well as hyperchaotic systems. Some of the method is widely used for synchronization of chaotic systems for the advantage that it needs only one controller to realize synchronization between chaotic systems and finally there are no derivatives in the controller [19].

In the present article, 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. In 2008, both the methods are used by Vincent [21] during synchronization of identical integer order systems already considered by A.M. Harb and M.A. Zohdy. This has inspired the authors to extend it in fractional order system. The main feature of the article is a comparative study of time of synchronization through numerical simulation and graphical presentation.

## 2 Definition and lemma

There are several definitions for fractional order derivative ([22–24]). Since the Caputo’s fractional derivative of a constant is zero, we choose Caputo’s definition [25] which is written as

*q*∈

*R*,

*n*∈

*N*and

*Γ*(⋅) is the Gamma function.

[26] Let *x*(*t*) ∈ *R* be a continuous and derivable function. Then, for any time instant *t* ≥ *a*,

## 3 Systems’ descriptions

### 3.1 Fractional order Chen system

The fractional order Chen system [27] of order *q*, is given by

*a*

_{1},

*a*

_{2},

*a*

_{3}are the parameters of the system. The phase portrait of the fractional order Chen system in

*x*

_{1}−

*x*

_{2}−

*x*

_{3}space is depicted through Fig. 1 at

*q*= 0.96 for the values of parameters

*a*

_{1}= 35,

*a*

_{2}= 3,

*a*

_{3}= 28 and the initial condition (10, 25, 36).

### 3.2 Fractional order Qi system

The fractional order Qi system [28] of order *q*, is given by

*b*

_{1},

*b*

_{2}and

*b*

_{3}are the parameters of the system.

The phase portrait of system (2) at *q* = 0.96 is depicted through the Fig. 2 at the values of the parameters

## 4 Synchronization of fractional order Chen and Qi systems using active control method

In this section, our aim is to achieve synchronization between fractional order Chen system and Qi system by using active control method. In this method, we assume that the Chen system (1) drives the Qi system (2). The response system is defined as

*u*

_{1}(

*t*),

*u*

_{2}(

*t*) and

*u*

_{3}(

*t*) are control functions.

Let us define the error *e* = (*e*_{1}(*t*), *e*_{2}(*t*), *e*_{3}(*t*))^{t} as

Since the derivative operator is linear, we get the error system as

To achieve our aim, we have to re-define the active control function as

Using above equations in system (5), we get the error system as

To control the linear error system (6) with control inputs *V*_{1}(*t*), *V*_{2}(*t*) and *V*_{3}(*t*) as functions of the error states *e*_{i}(*t*), *i* = 1, 2, 3, our aim is to find the feedback control function in such a way that *e*_{i}(*t*) → 0 as *t* → *∞*, *i* = 1, 2, 3, so that the systems (1) and (3) are globally synchronized. To achieve this, there are many choices for the control functions *V*_{1}(*t*), *V*_{2}(*t*) and *V*_{3}(*t*). We choose

*A*is a 3 × 3 constant matrix. In order to make the closed loop system stable, we choose the elements of the matrix

*A*in such way that the error system must have all the eigenvalues with negative real parts.

Let the matrix *A* is chosen in the form

In this particular choice, the closed loop system (6) has the eigenvalues −1, −1 and −1. This choice leads to a stable system and thus the synchronization between fractional order Chen system and Qi system is achieved.

## 5 Synchronization of fractional order Chen and Qi systems using backstepping approach

The drive system is taken as fractional order Chen system (1) and fractional order Qi system considered as response system (3).

Defining error states as *e*_{1} = *y*_{1} − *x*_{1}, *e*_{2} = *y*_{2} − *x*_{2}, *e*_{3} = *y*_{3} − *x*_{3}, the error dynamical system reduces to

Equation (7) can be considered as control problem where the system is to be controlled by the control functions *u*_{1}(*t*), *u*_{2}(*t*) and *u*_{3}(*t*) which are the functions of error vectors *e*_{1}(*t*), *e*_{2}(*t*) and *e*_{3}(*t*).

If the error states *e*_{1}(*t*), *e*_{2}(*t*) and *e*_{3}(*t*) converge to zero as time *t* becomes large, then the systems (1) and (3) are said to be synchronized. Now our aim is to design the control functions for synchronization of fractional order chaotic systems using backstepping approach.

