3. Switched Generalized Function Projective Synchronization between Two Hyperchaotic Systems

In this section, we investigate the adaptive SGFPS between systems (4) and (5) with fully unknown parameters.

Suppose that system (4) is the drive system whose four variables are denoted by subscript 1 and system (5) is the response system whose variables are denoted by subscript 2. Then the drive and response systems are described by the following equations, respectively,

$$\begin{cases} \dot{x}\_1 = a\_1(y\_1 - x\_1) \\ \dot{y}\_1 = b\_1 x\_1 - x\_1 z\_1 - c\_1 y\_1 + w\_1 \\ \dot{z}\_1 = x\_1 y\_1 - d\_1 z\_1 \\ \dot{w}\_1 = -k\_1 y\_1 - r\_1 w\_1 \end{cases} \tag{6}$$

and

$$\begin{cases} \dot{x}\_2 = a\_2 x\_2 - y\_2 z\_2 + w\_2 + u\_1 \\\\ \dot{y}\_2 = x\_2 z\_2 - b\_2 y\_2 + u\_2 \\\\ \dot{z}\_2 = x\_2 y\_2 - c\_2 z\_2 + x\_2 w\_2 + u\_3 \\\\ \dot{w}\_2 = -y\_2 + u\_4 \end{cases} \tag{7}$$

where a1, b1, c1, d1, k1, r1, a2, b<sup>2</sup> and c<sup>2</sup> are unknown parameters to be identified, and ui(i = 1, 2, 3, 4) are controllers to be determined such that the two hyperchaotic systems can achieve SGFPS, in the sense that

$$\begin{cases} \lim\_{t \to \infty} \|e\_1\| = \lim\_{t \to \infty} \|x\_2 - \phi\_1(x)z\_1\| = 0\\ \lim\_{t \to \infty} \|e\_2\| = \lim\_{t \to \infty} \|y\_2 - \phi\_2(x)w\_1\| = 0\\ \lim\_{t \to \infty} \|e\_3\| = \lim\_{t \to \infty} \|z\_2 - \phi\_3(x)x\_1\| = 0\\ \lim\_{t \to \infty} \|e\_4\| = \lim\_{t \to \infty} \|w\_2 - \phi\_4(x)y\_1\| = 0 \end{cases} \tag{8}$$

where φi(x)(i = 1, 2, 3, 4) are scaling functions.

So the SGFPS error dynamical system is determined as follows

$$\begin{cases} \dot{e}\_1 = a\_2 x\_2 - y\_2 z\_2 + w\_2 - \dot{\phi}\_1(x) z\_1 - \phi\_1(x) (x\_1 y\_1 - d\_1 z\_1) + u\_1 \\\\ \dot{e}\_2 = x\_2 z\_2 - b\_2 y\_2 - \dot{\phi}\_2(x) w\_1 - \phi\_2(x) (-k\_1 y\_1 - r\_1 w\_1) + u\_2 \\\\ \dot{e}\_3 = x\_2 y\_2 - c\_2 z\_2 + x\_2 w\_2 - \dot{\phi}\_3(x) x\_1 - \phi\_3(x) a\_1 (y\_1 - x\_1) + u\_3 \\\\ \dot{e}\_4 = -y\_2 - \dot{\phi}\_4(x) y\_1 - \phi\_4(x) (b\_1 x\_1 - x\_1 z\_1 - c\_1 y\_1 + w\_1) + u\_4 \end{cases} (9)$$

Without loss of generality, the scaling functions can be chosen as φ1(x) = m11x<sup>1</sup> +m12, φ2(x) = m21y1+m22, φ3(x) = m31z1+m<sup>32</sup> and φ4(x) = m41w1+m42, where mij (i = 1, 2, 3, 4; j = 1, 2) are constant numbers. And substituting systems (6) and (7) into system (9), yields the following form:

$$\begin{cases} \dot{e}\_1 = a\_2 x\_2 - y\_2 z\_2 + w\_2 - m\_{11} a\_1 (y\_1 - x\_1) z\_1 - \phi\_1(x) (x\_1 y\_1 - d\_1 z\_1) + u\_1 \\\\ \dot{e}\_2 = x\_2 z\_2 - b\_2 y\_2 - m\_{21} (b\_1 x\_1 - x\_1 z\_1 - c\_1 y\_1 + w\_1) w\_1 - \phi\_2(x) (-k\_1 y\_1 - r\_1 w\_1) + u\_2 \\\\ \dot{e}\_3 = x\_2 y\_2 - c\_2 z\_2 + x\_2 w\_2 - m\_{31} (x\_1 y\_1 - d\_1 z\_1) x\_1 - \phi\_3(x) a\_1 (y\_1 - x\_1) + u\_3 \\\\ \dot{e}\_4 = -y\_2 - m\_{41} (-k\_1 y\_1 - r\_1 w\_1) y\_1 - \phi\_4(x) (b\_1 x\_1 - x\_1 z\_1 - c\_1 y\_1 + w\_1) + u\_4 \end{cases} \tag{10}$$

