Bifurcation and synchronization of a new fractional-order system

Abstract

In this paper, bifurcation and adaptive synchronization of a new fractional-order system are investigated. Firstly, bifurcations for the system with the variation of a system parameter and a derivative order are studied and presented in a three-dimensional space. The dynamics of the system changes from regular to chaos with an increase of derivative orders. It is found that three derivative orders have different contributions to a critical total order for the system to change from regular to chaos, meaning that the nonlinearity in the system downgrades the critical order for transitions to chaos. By the analysis of bifurcations in a two-parameter space, we obtain that the minimal order is 2.62 for the system to remain chaos. Secondly, the entropy is analyzed to measure the level of chaos present in the fractional-order system. Finally, based on the stability theory of fractional-order systems, synchronization of the fractional-order system with fully uncertain parameters is realized by designing appropriate adaptive controllers and estimation laws. Numerical simulations are implemented to demonstrate the effectiveness and flexibility of the synchronization controllers and the estimation laws for the unknown parameters.

1 Introduction

It is well known that fractional calculus is not a new word due to almost the same long history with classical calculus. At the beginning of development, it is very slow because of lack of the geometrical interpretation and applications. In recent decades, researchers gradually realize that fractional calculus has the superior characteristics over conventional calculus in the modeling dynamics of natural phenomena [1]. In the field of nonlinear dynamics, it has become an important and hot topic of research, and been applied in many related fields [2,3,4,5,6]. Although chaos was first found and investigated in integer-order systems, its existence and characteristic in fractional-order systems has also been a subject of interest. Therefore, many fractional-order systems were proposed and investigated in detail, such as the fractional-order Duffing system, Chua circuit, Lorenz system, Chen system, and so on [7,8,9,10,11,12]. Compared with integer-order systems, this kind of systems has richer and more complex dynamics.

Dynamic analysis is a hot topic in nonlinear science, and concerns stability of equilibrium, symmetry, chaos, bifurcation, and so on. In which, bifurcation includes the changes in qualitative or topological structures of limiting motions with the variation of some system parameters or some derivative order. For a fractional-order system, investigation of bifurcation becomes complex due to the long memory property of fractional calculus. In Refs.[13,14,15,16,17,18], several kinds of bifurcations have been studied for fractional-order systems. However, these works mainly consider the bifurcation with the variation of a system parameter or a derivative order. As far as we know, few works reports the analysis of bifurcations in a two-parameter space for a fractional-order system [19, 20, 28, 35].

Pecora and Carroll proposed synchronization firstly in 1990 [21]. And since then synchronization has been a hot topic in nonlinear dynamics. For integer-order nonlinear systems, synchronization has been investigated in detail and applied to secure communication. Meanwhile, a great number of synchronization methods, such as master slave method, adaptive control, active control, and back-stepping approach, have been analyzed in detail [22,23,24,25]. It is well known that fractional-order nonlinear systems can provide high security in secure communication for the complexity [26, 27]. Therefore, it is very meaningful to investigate the synchronization for a new fractional-order system.

In [28], we proposed a new fractional-order system with two system parameters. And bifurcations of the system were studied when a parameter and a derivative order are varied in commensurate-order and incommensurate-order cases in detail. Based on this, a new system parameter is introduced into the system of the Ref. [28] in this paper. Therefore, we obtain a more flexible fractional-order system with three parameters. Firstly, bifurcations for the system with the variation of a system parameter and a derivative order are studied in a three-dimensional space. The dynamics of the system changes from regular to chaos with an increase of derivative orders. Meanwhile, the results represent that the three derivative orders of the system are very important parameters which have different influence intensity for the dynamics of the fractional-order system. The entropy is analyzed to measure the level of chaos for the system. Adaptive synchronization for the fractional-order system is realized according to the stability theory.

This paper is structured as follows: the definitions for fractional calculus and related preliminaries are introduced in Sect. 2. A fractional-order system with three system parameters is presented in Sect. 3. Bifurcations of the system as a parameter and a derivative order vary are analyzed. The entropy analysis for the system is included in Sect. 4. Section 5 contains the analysis of adaptive synchronization for the system with uncertain parameters. Some summarizing remarks of the last section conclude the paper.

2 Fractional derivatives and preliminaries

In this subsection, the operator and three definitions of fractional calculus will be introduced. The operator ${}_{a}D_{t}^{q}$ of fractional calculus is defined as follows [1, 29]:

$$ {}_{a}D_{t}^{q} = \left\{ \begin{gathered} \frac{{d^{q} }}{{dt^{q} }}\quad R(q) > 0 \hfill \\ 1\quad R(q) = 0 \hfill \\ \int_{a}^{t} {(d\tau )^{q} \quad R(q) < 0} \hfill \\ \end{gathered} \right. $$

(1)

here, $q$ is the fractional order, $R(q)$ the real part of the order $q$, $a$ and $t$ are the limits of the operator.

