Recently, the chaos synchronization of fractional-order systems gained a lot of attention, such as [1-11]. Sun in [12] addressed the sliding mode synchronization problem of fractional-order uncertainty systems, which the master-slave systems can realize project synchronization. The authors in [13] studied the problem of self-adaptive sliding mode synchronization of a class of fractional-order chaos systems, which the drive-response systems achieved chaos synchronization. Chaos synchronization control problem was investigated for fractional-order systems in [14]. Zhang in [15] considered the self-adaptive trace project synchronization problem of the fractional-order Rayleigh-Duffing-like systems. Since the Victor-Carmen chaos systems involving lots of secreted key parameters and getting extensive use in communications, some results on this topic were investigated. For example, a novel chaotic systems was studied for random pulse generation in [16], and in [17], the terminal sliding mode chaos control of fractional-order systems was studied. In this paper, the problem of sliding mode synchronization of a class of fractional-order uncertain Victor-Carmen systems is tackled using self-adaptive sliding mode control approach, and sufficient conditions on sliding mode synchronization are derived for the fractional-order systems.

Definition 1.1   (see [18]) The fractional derivative of Caputo is given as follows

$ {}{_c}D{_{t_0, t}^{\alpha}}x(t) = D {_{t_0, t}^{-(n-\alpha)}} \frac {d^n}{dt^n}x(t) = \frac 1 {\Gamma(n-\alpha) } \int ^t_{t_0} (t-\tau )^{n-\alpha-1}x^{(n)}(\tau)d\tau, n-1<\alpha<n\in Z^+. $

Consider the following integer-order Victor-Carmen systems

$ \begin{equation} \left\{ \begin{aligned} \dot {x_1}& = -x_1-\alpha x_2x_3 , \\ \dot{x_2}& = -x_2+ax_3-\beta x_1x_3, \\ \dot{x_3}& = -bx_1-a x_2+ x_3+\gamma x_1x_2, \end{aligned} \right. \end{equation} $

(2.1)

where $ x_1, x_2, x_3\in R^3 $ are system states, $ a, b, \alpha, \beta, \gamma $ are constant parameters.

The responsive systems are as follows

$ \begin{equation} \left\{ \begin{aligned} \dot{y_1}& = -y_1-\alpha y_2y_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ \dot{y_2}& = -y_2+ay_3-\beta y_1y_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ \dot {y_3}& = -by_1-a y_2+ y_3+\gamma y_1y_2+\Delta f_3(y)+d_3(t)+u_3(t), \end{aligned} \right. \end{equation} $

(2.2)

where $ \Delta f_i(y) $ is uncertain, $ d_i(t) $ is bounded disturbance, $ u_i $ is controller, subtracting (2.2) to (2.1), we get

$ \begin{equation} \left\{ \begin{aligned} \dot {e_1}& = -e_1-\alpha y_2y_3+\alpha x_2x_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ \dot {e_2}& = -e_2+ae_3-\beta y_1y_3+\beta x_1x_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ \dot{e_3}& = -be_1-a e_2+ e_3+\gamma y_1y_2-\gamma x_1x_2+\Delta f_3(y)+d_3(t)+u_3(t). \end{aligned} \right. \end{equation} $

(2.3)

Assumption 2.1   $ \Delta f_i(y) $ and $ d_i(t) $ are bounded, $ m_i, n_i>0, |\Delta f_i(y)|<m_i, |d_i(t)|<n_i. $

Assumption 2.2   $ \; m_i $ and $ n_i $ are unknown for all $ i = 1, 2, 3 $.

Assumption 2.3   Definite $ \Delta f_i(y)+d_i(t) = g_i(t), i = 1, 2, 3. $

Assumption 2.4   $ g_i(t) $ satisfies the condition $ |g_i(t)|\leq \varepsilon |e_i(t)|, $ where $ 0<\varepsilon<1. $

Assumption 2.5   If $ e_i(t) = 0 $, then $ g_i(t) = 0 $ and if $ e_i(t)\neq 0 $, then $ g_i(t)\neq 0. $

Lemma 2.6   (Barbalat's lemma, see [19]) If $ f(t) $ is uniform continuity in $ [0, +\infty) $, and $ \int ^{+\infty}_0 f(t)dt $ is exist, then $ \lim\limits_{t\rightarrow \infty}f(t) = 0. $

Lemma 2.7   (see [19]) If there exists a symmetric and positive-definite matrix $ \bf P $ such that $ {\boldsymbol{\rm{J}}}({\boldsymbol{\rm{x}}}(t)) = {{\boldsymbol{\rm{x}}}^T}(t){\boldsymbol{\rm{P}}}D_t^\alpha {\boldsymbol{\rm{x}}}(t) < 0 $, where the systems order number $ 0<\alpha \leq 1 $, then general fractional-order autonomous systems $ D_t^\alpha \bf x(t) = \bf {f}(\bf x(t)) $ is asymptotic stable.