If the control function are chosen as

*w*

_{1}=

*e*

_{1},

*w*

_{2}=

*e*

_{2},

*w*

_{3}=

*e*

_{3}, then the drive system (1) will be synchronized with the response system (3).

To achieve our results we use backstepping procedure which has three steps.

**Step-I**: Defining *w*_{1} = *e*_{1}, we have

*e*

_{2}=

*α*

_{1}(

*w*

_{1}) is regarded as an virtual controller. For the design of

*α*

_{1}(

*w*

_{1}) to stabilize

*w*

_{1}-subsystem, choose the Lyapunov function as

The *q*-th order derivative of *v*_{1} is

If we choose *u*_{1}(*t*) = − *φ*_{1} − *e*_{3}*x*_{2} and *α*_{1}(*w*_{1}) = 0, then *w*_{1}-subsystem (8) is asymptotically stable.

Defining the error variable between *e*_{2} and the estimative virtual controller *α*_{1}(*w*_{1}) as *w*_{2} = *e*_{2} − *α*_{1}(*w*_{1}), we can obtain the following (*w*_{1}, *w*_{2})-subsystem as

*e*

_{3}=

*α*

_{2}(

*w*

_{1},

*w*

_{2}) is a virtual controller.

**Step-II**: In this step, we define the following Lyapunov function *v*_{2} to stabilize (*w*_{1}, *w*_{2})-subsystem (9) as

The *q*-th order derivative of *v*_{2} w.r. to *t* is

(using lemma), i.e.,

If we choose *α*_{2}(*w*_{1}, *w*_{2}) = 0 and *u*_{2}(*t*) = − *φ*_{2} − *b*_{3}*w*_{1} − *b*_{1}*w*_{1}, then *w*_{1}, *w*_{2})-subsystem (9) is asymptotically stable.

Similarly considering *w*_{3} = *e*_{3} − *α*_{2}(*w*_{1}, *w*_{2}), we get the following (*w*_{1}, *w*_{2}, *w*_{3})-system as

**Step-III**: In order to stabilize (

*w*

_{1},

*w*

_{2},

*w*

_{3})-system, define the Lyapunov function

*v*

_{3}as

The *q*-th order fractional order derivative of *v*_{3} w.r. to *t* is

If we choose *u*_{3}(*t*) = − *φ*_{3} − *w*_{1}*x*_{2} − *w*_{1}*w*_{2}, then *w*_{1}, *w*_{2}, *w*_{3})-system is asymptotically stable. Thus for *w*_{1} = *e*_{1}, *w*_{2} = *e*_{2} − *α*_{1}(*w*_{1}) = *e*_{2}, *w*_{3} = *e*_{3} − *α*_{2}(*w*_{1}, *w*_{2}) = *e*_{3}, the state errors *e*_{1}, *e*_{2} and *e*_{3} converge to zero after a finite period of time, confirm the synchronization between fractional order Chen system and Qi system. □

## 6 Numerical simulation and results

In this section, we take the earlier considered values of the parameters of both the systems. The initial conditions of drive and response systems are taken as (*x*_{1}(0), *x*_{2}(0), *x*_{3}(0)) = (10, 25, 36) and (*y*_{1}(0), *y*_{2}(0), *y*_{3}(0)) = (3, 2, 1), respectively. Hence the initial conditions of error system is (*e*_{1}(0), *e*_{2}(0), *e*_{3}(0)) = (− 7, − 23, − 35). During synchronization of the systems, the time step size is taken as 0.005. The synchronization between *x*_{1} − *y*_{1}, *x*_{2} − *y*_{2} and *x*_{3} − *y*_{3} are depicted through Fig. 3 and Fig. 5 at *q* = 0.96, for active control method and backstepping approach respectively. The error functions are depicted through Fig. 4 and Fig. 6 for active control method and backstepping approach respectively at *q* = 0.92, 0.96, 1. It is clear from the figures that in both the cases it takes less time to synchronize as the systems pair approaches from standard order to fractional order. Also it is found from the figures that it takes less time for synchronization for the backstepping method compared to active control method.

## 7 Conclusion

The theme of the present research article is to investigate the synchronization between two non-identical fractional order chaotic systems using active control method and backstepping method. Based on stability analysis, the required synchronization of the chaotic systems, viz., Chen and Qi systems has been achieved. The components of error systems tending to zero as time progresses is attempt through proper choices of control functions. This helps to get the time required for synchronization. The novelty of the article is the finding that less time is necessary for synchronization, when the system pair approaches to fractional order from the standard order upon application of both the methods, exhibited through graphical presentations. Additionally a comparison of time requirement for both standard and fractional order derivatives applying the active control method and backstepping method has been done.