For the definitions of fractional calculus, the Grunwald–Letnikov definition, the Riemann–Liouville and the Caputo definitions are employed frequently[1, 29]. The specific definitions will be given in the following.

The Grunwald–Letnikov definition (GL) derivative with fractional order $q$ is given by

$$ {}_{a}^{{{\text{GL}}}} D_{t}^{q} = \mathop {\lim }\limits_{h \to 0} \frac{1}{{h^{q} }}\sum\limits_{j = 0}^{{\left[ {\frac{t - a}{h}} \right]}} {( - 1)^{q} } \left( {\begin{array}{*{20}c} q \\ j \\ \end{array} } \right)f(t - jh) $$

(2)

here, the symbol $\left[ \cdot \right]$ denotes the integer part.

The Riemann–Liouville (RL) definition is defined as:

$$ {}_{a}^{{{\text{RL}}}} D_{t}^{q} f(t) = \frac{{d^{n} }}{{dt^{n} }}\frac{1}{\Gamma (n - q)}\int_{a}^{t} {\frac{f(\tau )}{{(t - \tau )^{q - n + 1} }}} d\tau, \quad n - 1 < q < n, $$

(3)

here, $\Gamma ()$ is Euler’s gamma function.

The Caputo (C) definition is:

$$ {}_{a}^{C} D_{t}^{q} f(t) = \frac{1}{\Gamma (n - q)}\int_{a}^{t} {(t - \tau )^{n - q - 1} f^{(n)} (\tau )} d\tau, \quad n - 1 < q < n. $$

(4)

Because the initial condition has the same form as those for the integer-order ones, the Caputo definition will be used in this paper.

The following lemma can be used to analyze the stability of equilibrium points for a fractional-order system.

Lemma 1

([29]) For a commensurate fractional-order system, the equilibrium points of the system are asymptotically stable if all the eigenvalues at the equilibrium ${\text{E}}^{*}$, satisfy the following condition:

$$ \left| {\arg ({\text{eig}}({\mathbf{J}}))} \right| = \left| {\arg (\lambda_{i} )} \right| > \frac{\pi }{2}q, \, j = 1,2, \cdots ,n, $$

(5)

where ${\mathbf{J}}$ is the Jacobian matrix of the system evaluated at the equilibria ${\text{E}}^{*}$.

The remarkable property of the operator for fractional calculus is long memory. This property makes it difficult to compute the numerical solutions for fractional differential equations. Generally speaking, there are two main numerical computation methods for determining numerical solutions for this kind of systems, that is, an improved version of Adams–Bashforth-Moulton algorithm based on the predictor-correctors scheme and the frequency domain approximation [30,31,32,33]. Considering the nonlinearity of the fractional-order system in this paper and the accuracy of the numerical solutions, the improved predictor–corrector algorithm will be used in the rest of the paper [34, 35].

3 Bifurcations of the fractional-order system

Firstly, a new fractional-order system with three parameters, which contains two linear equations (first two equations) and one nonlinear equation (third equation), is proposed based on the Ref.[28]. It can be described by the following fractional differential equations:

$$ \left\{ \begin{gathered} D^{{q_{{_{1} }} }} x = ax - y \hfill \\ D^{{q_{{_{2} }} }} y = x - z - by \hfill \\ D^{{q_{{_{3} }} }} z = ax + 4z(y - c) \hfill \\ \end{gathered} \right. $$

(6)

here, $x,y,z$ are state variables, $a,b,c$ the system parameters, and $q_{1} ,q_{2} ,q_{3}$ derivative orders. When $q_{1} = q_{2} = q_{3} = q = 0.99$ and $a = 0.3,b = 0.02,c = 2$, the corresponding Lyapunov exponents are $\lambda_{1} = 0.117,\lambda_{2} = 0,\lambda_{3} = - 10.49$. It means the system (6) is chaotic, and the chaotic attractor is depicted in Fig. 1.

Fig. 1

The chaotic attractor. a the attractor in three-dimensional space; b the attractor in $x - y$ plane

In the following, the bifurcation of the system (6) when a parameter and an order are varied simultaneously is studied with the initial condition (− 0.5,1,1). The time for numerical simulation is 300, and the step size is fixed as 0.005.

