1 Introduction

Fractional calculus is a much older classical mathematical notion with the same 300-year history as integer calculus. In recent years, it has found application in many areas of physics [1] and engineering [2]. At the same time, control and synchronization of fractional-order chaotic systems have made great contributions. Some papers discuss the synchronization of general fractional-order chaotic systems [3–5], while others consider special classes of fractional-order chaotic systems [6–8].

Chaos synchronization is the concept of closeness of the frequencies between different periodic oscillations generated by two chaotic systems, which is fist proposed by Pecora and Carroll [9]. And it is widely explored in a various field’s chemical, ecological, and physical system [10–12]. Chaos synchronization is a very active topic in nonlinear science and has been extensively studied in the past decades. Therefore, various synchronization scheme such as sliding mode control [13–16], linear feedback control [17, 18], adaptive control theory [19, 20], back-stepping control [21], active control [22–24], fuzzy control [25, 26], and fuzzy sliding mode control [27] have been successfully applied to chaos synchronization.

However, there are few previous papers considering synchronization between integer-order chaotic systems and fractional-order chaotic systems with different structure and dimensions [28, 29]. Obviously, it is more difficult to achieve synchronization between integer-order chaotic systems and fractional-order chaotic systems. On the other hand, it is more interesting and more valuable for the application of fractional-order nonlinear systems.

Motivated by the above discussion, fuzzy sliding mode control is utilized here to realize the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system because its robustness and stability. There are three advantages of our approach. First, based on fuzzy theory and sliding mode control (SMC), a new method for chaos synchronization between integer-order chaotic systems and a class of fractional-order chaotic system is presented. Second, the performance of the system in the sense of removing chattering is improved with the utilization of fuzzy logic. Last, two chaotic systems are synchronized with different structure and dimension.

The rest of the paper is outlined as follows. Section 2 introduces the integer-order chaotic systems and a class of fractional-order chaotic systems. Section 3 proposes a compensation controller and vector controller based on fuzzy sliding mode control theory. Furthermore, the controller design scheme and the stability analysis of the closed loop system are included in this section. Section 4 provides results of numerical simulations, and Sect. 5 gives brief comments and conclusions.

2 System description

Consider the n-dimensional, integer-order chaotic drive system

$$ \frac{d^{d}x}{dt^{d}} = f(x) $$
(1)

where x∈R n,f:R n→R n are differentiable functions.

Then consider the n-dimensional, fractional-order chaotic response system

(2)

where y∈R n, g:R n−1→R n−1 are differentiable functions. The dimensions q=(q 1,q 2,…,q n )T (0<q i <1) may be equal or not, and the response system (2) is an integer-order system if q i =1 (i∈[1,n]). The constant a is positive.

3 Problem formulation and control design

System (1) represents the drive system, and the controller u(t)∈R n is added into the response system (2) according to

$$ \frac{d^{q}y}{dt^{q}} = g(y) + u(t) $$
(3)

We define the synchronization errors as e=y−x. The aim is to choose suitable control signals u(t)∈R n such that the states of the master and response systems are synchronized (i.e., lim t→∞∥e∥=0, where ∥⋅∥ is the Euclidean norm).

Now let the controller u(t) be

$$ u(t) = u_{1}(t) + u_{2}(t) $$
(4)

where u 2(t)∈R n−1 is a vector control function that will be designed later. The u 1(t)∈R n is a compensation controller, and \(u_{1} = \frac{d^{q}x}{dt^{q}} - g(x)\). Using Eq. (4), the response system (3) can be rewritten as

(5)

To control the chaotic systems easily, the modified compensation controller u 1 can be represented as

$$ u_{1} = \frac{d^{q}x}{dt^{q}} - g(y - e) $$
(6)

and the modified error dynamics (5) can be represented as

$$ \frac{d^{q}e}{dt^{q}} = h(e,y) + u_{2} $$
(7)

where h(e,y)=g(y)−g(e−y).

Two steps are required to design a sliding mode controller. First, we construct a sliding surface that represents a desired system dynamic. Then we develop a switching control law such that a sliding mode exists on every point of the sliding surface, and any states outside the surface are driven to reach the surface in a finite time [30]. As a choice for the sliding surface, we take

(8)

where r∈[1,n], r∉(i,j), and k 1,k p (p∈[2,n−1]) is a positive constant vector. For the sliding mode method, the sliding surface and its derivative must satisfy

$$ s(t) = 0, \qquad \dot{s}(t) = 0 $$
(9)

Consider

$$ \dot{s}(t) = D^{1-q} \bigl(D^qs(t)\bigr) = 0 \quad \Rightarrow \quad D^qs(t) = 0 $$
(10)

from which it follows that

(11)

and

(12)

In accordance with active control design procedure, the nonlinear part of the error dynamics is eliminated by the following choice of the input vector:

(13)
(14)

where k f is the normalization factor of the output variable, and u fl is the output of the fuzzy logic, which is determined by the normalized s and \(\dot{s}\).