Theorem 2.8   Under Assumptions 2.1–2.5, choosing sliding mode function $ s(t) = e_1+e_2+e_3, $ and the following controllers

$ \begin{equation} \begin{array}{ll} u_1 = \alpha y_2y_3-\alpha x_2x_3-(\hat m_1+\hat n_1){\rm sgn}(s) , \\ u_2 = -ae_3+\beta y_1y_3-\beta x_1x_3-(\hat m_2+\hat n_2){\rm sgn}(s) , \\ u_3 = (b+1)e_1+(a+1)e_2- e_3+\gamma (x_1x_2- y_1y_2)-(\hat m_3+\hat n_3-\eta){\rm sgn}(s), \end{array} \end{equation} $

(2.4)

where $ \eta>0, $ $ \hat m_i $ and $ \hat n_i $ are the estimate values of $ m_i $ and $ n_i $, and for all $ i = 1, 2, 3 $, designing self-adaptive laws

$ \begin{equation} \left\{ \begin{array}{ll} \dot{ \hat m}_i = |s|, \hat m_i(0) = \hat m_{i0}, \\ \dot{ \hat n}_i = |s|, \hat n_i(0) = \hat n_{i0}, \end{array} \right. i = 1, 2, 3.\nonumber \end{equation} $

Then the master-slave systems (2.1) and (2.2) of integer-order Victor-Carmen systems are self-adaptive sliding mode synchronization.

Proof   When the systems state moving on the sliding mode surface, then we can get $ s(t) = 0, \dot s(t) = 0 $, because

$ \begin{equation} s(t) = e_1+e_2+e_3 = 0. \end{equation} $

(2.5)

If we substitute (2.4) to (2.3), then $ \dot{e_i} = -e_i+ g_i(t)-(\hat m_i+\hat n_i){\rm sgn}(s), i = 1, 2 $ for $ s(t) = 0, $ it is easy to get $ \dot{e_i} = -e_i+ g_i(t), i = 1, 2 $. On the other hand, for $ \dot{e_3} = e_1+e_2+g_3(t)-(\hat m_3+\hat n_3){\rm sgn}(s)-\eta {\rm sgn}(s), $ from (2.5), it is easy to get $ e_1+e_2 = -e_3 $, so we get $ \dot{e_3} = -e_3+g_3(t) $, so $ \dot{e_i} = -e_i+ g_i(t), i = 1, 2, 3. $ According to Lyapunov stability theory, when $ e_i(t)\neq 0 $, found Lyapunov function $ V(t) = \frac 1 2 e^2(t), $ we get

$ \dot{V}(t) = e^T(t)\dot e(t) = \sum \limits^3_{i = 1}e_i \dot{e_i} = \sum \limits^3_{i = 1}e_i( -e_i+ g_i(t)) \leq -(1-\varepsilon )\sum \limits^3_{i = 1}|e_i(t)|^2<0. $

So the solution of $ \dot{e_i} = -e_i+ g_i(t) $ convergence to zero, which is $ e_i(t)\rightarrow 0, i = 1, 2, 3. $ For the systems state moving on the sliding mode surface, so the solution of errors equation (2.3) is asymptotic stable, then $ e_i(t)\rightarrow 0, i = 1, 2, 3. $

When the systems aren't moving on the sliding mode surface, we found Lyapunov function as $ V(t) = \frac 1 2{s}^2(t)+ \frac 1 2\sum \limits^3_{i = 1}\left((\hat m_i- m_i)^2+(\hat n_i-n_i)^2\right ), $ so it has