3.1 Bifurcation with the variation of the parameter $a$ and a derivative order

When the other parameters are fixed as $b = 0.02,c = 2$ and the other two orders $q_{2} = q_{3} = 1$, the bifurcation diagram with the variation of the system parameter $a$ and the derivative order $q_{1}$ is depicted, see Fig. 2. The interval of the order $q_{1}$ is [0.86, 0.99] and the step size is set as 0.01. From Fig. 2, we can see that chaotic motion appears firstly in the system (6) when the order $q_{1} { = }0.91$ and the parameter $a$ varies from 0.1 to 0.3. It implies the total order for the system to remain chaos is 2.91 in this case. Furthermore, the system only exists period-1 solutions for $q_{1} \le 0.87$, period-2 solutions for $q_{1} { = }0.88$, and period-4 for $q_{1} { = }0.90$.

Fig. 2

The bifurcation diagram of the system (6) when the parameter $a \in [0.1,0.3]$ and the order $q_{1} \in [0.86,0.99]$

Form Fig. 2, it is clear that the period-1 solutions whose orbit likes a tail continues for a long duration. The area of the period-1 solutions becomes smaller and smaller while the region of chaotic motion becomes larger and larger gradually as the order $q_{1}$ increases. Chaotic motion and periodic windows appear alternately as the order $q_{1}$ increases from 0.94 to 0.99. Based on these, it can be determined that complexity of the dynamics for the system (6) enhances as the derivative order $q_{1}$ approaches to 1. In other words, the dynamics of the system becomes complex as the derivative order $q_{1}$ increases, and regular as the order $q_{1}$ decreases. Besides, it can be seen that the route to chaos is the typical period-doubling bifurcation in this case.

Figure 3 displays the bifurcation of the system when the parameter $a$ and the derivative order $q_{2}$ are varied with the other two orders $q_{1} = q_{3} = 1$. The interval of the order $q_{2}$ is [0.86, 0.99] and the step size is set as 0.01. From the figure, it is clear that chaotic motion appears in the system (6) when the order $q_{2} { = }0.94$ and the parameter $a$ varies from 0.1 to 0.3. It means the total order for the system to remain chaos is 2.94 in this case, see the fourth diagram in the Fig. 3. Besides, the system only exists period-1 solutions when the order $q_{2} \le 0.88$, period-2 solutions when the order $q_{2} { = }0.89$, and period-4 when the order $q_{2} { = }0.92$. The system (6) is complete periodic for $q_{2} \le 0.93$, and typical evolution of period-doubling can also be obtained from the first seven bifurcation diagrams in the Fig. 3. The region of the chaos increases while the area of the periodic motions decreases when the order $q_{2}$ continues to increase. Meanwhile, chaotic motion and periodic windows appear alternately as the order $q_{2}$ increases to 0.99.

Fig. 3

The bifurcation diagram of the system (6) when the parameter $a \in [0.1,0.3]$ and the order $q_{2} \in [0.87,0.99]$

The bifurcation diagram as the parameter $a$ and derivative order $q_{3}$ vary when the other two orders $q_{1} = q_{2} = 1$ is plotted in Fig. 4. The interval of the order $q_{3}$ is [0.50, 0.99] and the step size is set as 0.05. The similar phenomena can also be observed from the figure. In order to obtain the order for the system to remain chaos, Fig. 5 is plotted to show the appearance of the chaos for the first time. From which we can get the total order of the system to remain chaos is 2.66 in this case.

Fig. 4

The bifurcation diagram of the system (6) when the parameter $a \in [0.1,0.3]$ and the order $q_{3} \in [0.50,0.99]$

Fig. 5

The bifurcation diagram of the system (6) when the order $q_{3}$ increases from 0.64 to 0.66

By comparing the Figs. 2, 3 and 4, we can get that the dynamics of the system (6) with the variation of the order $q_{3}$ is more complex than those of the system with the variation of the orders $q_{1}$ or $q_{2}$. It means that the influence intensity of the order $q_{3}$ for the complexity of the dynamics of the system (6) is larger than those of the orders $q_{1}$ or $q_{2}$ when the system parameter $a$ is varied.

3.2 Bifurcation with the variation of the parameter $b$ and a derivative order

When the other parameters are fixed as $a = 0.3,c = 2$ and the orders $q_{2} = q_{3} = 1$, bifurcation with the variation of the system parameter $b$ and the derivative order $q_{1}$ is studied, see Fig. 6. The interval of the order $q_{1}$ is [0.84, 0.99] and the step size is set as 0.01. From Fig. 6, we can see that chaotic motion appears in the system (6) when the order $q_{1} { = }0.88$ and the parameter $b$ varies from -0.03 to 0.02, which implies the total order for the system to remain chaos is 2.88 in this case. Furthermore, the system only exists period-1 solutions for the order $q_{1} \le 0.84$, period-2 solutions for the order $q_{1} { = }0.85$, and period-4 for the order $q_{1} { = }0.87$. The system (6) exists the period-3 solutions when the order $0.91 \le q_{1} \le 0.94$. The typical period-doubling bifurcations can be found from them. Chaotic motions and periodic windows appear alternately as the order $q_{1}$ increases.