In the above vector, a fuzzy inference engine is used for reaching phase instead of sign function. One major feature of fuzzy logic is its ability to express the amount of ambiguity in human thinking. The fuzzy control rules can be represented as the mapping of the input linguistic variables s and \(\dot{s}\) to the output linguistic variable u fl as follows:

(15)

The membership function of input linguistic for each set of variables s i and \(\dot{s}_{i}\), and the membership functions of the output linguistic variable u fl(i) (i=1,2,…,6), are shown in Fig. 1, respectively. Here, u fl(i) is denoted as:

$$ u_{fl(i)} = \mathit{FL}(s_{i},\dot{s}_{i}) $$
(16)
Fig. 1
figure 1

The membership function for inputs s, \(\dot{s}\), and output u fl

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As usual, the dynamical behavior of a FLC scheme is governed by a set of linguistic rules derived from expert knowledge. By referencing these rules, the inference mechanism of the FLC is able to instruct an appropriate fuzzy control action in response to any change in the input signal. Suppose the rules of fuzzy controller are based on SMC, and then it is called the FSMC [31].

As is described above, our proposed FLC has two inputs and one output. These are s, \(\dot{s}\), and the control signal, respectively. Linguistic variables which imply inputs and output have been classified as [32]: NB, NM, NS, ZE, PS, PM, PB. As is shown in Fig. 1, inputs are all normalized in the interval of [−3,3] and output is normalized in the interval of [−1,1], all with equal span. The linguistic labels used to describe the fuzzy sets were “Negative Big” (NB), “Negative Medium” (NM), “Negative Small” (NS), “Zero” (ZE), “Positive Small” (PS), “Positive Medium” (PM) and “Positive Big” (PB). It is possible to assign a set of decision rules as shown in Table 1. These rules contain the input/output relationships that define the control strategy. Each control input has seven fuzzy sets so that there are at most 49 fuzzy rules.

Table 1 Rule base of FSMC
Full size table

According to function (2) and control (6) and (14), the error is given by

$$ \frac{d^{q}e}{dt^{q}} = Ae + k_{f}u_{fl} $$
(17)

Theorem

Consider the error function (14). The error between response system (3) and drive system (1) can be determined if and only if |arg(λ i )|>qπ/2 is satisfied for all eigenvalues λ i of matrix A. Besides, this error system is stable if and only if |arg(λ i )|≥qπ/2 is satisfied for all eigenvalues λ i of matrix A and those critical eigenvalues which satisfies the condition |arg(λ i )|=qπ/2, have geometric multiplicity one.

Proof

According to functions (2), (6), (14), and (17), A is given by the matrix

(18)

whose eigenvalues are

(19)

Thus, all eigenvalues of matrix A satisfy Re(λ)<0, which implies \(| \arg\lambda| > \frac{\pi}{2} > \frac{q\pi}{2}\). According to the stability theory of fractional-order systems [33–35], the equilibrium point e=0 in function (17) is asymptotically stable:

$$ \lim_{t \to + \infty} e = \lim_{t \to + \infty} (y - x) = 0 $$
(20)

Remark 1

If chaotic orders in drive system (10) meet α i =1, like the system \(\dot{x} = Ax + f( x )\), then the synchronization between a fractional-order system and an integer-order system can be achieved using the controller (4).

Remark 2

Most system parameters change stochastically within a certain range. As we can see, the fuzzy logic output (16), which vary with system parameters, that is, when the system parameter change, the controller (4) changes in certain regular way as well. Especially for the system with stochastic parameters, the controller is extremely effective.

4 Numerical simulation

This section of the paper presents three illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. In Case I, a three-dimensional integer-order system is synchronized with a fractional-order system having a different structure. In Case II, a four-dimensional integer-order system is synchronized with a fractional-order system. In Case III, a three-dimensional integer-order system is synchronized with a four-dimensional fractional-order system. The numerical simulation results were carried out in MATLAB using the Caputo version and a predictor-corrector algorithm with a fixed step size of 0.01, which is the generalization of Adams–Bashforth–Moulton one.

Case I

Synchronization between a fractional-order \(L\ddot{u}\) chaotic system and an integer-order Lü chaotic system.

The integer-order Liu chaotic system [36] is described by

(21)

The Liu system exhibits chaotic behavior for the parameters (a,b,c,k,h)=(10,40,2.5,1,4) with initial conditions [x d ,y d ,z d ]T=[10,0,30]T and a chaotic attractor as shown in Fig. 2(a).

Fig. 2
figure 2

Chaotic attractors of integer-order Liu chaotic system and fractional-order Lü chaotic system

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The fractional-order Lü chaotic system [37, 38] is

(22)

and exhibits chaotic behavior for q r1=q r2=q r3=0.90 and (a,b,c)=(35,3,28) with initial conditions [x r ,y r ,z r ]T=[0,3,9]T and a chaotic attractor as shown in Fig. 2(b).