$ \begin{aligned} \dot{V} = &s\dot{s}+\sum \limits^3_{i = 1}(\hat m_i- m_i)|s|+\sum \limits^3_{i = 1}(\hat n_i-n_i)|s|\\ = &s[ -e_1+\Delta f_1(y)+d_1(t)-(\hat m_1+\hat n_1){\rm sgn}(s) -e_2+\Delta f_2(y) +d_2(t)-(\hat m_2+\hat n_2){\rm sgn}(s) \\ &+e_1+ e_2+\Delta f_3(y)+d_3(t)-(\hat m_3+\hat n_3){\rm sgn}(s)-\eta {\rm sgn}(s)]\\ \leq& \sum \limits^3_{i = 1}(m_i+n_i)|s|-\sum \limits^3_{i = 1}(\hat m_i+\hat n_i)|s|+\sum \limits^3_{i = 1}(\hat m_i- m_i)|s|+\sum \limits^3_{i = 1}(\hat n_i-n_i)|s|-\eta |s|\\ = &-\eta |s|<0. \end{aligned} $

For $ \dot{V} \leq -\eta |s| $, integral on the both sides

$ \int ^t_0 |s(\tau)| d\tau \leq \frac {-1} \eta \int ^t_0 \dot{V}(\tau) d\tau \leq \frac {V(0)-V(\infty)}{\eta}\leq \frac {V(0)}{\eta}<\infty, $

so $ s(t) $ is bounded and integrable. From Lemma 2.6, we get $ s(t)\rightarrow 0\Rightarrow e_i(t)\rightarrow 0 $, so the errors converge to zero.

Consider the master systems of fractional-order Victor-Carmen systems

$ \begin{equation} \left\{ \begin{aligned} D_t^q {x_1}& = -x_1-\alpha x_2x_3, \\ D_t^q {x_2}& = -x_2+ax_3-\beta x_1x_3, \\ D_t^q {x_3}& = -bx_1-a x_2+ x_3+\gamma x_1x_2. \end{aligned} \right. \end{equation} $

(2.6)

Design the slave systems as following

$ \begin{equation} \left\{ \begin{aligned} D_t^q {y_1}& = -y_1-\alpha y_2y_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ D_t^q {y_2}& = -y_2+ay_3-\beta y_1y_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ D_t^q {y_3}& = -by_1-a y_2+ y_3+\gamma y_1y_2+\Delta f_3(y)+d_3(t)+u_3(t), \end{aligned} \right. \end{equation} $

(2.7)

where $ \Delta f_i(y) $ is uncertainty, $ y = [y_1\; \; y_2\; \; y_3], d_i(t) $ is bounded disturbance, $ u_i $ is controller, subtract (2.7) to (2.6), we get the following errors equation

$ \begin{equation} \left\{ \begin{aligned} D_t^q {e_1}& = -e_1-\alpha y_2y_3+\alpha x_2x_3+\Delta f_1(y)+d_1(t)+u_1(t), \\ D_t^q {e_2}& = -e_2+ae_3-\beta y_1y_3+\beta x_1x_3+\Delta f_2(y)+d_2(t)+u_2(t), \\ D_t^q {e_3}& = -be_1-a e_2+ e_3+\gamma y_1y_2-\gamma x_1x_2+\Delta f_3(y)+d_3(t)+u_3(t). \end{aligned} \right. \end{equation} $

(2.8)

Theorem 2.9   Under Assumptions 2.1–2.5, design sliding mode function $ s(t) = D_t^{q-1} (e_1+e_2+e_3) $, choosing controller

$ \begin{equation} \begin{array}{ll} u_1 = \alpha y_2y_3-\alpha x_2x_3-(\hat m_1+\hat n_1){\rm sgn}(s) , \\ u_2 = -ae_3+\beta y_1y_3-\beta x_1x_3-(\hat m_2+\hat n_2){\rm sgn}(s) , \\ u_3 = (b+1)e_1+(a+1)e_2- e_3-\gamma y_1y_2+\gamma x_1x_2-(\hat m_3+\hat n_3-\eta){\rm sgn}(s), \end{array} \end{equation} $

(2.9)

where $ \eta>0, $ $ \hat m_i, \hat n_i $ are the estimate values of $ m_i, n_i $, design self-adaptive laws

$ \begin{equation} \left\{ \begin{array}{ll} \dot{ \hat m}_i = |s|, \hat m_i(0) = \hat m_{i0}, \\ \dot{ \hat n}_i = |s|, \hat n_i(0) = \hat n_{i0}, \end{array} \right. i = 1, 2, 3.\nonumber \end{equation} $

Then the master-slave systems (2.6) and (2.7) of fractional-order Victor-Carmen systems are self-adaptive sliding mode synchronization.