Fig. 6

The bifurcation diagram of the system (6) when the parameter $b \in [ - 0.03,0.02]$ and the order $q_{1} \in [0.84,0.99]$

When the other orders are taken as $q_{1} = q_{3} = 1$, Fig. 7 plots the bifurcation of the system with the variation of the parameter $b$ and the derivative order $q_{2}$. The interval of the order $q_{2}$ is [0.84, 0.99] and the step size is set as 0.01. The chaotic motion for the system is firstly observed when the order $q_{2} { = }0.93$, which means the total order for the system to remain chaos is 2.93 in this case. From the figure, it can be seen that the system (6) is complete periodic for $q_{2} \le 0.92$, and the typical period-1 and period-2 solutions can be observed from the first nine bifurcation diagrams in the Fig. 7. The dynamics of the system is almost chaotic as the order increases from 0.96 to 0.97. As the order $q_{2}$ continues to increase, the chaotic motion and periodic windows appear alternately.

Fig. 7

The bifurcation diagram of the system (6) when the parameter $b \in [ - 0.03,0.02]$ and the order $q_{2} \in [0.84,0.99]$

When the other orders $q_{1} = q_{2} = 1$, the bifurcation with the variation of the parameter $b$ and the derivative order $q_{3}$ is studied, see Fig. 8. The interval of the order $q_{3}$ is [0.50, 0.99] and the step size is set as 0.05. From Fig. 8, we can see that the dynamics of the system (6) is periodic completely when the order $q_{3}$ increases from 0.5 to 0.65. The chaotic motion and periodic windows appear alternately as the order $q_{3}$ increases from 0.7 to 0.99. Figure 9 is depicted to show the critical order which the chaotic motion appears firstly for the system. From which we can get the total order for the system to remain chaos is 2.70 in this case.

Fig. 8

The bifurcation diagram of the system (6) when the parameter $b \in [ - 0.03,0.02]$ and the order $q_{3} \in [0.50,0.99]$

Fig. 9

The bifurcation diagram of the system (6) when the order $q_{3}$ increases from 0.66 to 0.70

Comparing the Figs. 6, 7 and 8, we can see that the dynamics of the system (6) with the variation of the order $q_{3}$ is more complex than those of the system with the variation of the orders $q_{1}$ or $q_{2}$, which is similar to the case of the variation of the system parameter $a$.

3.3 Bifurcation with the variation of the parameter $c$ and a derivative order

When the other parameters are taken as $a = 0.3,b = 0.02$ and the orders $q_{2} = q_{3} = 1$, Fig. 10 displays the bifurcation of the system (6) as the parameter $c$ and the derivative order $q_{1}$ vary. The interval of the order $q_{1}$ is [0.84, 0.99] and the step size is set as 0.01. From the figure, we can see that chaotic motion appears in the system (6) when the order $q_{1} { = }0.88$, which means the total order for the system to remain chaos is 2.88 in this case. Furthermore, the system only exists period-1 and period-2 solutions for the order $q_{1} { = }0.84$ and $q_{1} { = }0.85$, and period-4 solutions for $q_{1} { = }0.86$. Besides, period-doubling and tangent bifurcations can be obtained from these bifurcation diagrams in Fig. 10. Meanwhile, compared the every diagram in the figure, we can see that the region of chaos motions increases while the area of periodic windows decreases as the order $q_{1}$ changes from 0.84 to 0.99.

Fig. 10

The bifurcation diagram of the system (6) when the parameter $c \in [1,10][1,10]$ and the order $q_{1} \in [0.84,0.99]$

The bifurcation with the variation of the system parameter $c$ and the derivative order $q_{2}$ is investigated when the other orders $q_{1} = q_{3} = 1$, see Fig. 11. The interval of the order $q_{1}$ is [0.85, 0.99] and the step size is set as 0.01. The system (6) is complete periodic when the order $q_{2} \le 0.93$. Besides, the chaotic motion is firstly observed when the order, which means the total order for the system to remain chaos is 2.94 in this case. As the order $q_{2}$ continues to increase, more and more chaotic motion can be seen from the figure.