The authors are extending their heartfelt thanks to the revered reviewer for their valuable comments towards the up-gradation of the article. First and second authors acknowledge the financial support from the UGC, New Delhi, India under the SRF-NET scheme.

## References

- [1]↑
Miller K.S., Ross B., An Introduction to the fractional calculus and fractional differential equations. John Wiley & Sons, 1993.

- [4]↑
Yassen M.T., Controlling, synchronization and tracking chaotic Liu system using active backstepping design. Physics Letters A, 2007, 360, 582–587.

- [5]
Wu Z., Fu X., Combination synchronization of three different order nonlinear systems using active backstepping design. Nonlinear Dyn, 2013; 73, 1863–1872.

- [6]
Park J.H., Synchronization of Genesio chaotic system via backstepping approach. Chaos, Solitons and Fractals, 2006 27, 1369–1375.

- [7]
Tan X., Zhang J., Yang Y., Synchronizing chaotic systems using backstepping design. Chaos, Solitons and Fractals, 2003, 16, 37–45.

- [8]
Njah A.N., Vincent U.E., Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control. Chaos, Solitons and Fractals, 2008, 37, 1356–1361.

- [9]
Yassen M.T., Chaos synchronization between two different chaotic systems using active control. Chaos Solitons and Fractals, 2005, 23, 131–140.

- [10]
Agrawal S.K., Srivastava M., Das S., Synchronization of fractional order chaotic systems using active control method. Chaos, Solitons and Fractals, 2012, 45, 737–752.

- [11]
Vincent U.E., Synchronization of Rikitake chaotic attractor using active control. Physics Letters A, 2005, 343, 133–138.

- [12]↑
Ho M.C., Hung Y.C., Synchronization of two different systems by using generalized active control. Physics Letters A, 2002, 301, 424–428.

- [13]↑
Bai E.W., Lonngren K.E., Synchronization of two Lorenz systems using Active control. Chaos, Solitons and Fractals, 1997, 8, 51–58.

- [14]↑
Ho M.C., Hung Y.C., Synchronization of two different systems by using generalized active control. Phys Lett A, 2002, 301, 424–428.

- [15]↑
Vincent U.E., Laoye J.A., Synchronization and control of directed transport in inertial ratchets via active control. Phys Lett A, 2007, 363, 91–95.

- [16]↑
Zhang H., Ma X., Li M., Zou J., Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos Solitons and Fractals 2005, 26, 353–361.

- [17]
Kokotovic P.V., The joy of feedback: nonlinear and adaptive. IEEE Control Syst Mag, 1992, 6, 7–17.

- [18]↑
Krstic M., Kanellakopoulus I, Kokotovic P. Nonlinear and adaptive control design John Wiley, New York, 1995.

- [19]↑
Tan X., Zhang J., Yang Y., Synchronizing chaotic systems using backstepping design. Chaos Solitons and Fractals, 2003, 16, 37–45.

- [20]
Vincent U.E., Njah A.N., Laoye J.A., Controlling chaotic motion and deterministic directed transport in Chaotic Ratchets using Backstepping nonlinear controller. Physica D, 2007, 231, 130.

- [21]↑
Vincent U.E., Chaos synchronization using active control and backstepping control: a comparative analysis. Nonlinear Analysis: Modelling and Control, 2008, 13(2), 253–261.

- [23]
Kilbas A.A., Srivastava H.M., Trujllo J.J., Theory and applications of fractional differential equations. Elsevier, 2006.

- [24]↑
Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives: Theory and Applications. Gordon and Breach Science, 1993.

- [25]↑
Caputo M., Linear models of dissipation whose Q is almost frequency independent-II Geophysical Journal International, 1967, 13(5), 529–539.

- [26]↑
Norelys A.C., Manuel A.D.M., Javier A.G., Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simulat, 2014, 19, 2951–2957.

- [27]↑
Li C., Peng G., Choas in Chen’s system with a fractional order. Chaos Solitons and Fractals, 2004, 22, 443–450.

- [28]↑
Wang Z., Sun Y., Qi G., Wyk B.J.V., The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor. Nonlinear Dyn, 2010, 62,139–150.