Proof   When the systems state moving on the sliding mode surface, $ s(t) = 0, \dot s(t) = 0 $, then $ s(t) = D_t^{q-1} (e_1+e_2+e_3) = 0 $, so we get $ D_t^{1-q}D_t^{q-1} (e_1+e_2+e_3) = 0 $, such that we have

$ \begin{equation} e_1+e_2+e_3 = 0. \end{equation} $

(2.10)

Substitute controller (2.9) to (2.8), we get $ D_t^{q}{e_i} = -e_i+ g_i(t)-(\hat m_i+\hat n_i){\rm sgn}(s), i = 1, 2, $ when the systems state moving on the sliding mode surface $ s(t) = 0 $, so it is easy to get

$ D_t^{q}{e_i} = -e_i+ g_i(t), i = 1, 2. $

On the other hand, $ D_t^{q}{e_3} = e_1+e_2+g_3(t)-(\hat m_3+\hat n_3){\rm sgn}(s)-\eta {\rm sgn}(s), $ according to (2.10), such we get $ e_1+e_2 = -e_3 $ and $ D_t^{q}{e_3} = -e_3+g_3(t), $ so it has $ D_t^{q} e_i = -e_i+ g_i(t), i = 1, 2, 3. $

According to Lemma 2.7, if $ e_i(t)\neq 0 $,

$ J = e^T(t)D_t^{q}e(t) = \sum \limits^3_{i = 1}e_i D_t^{q}{e_i} = \sum \limits^3_{i = 1}e_i( -e_i+ g_i(t)) \leq -(1-\varepsilon )\sum \limits^3_{i = 1}|e_i(t)|^2<0. $

According to Lemma 2.7, the solution of following equation $ D_t^{q} e_i = -e_i+ g_i(t), $ so $ e_i(t)\rightarrow 0, i = 1, 2, 3. $ When the systems state moving on the sliding mode surface $ s(t) = 0 $, then the solution of errors equation (2.8) is asymptotic stable such that we get $ e_i(t)\rightarrow 0, i = 1, 2, 3. $

When the systems aren't moving on the sliding mode surface, we found Lyapunov function $ V(t) = \frac 1 2{s}^2(t)+ \frac 1 2\sum \limits^3_{i = 1}\left((\hat m_i- m_i)^2+(\hat n_i-n_i)^2\right ) $ such that we get

$ \dot{V} = s\dot{s}+\sum \limits^3_{i = 1}(\hat m_i- m_i)|s|+\sum \limits^3_{i = 1}(\hat n_i-n_i)|s| \leq-\eta |s|<0. $

According to Lemma 2.6, $ s(t)\rightarrow 0 $, so we get $ e_i(t)\rightarrow 0. $

In this section, the example is provided to verify the effectiveness of the proposed method. The systems appears chaos attractors, when

$ \begin{eqnarray*} &&\alpha = 50, \beta = 20, \gamma = 4.1, a = 5, b = 9, q = 0.873, \\ &&\Delta f_1(y) = \cos(2\pi y_2 ), \Delta f_2(y) = 0.5\cos(2\pi y_3 ), \Delta f_3(y) = 0.3\cos(2\pi y_2 ), \end{eqnarray*} $

the disturbance is bounded

$ \begin{eqnarray*} &&d_1(t) = 0.2\cos t , d_2(t) = 0.6\sin t, d_3(t) = \cos 3t, \\ &&(\hat m_1, \hat m_2, \hat m_3) = (0.3, 0.5, 1), (\hat n_1, \hat n_2, \hat n_3) = (0.8, 0.6, 0.3). \end{eqnarray*} $

From Figure 1, we see that the systems aren't getting synchronization without controller. From Figure 2, we see the systems getting rapidly synchronization with controller. From Figure 3, we see that the errors approaching zero, which verifies the systems getting chaos synchronization rapidly.

In Theorem 2.8, $ g_1(t) = \cos(2\pi y_2 )+0.2\cos t, g_2(t) = 0.5\cos(2\pi y_3 )+0.6\sin t, g_3(t) = 0.3\cos(2\pi y_2 )+\cos 3t, \eta = 2.5 . $ The uncertainty and outer disturbance as Theorem 2.9, $ \eta = 3, q = 0.873, $ the systems errors as Figure 4.

In this paper, we study the self-adaptive sliding mode synchronization problem of a class of fractional-order Victor-Carmen systems based on fractional-order calculus. The conclusion indicates that the systems are self-adaptive synchronization if designing appropriate controller and sliding mode function. We give out the strict proof in mathematics, and the numerical simulation demonstrates the effectiveness of the proposed method.