Fig. 11

The bifurcation diagram of the system (6) when the parameter $c \in [1,10]$ and the order $q_{2} \in [0.85,0.99]$

Figure 12 displays the bifurcation of the system (6) as the system parameter $c$ and the derivative order $q_{3}$ vary when the other orders $q_{1} = q_{2} = 1$. The interval of the order $q_{3}$ is fixed as [0.55, 0.99] and the step size is 0.05. From the Fig. 12, it can be seen that the dynamics of the system (6) is periodic when the order $q_{3} \le 0.6$, and chaotic when the order $q_{3}$ increases from 0.65 to 0.99. Figure 13 is plotted to show the appearance of chaos firstly for the system. From which we can get the total order for the system to remain chaos is 2.62 in this case.

Fig. 12

The bifurcation diagram of the system (6) when the parameter $c \in [1,10]$ and the order $q_{3} \in [0.55,0.99]$

Fig. 13

The bifurcation diagram of the system (6) when the order $q_{3}$ increases from 0.60 to 0.62

Compared Figs. 10, 11 and 12, the dynamics of the system (6) with the variation of the order $q_{3}$ is more complex than those of the system (6) with the variation of the orders $q_{1}$ or $q_{2}$. It can be determined that the influence intensity of the order $q_{3}$ for the complexity of the dynamics of the system (6) is larger than those of the orders $q_{1}$ or $q_{2}$ when the system parameter $c$ is varied.

By comparing Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, we can conclude that dynamics of the system (6) becomes regular as a derivate order decreases, and complex as a derivative order increases. It is found that the three derivative orders are very important parameters which have different contributions to a critical total order for the system to change from regular to chaos. It means that the nonlinearity in the system (6) downgrades the critical order for transitions to chaos. A table is given to show the critical order of the system (6) remains chaos in every case, see Table 1. From which, we can obtain the minimal order of the system to remain chaos is 2.62. In particular, the obtained value of the orders reflect that the nonlinear differential equation (third equation) in the system (6) seems to be more “sensitive” to the critical total order introduced by a fractional derivative than two other linear equations [12], under which the system undergoes a transition from chaotic motion to regular one.

Table 1 Total order of the system (6) remains chaos in every case

4 Entropy analysis

In information theory, entropy can quantify the rate of transfer or generation of information in a dynamical system [36]. Entropy analysis is an effective way of quantifying the level of chaos for a particular system. The main idea of entropy is to measure the information generated in a sequence. Generally speaking, the exact entropy for a dynamical system is difficult to determine. Therefore, the approximate entropy (ApEn for short) is used frequently. In [37], an important approximate entropy measure was presented to indicate the complexity for a dynamical system by a non-negative value. The calculation process of the ApEn is shown as follows:

• Step 1: A time-series $x(1),x(2), \cdots ,x(N)$ is divided into an m-dimension vector

  $$ X(i) = (x(i),x(i + 1), \cdots ,x(i + m + 1)) $$

  (7)

  where $1 \le i \le N - m + 1$.

• Step 2: The distance $d(x(i),x(j))$ is defined as

  $$ d(x(i),x(j)) = \mathop {\max }\limits_{k = 1,2, \cdots ,m} (\left| {x(i + k - 1) - x(j + k - 1)} \right|) $$

  (8)

• Step 3: Setting a threshold value $r > 0$, we can get the statistics $C_{i}^{m} (r)$ for $1 \le i \le N - m + 1$,

  $$ C_{i}^{m} (r) = \frac{number \, of \, j \, such \, that \, d[x(i),x(j)] \le r}{{N - m + 1}} $$

  (9)

• Step 4: The mean of logarithm of $C_{i}^{m} (r)$ is denoted as $\phi^{m} (r)$, which can be described as

  $$ \phi^{m} (r) = \frac{1}{N - m + 1}\sum\limits_{i = 1}^{N - m + 1} {\log C_{i}^{m} (r)} $$

  (10)

• Step 5: Repeating the steps 1 to 4, the ApEn can be determined

  $$ {\text{ApEn}}(m,r) = \mathop {\lim }\limits_{N \to \infty } (\phi^{m} (r) - \phi^{m + 1} (r)) $$

  (11)

For $N$ data points, formula (11) can be changed into

$$ {\text{ApEn}}(m,r,N) = \phi^{m} (r) - \phi^{m + 1} (r) $$

(12)

When $N{ = }2000$ and $m = 2$, the ApEn for the fractional-order system (6) with different values of fractional order in commensurate-order case $(q_{1} = q_{2} = q_{3} = q)$ when $a = 0.3,b = 0.02,c = 2$ are listed in Table 2. From which, it is clear that the ApEn decreases as the order changes from 1 to 0.5, which implies that the complexity of the system (6) decreases. The result matches with the conclusion in Sect. 3, namely, dynamics of the system (6) becomes regular as a derivate order decreases, and complex as a derivative order increases.