Here, the controller parameters K 1=K 2=10 and k f =1 are chosen, and the eigenvalues (λ 1,λ 2,λ 3)=(−22.5+32.6917i,−22.5−32.6917i,−10) are located in the stable region. As described above, we can obtain the controller u(t) for the response system (6) and (14) as follows:

  1. i.

    Compensation controller

    (23)
  2. ii.

    Vector controller

    (24)

The synchronization errors are shown in Fig. 3, which demonstrates that the proposed method is successful in synchronizing the two systems.

Fig. 3
figure 3

Synchronization errors between integer-order Liu chaotic system and fractional-order Lü chaotic system (e 1=x r −x d , e 2=y r −y d , e 3=z r −z d )

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Case II

Synchronization between a fractional-order hyperchaotic system based on Chen’s system and an integer-order hyperchaotic system based on the Lorenz system.

The integer-order hyper-chaotic system based on the Lorenz system [39] is given by

(25)

This system exhibits chaotic behavior for the parameters (a,b,c,k)=(10,28,8/3,0.03) with initial conditions [x d ,y d ,z d ,w d ]T=[5,15,2,20]T and a chaotic attractor as shown in Fig. 4(a).

Fig. 4
figure 4

Chaotic attractors of integer-order hyperchaotic based upon Lorena system and fractional-order hyperchaotic system based upon Chen system

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The fractional-order hyperchaotic system based on Chen’s system [40] is

(26)

This system exhibits chaos for q r1=q r2=q r3=0.90 and (a,b,c,d,k)=(36,3,28,−16,0.5) with initial conditions [x r ,y r ,z r ,w r ]T=[0,1,9,7]T and a chaotic attractor as shown in Fig. 4(b).

The controller parameters K 1=K 2=K 3=10 and k f =1 are chosen, and the eigenvalues (λ 1,λ 2,λ 3,λ 4)=(−23+33.5708i,−23−33.5708i,−10,−10) are located in the stable region. As before, we can obtain the controller u(t) for the response system (6) and (14) as follows:

  1. i.

    Compensation controller

    (27)
  2. ii.

    Vector controller

    (28)

The synchronization errors are shown in Fig. 5, which demonstrates that the proposed method is successful in synchronizing the two systems.

Fig. 5
figure 5

Synchronization errors between integer-order hyperchaotic based upon Lorena system and fractional-order hyperchaotic system based upon Chen system

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Case III

Synchronization between a fractional-order hyperchaotic system based on Chen’s system and an integer-order Liu chaotic system.

The Liu chaotic system [36] is described by

(29)

and exhibits chaotic behavior for the parameters (a,b,c,k,h)=(10,40,2.5,1,4) with initial conditions [x d ,y d ,z d ]T=[5,10,15]T and a chaotic attractor as shown in Fig. 6(a).

Fig. 6
figure 6

Chaotic attractors of integer-order Liu chaotic system and fractional-order hyperchaotic system based upon Chen system

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The fractional-order hyperchaotic system based on Chen’s system [40] is

(30)

and exhibits chaotic behavior for q r1=q r2=q r3=0.90 and (a,b,c,d,k)=(36,3,28,−16,0.5) with initial conditions [x r ,y r ,z r ,w r ]T=[0,3,9,17]T and a chaotic attractor as shown in Fig. 6(b).

The controller parameters K 1=K 2=K 3=7, k f =1 are chosen, and the eigenvalues are located in the stable region. As mentioned above, we can obtain the controller u(t) for the response system (6) and (14) as follows:

  1. i.

    Compensation controller

    (31)
  2. ii.

    Vector controller

    (32)

The synchronization errors are shown in Fig. 7, which demonstrates that the proposed method is successful in synchronizing the two systems.

Fig. 7
figure 7

Synchronization errors between integer-order Liu chaotic system and fractional-order hyperchaotic system based upon Chen system (e 1=x r −x d , e 2=y r −y d , e 3=z r −z d , e 4=w r −w d )

Full size image

5 Conclusions and discussion

In this paper, the synchronization between fractional-order chaotic systems and integer-order chaotic systems was achieved based on fuzzy sliding mode control. The proposed synchronization approach is theoretically rigorous and pervasive. Furthermore, three typical examples were shown: (1) the synchronization between different three-dimensional chaotic systems, (2) between different four-dimensional chaotic systems, and (3) between a three-dimensional chaotic system and a four-dimensional chaotic system. Numerical results illustrated the effectiveness of the proposed scheme. These theoretical and numerical results provide a bridge between integer-order chaotic system and fractional-order chaotic systems and lend theoretical support for fractional-order chaotic systems.

More and better methods for the synchronization between integer-order chaotic systems and fractional-order chaotic systems should be studied. Moreover, this knowledge should be applied in engineering to fields such as communications, and its circuit design will be a subject of our future work.