Table 2 ApEn calculation of the system (5) with different values of the derivative order

5 Synchronization

Adaptive synchronization for the system (6) with unknown parameters will be investigated in this section. Firstly, the scheme of adaptive synchronization is given here. The drive system is taken as

$$ D^{q} x = f(x) + F(x)\alpha $$

(13)

where $x \in R^{n}$ is the state variable vector, $q \in R^{n}$ is the vector of derivative orders, $\alpha \in R^{m}$ is the uncertain parameters vector.$f(x)$ is an $n \times 1$ matrix, and $F(x)$ is an $n \times m$ matrix. The response system is denoted by the following fractional differential equations:

$$ D^{q} y = f(y) + F(y)\tilde{\alpha } + u $$

(14)

where $y \in R^{n}$ is the state vector, $\tilde{\alpha } \in R^{m}$ is the estimations vector of the unknown parameters, $u$ is the adaptive synchronization controller.

We define the synchronization error vector $e = y - x$, design the synchronization controller as $u = f(x) + F(x)\tilde{\alpha } - f(y) - F(y)\tilde{\alpha } - ke = - F(e)\tilde{\alpha } - f(e) - ke$ ($k$ is a positive constant), and choose the adaptive identification laws for unknown parameters as $D^{q} \tilde{\alpha } = - \left[ {F(x)} \right]^{T} e$. Based on this, the trajectory of the response system (14) with an initial conditions $y_{0}$ can asymptotically approaches the drive system (13) with an initial condition $x_{0}$, namely, $\mathop {\lim }\limits_{t \to \infty } \left\| e \right\| = \mathop {\lim }\limits_{t \to \infty } \left\| {y(t,y_{0} ) - x(t,x_{0} )} \right\| = 0$, and $\left\| \cdot \right\|$ denotes the Euclidean norm.

The system (6) is taken as the drive system, and can be rewritten as follows:

$$ \left\{ \begin{gathered} D^{q} x_{1} = ax_{1} - x_{2} \hfill \\ D^{q} x_{2} = x_{1} - x_{3} - bx_{2} \hfill \\ D^{q} x_{3} = ax_{1} + 4x_{3} (x_{2} - c) \hfill \\ \end{gathered} \right. = f(x) + F(x)\alpha $$

(15)

where $f(x) = [ - x_{2} , - x_{1} - x_{3} ,4x_{2} x_{3} ]^{T}$, $F(x) = \left[ {\begin{array}{*{20}c} {x_{1} } & 0 & 0 \\ 0 & { - x_{2} } & 0 \\ {x_{1} } & 0 & { - 4x_{3} } \\ \end{array} } \right]$, $x{ = }\left[ {x_{1} ,x_{2} ,x_{3} } \right]^{T}$ is the state variables, vector, $\alpha { = }\left[ {a,b,c} \right]^{T}$ denotes the unknown parameters vector.

The corresponding response system is

$$ \left\{ \begin{gathered} D^{q} y_{1} = \tilde{a}y_{1} - y_{2} + u_{1} \hfill \\ D^{q} y_{2} = y_{1} - y_{3} - \tilde{b}y_{2} + u_{2} \hfill \\ D^{q} y_{3} = \tilde{a}y_{1} + 4y_{3} (y_{2} - \tilde{c}) + u_{3} \hfill \\ \end{gathered} \right. = f(y) + F(y)\tilde{\alpha } + u $$

(16)

where $f(y) = [ - y_{2} , - y_{1} - y_{3} ,4y_{2} y_{3} ]^{T}$, $F(y) = \left[ {\begin{array}{*{20}c} {y_{1} } & 0 & 0 \\ 0 & { - y_{2} } & 0 \\ {x_{1} } & 0 & { - 4y_{3} } \\ \end{array} } \right]$, $\tilde{\alpha } = \left[ {\tilde{a},\tilde{b},\tilde{c}} \right]^{T}$ denotes the estimations vector of the unknown parameters $a,b,c$, and $u = \left[ {u_{1} ,u_{2} ,u_{3} } \right]^{T}$ is the synchronization controller vector.

Now, the synchronization errors vector is denoted as $e = [e_{1} ,e_{2} ,e_{3} ]^{T} = [y_{1} - x_{1} ,y_{2} - x_{2} ,y_{3} - x_{3} ]^{T}$, and the estimation errors vector of uncertain parameters is defined as $e_{\alpha } = [e_{a} ,e_{b} ,e_{c} ]^{T} = [\tilde{a} - a,\tilde{b} - b,\tilde{c} - c]^{T}$. By simple computation, we can obtain the error dynamical system

$$ \left\{ \begin{gathered} D^{q} e_{1} = \tilde{a}e_{1} + e_{a} x_{1} - e_{2} + u_{1} \hfill \\ D^{q} e_{2} = e_{1} - e_{3} - \tilde{b}e_{2} - e_{b} y_{1} + u_{2} \hfill \\ D^{q} e_{3} = \tilde{a}e_{1} + e_{a} x_{1} + 4z_{2} e_{2} + 4e_{3} y_{1} - 4\tilde{c}e_{3} - 4e_{c} z_{1} + u_{3} \hfill \\ \end{gathered} \right. $$

(17)

The synchronization between the drive and response systems (15) and (16) can be realized by designing the suitable synchronization controllers and parameter estimation rules. Hence, the following theorem is given to ensure the drive system effectively synchronizes the response system.

Theorem 1

The drive system (15) can adaptive synchronizes the response system (16) when the controllers $u_{1} ,u_{2} ,u_{3}$ and laws of the unknown parameters $e_{a} ,e_{b} ,e_{c}$ are defined as follows:

$$ \begin{aligned} u & = f(x) + F(x)\tilde{\alpha } - f(y) - F(y)\tilde{\alpha } - ke = - F(e)\tilde{\alpha } - f(e) - ke \\ & = \left[ {\begin{array}{*{20}c} {u_{1} } \\ {u_{2} } \\ {u_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \tilde{a}e_{1} + e_{2} - e_{1} } \\ { - e_{1} + e_{3} + \tilde{b}e_{2} - e_{2} } \\ { - \tilde{a}e_{1} - 4y_{3} e_{2} - 4e_{3} x_{2} + 4\tilde{c}e_{3} } \\ \end{array} } \right] \\ \end{aligned} $$

(18)

$$ D^{q} \tilde{\alpha } = D^{q} e_{\alpha } = - \left[ {F(x)} \right]^{T} e = \left[ {\begin{array}{*{20}c} { - x_{1} (e_{1} + e_{3} )} & {x_{2} e_{2} } & {4x_{3} e_{3} } \\ \end{array} } \right]^{T} $$

(19)

Proof

The controllers (18) are substituted into the system (17), and then the error dynamical system can be rewritten as:

$$ \left\{ \begin{gathered} D^{q} e_{1} = e_{a} x_{1} - e_{1} \hfill \\ D^{q} e_{2} = - e_{b} x_{2} - e_{2} \hfill \\ D^{q} e_{3} = e_{a} x_{1} - 4e_{c} x_{3} - e_{3} \hfill \\ \end{gathered} \right. $$

(20)

By combining (19) and (20), we get

$$ \left( {D^{q} e_{1} ,D^{q} e_{2} ,D^{q} e_{3} ,D^{q} e_{a} ,D^{q} e_{b} ,D^{q} e_{c} } \right)^{{\text{T}}} = {\mathbf{A}}\left( {e_{1} ,e_{2} ,e_{3} ,e_{a} ,e_{b} ,e_{c} } \right)^{{\text{T}}} $$

(21)

here, ${\mathbf{A}} = \left[ {\begin{array}{*{20}c} {{\mathbf{ - I}}} & {{\mathbf{A}}_{{\mathbf{1}}} } \\ {{\mathbf{ - A}}_{1}^{{\text{T}}} } & {\mathbf{O}} \\ \end{array} } \right],{\mathbf{I}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right],{\mathbf{A}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} {x_{1} } & 0 & 0 \\ 0 & { - x_{2} } & 0 \\ {x_{1} } & 0 & { - 4x_{3} } \\ \end{array} } \right],{\mathbf{O}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]$.

We assume that $\lambda$ is one of eigenvalues of matrix ${\mathbf{A}}$, and ${{\varvec{\upxi}}} = \left( {\xi_{1} ,\xi_{2} ,\xi_{3} ,\xi_{4} ,\xi_{5} ,\xi_{6} } \right)^{{\text{T}}}$ is the corresponding nonzero eigenvector, then we can get

$$ A{\varvec{\upxi }} = \lambda {\mathbf{\xi }} $$

(22)

Based on this, the following relation can be obtained when the conjugate transpose is taken on both sides of (22)

$$ \overline{{({\mathbf{A}}{\varvec{\upxi }})^{{\text{T}}} }} = \overline{\lambda }{{\varvec{\upxi}}}^{{\text{H}}} $$

(23)

By multiplying left (22) by $\frac{1}{2}{{\varvec{\upxi}}}^{{\text{H}}}$ plus (23) multiplied right by $\frac{1}{2}{{\varvec{\upxi}}}$, we can get

$$ {{\varvec{\upxi}}}^{{\text{H}}} \left( {\frac{1}{2}{\mathbf{A}} + \frac{1}{2}{\mathbf{A}}^{{\text{H}}} } \right){{\varvec{\upxi}}} = \frac{1}{2}\left( {\lambda + \overline{\lambda }} \right){{\varvec{\upxi}}}^{{\text{H}}} {{\varvec{\upxi}}} $$

(24)

Then

$$ \frac{1}{2}\left( {\lambda + \overline{\lambda }} \right) = \frac{{{{\varvec{\upxi}}}^{{\text{H}}} \left( {\frac{1}{2}{\mathbf{A}} + \frac{1}{2}{\mathbf{A}}^{{\text{H}}} } \right){{\varvec{\upxi}}}}}{{{{\varvec{\upxi}}}^{{\text{H}}} {{\varvec{\upxi}}}}} $$

(25)

By substituting the matrix ${\mathbf{A}}$ into (25), we have

$$ \frac{1}{2}\left( {\lambda + \overline{\lambda }} \right) = \frac{{{{\varvec{\upxi}}}^{{\text{H}}} {\mathbf{B\xi }}}}{{{{\varvec{\upxi}}}^{{\text{H}}} {{\varvec{\upxi}}}}} $$

(26)

where ${\mathbf{B}} = \left[ {\begin{array}{*{20}c} { - {\mathbf{I}}} & {\mathbf{O}} \\ {\mathbf{O}} & {\mathbf{O}} \\ \end{array} } \right]$.

Therefore, any eigenvalue of the matrix ${\mathbf{A}}$ satisfies the relationship $\left| {\arg (\lambda )} \right| \ge \frac{\pi }{2} > \frac{\pi }{2}q,{ (0 < }q < 1{)}$ for $\lambda + \overline{\lambda } \le 0$[38]. Based on the stability theory of fractional-order systems (Lemma 1), the zero equilibrium of error dynamical system (21) is asymptotically stable, which means the adaptive synchronization between the drive and response systems (15) and (16) is realized. The proof is completed.

In numerical simulations, the derivative orders are fixed as $q = 0.99$, and the real values of the unknown parameters are $a = 0.3,b = 0.02,c = 2$. The initial conditions of the systems (15) and (16) are $(1,2,3)$ and $(3,2,1)$, respectively. The simulation results are depicted in Figs.14 and 15. In Fig. 15, the red denotes the phase trajectory of the drive system (15), and blue the phase trajectory of the response system (16). From the two figures, we can see clear that the errors variables $e_{1} ,e_{2}$ and $e_{3}$ converge to zero, the estimations converge to the real values of the unknown parameters as $t \to 100 \,$, and the phase trajectories of the drive and response systems evolve synchronously. It implies that the synchronization between the two systems is realized under the controllers.

Fig. 14

The result of numerical simulations. a the curve of synchronization error $e_{1}$; b the curve of synchronization error $e_{2}$; c the curve of synchronization error $e_{3}$; d the curve of estimations for the unknown parameters

Fig. 15

The evolution of the state variables for the drive and response systems

6 Conclusions

In this paper, a new fractional-order system with three system parameters which has a one-scroll chaotic attractor is proposed. Bifurcations for the fractional-order system with the variation simultaneously of a system parameter and a derivative order are investigated in detail. In a three-dimensional space, different bifurcation diagrams are displayed and compared. Form which, it can be seen that the system occurs a series of period-doubling and tangent bifurcations. Meanwhile, the dynamics of the system becomes regular as the derivate order decreases, and complex as the derivate order increases. Besides, the results represent that three derivative orders have different contributions to a critical total order for the system to change from regular to chaos, meaning that the nonlinearity in the system downgrades the critical order for transitions to chaos. By the analysis of bifurcations in a two-parameter space, we obtain that the minimal order is 2.62 for the system to remain chaos. The entropy is analyzed to measure the level of chaos present in the fractional-order system.

Adaptive synchronization of the fractional-order system with fully unknown parameters is realized based on the stability theory of fractional-order systems. The designed synchronization controllers and estimation laws guarantee the convergence of all error states toward zero. Numerical simulations are presented to confirm the effectiveness and verify the flexibility of the synchronization controllers and the estimation laws for the unknown parameters.