Our goal is to find the appropriate controllers ui(i = 1, 2, 3, 4) to stabilize the error variables of system (10) at the origin. For this purpose, we propose the following controllers for system (10)

$$\begin{cases} u\_1 = -\bar{a}\_2 x\_2 + y\_2 z\_2 - w\_2 + m\_{11} \bar{a}\_1 (y\_1 - x\_1) z\_1 + \phi\_1(x) (x\_1 y\_1 - \bar{d}\_1 z\_1) - l\_1 e\_1 \\\\ u\_2 = -x\_2 z\_2 + \bar{b}\_2 y\_2 + m\_{21} (\bar{b}\_1 x\_1 - x\_1 z\_1 - \bar{c}\_1 y\_1 + w\_1) w\_1 + \phi\_2(x) (-\bar{k}\_1 y\_1 - \bar{r}\_1 w\_1) - l\_2 e\_2 \\\\ u\_3 = -x\_2 y\_2 + \bar{c}\_2 z\_2 - x\_2 w\_2 + m\_{31} (x\_1 y\_1 - \bar{d}\_1 z\_1) x\_1 - \phi\_3(x) \bar{a}\_1 (y\_1 - x\_1) - l\_3 e \\\\ u\_4 = y\_2 + m\_{41} (-\bar{k}\_1 y\_1 - \bar{r}\_1 w\_1) y\_1 + \phi\_4(x) (\bar{b}\_1 x\_1 - x\_1 z\_1 - \bar{c}\_1 y\_1 + w\_1) - l\_4 e\_4 \end{cases} \tag{11}$$

where L = diag(l1, l2, l3, l4) is a positive gain matrix for each state controller. In practical applications the synchronization process can be sped up by increasing the gain matrix L.

The update laws for the unknown parameters a1, b1, c1, d1, k1, r1, a2, b<sup>2</sup> and c<sup>2</sup> are given as follows

$$\begin{cases} \dot{a}\_1 = -m\_{11}(y\_1 - x\_1)z\_1e\_1 - \phi\_3(x)(y\_1 - x\_1)e\_3 + (a\_1 - \bar{a}\_1) \\ \dot{b}\_1 = -m\_{21}x\_1w\_1e\_2 - \phi\_4(x)x\_1e\_4 + (b\_1 - \bar{b}\_1) \\ \dot{c}\_1 = \phi\_4(x)y\_1e\_4 + m\_{21}y\_1w\_1e\_2 + (c\_1 - \bar{c}\_1) \\ \dot{d}\_1 = \phi\_1(x)z\_1e\_1 + m\_{31}z\_1x\_1e\_3 + (d\_1 - \bar{d}\_1) \\ \dot{k}\_1 = \phi\_2(x)y\_1e\_2 + m\_{41}y\_1^2e\_4 + (k\_1 - \bar{k}\_1) \\ \dot{r}\_1 = \phi\_2(x)w\_1e\_2 + m\_{41}w\_1y\_1e\_4 + (r\_1 - \bar{r}\_1) \\ \dot{a}\_2 = x\_2e\_1 + (a\_2 - \bar{a}\_2) \\ \dot{\bar{b}}\_2 = -y\_2e\_2 + (b\_2 - \bar{b}\_2) \\ \dot{\bar{c}}\_2 = -z\_2e\_3 + (c\_2 - \bar{c}\_2) \end{cases} \tag{12}$$

where a¯1, ¯b1, c¯1, ¯ d1, ¯ k1, r¯1, a¯2, ¯b<sup>2</sup> and c¯<sup>2</sup> are the estimate values for these unknown parameters, respectively. Then, we have the following main result.

Theorem 1. For a given continuous differential scaling function matrix φ(x) = diag{φ1(x), φ2(x), φ3(x), φ4(x)}, and any initial values, the SGFPS between systems (6) and (7) can be achieved by the adaptive controllers (11) and the parameter update laws (12).

Proof. Choose the following Lyapunov function,

$$\begin{split} V &= \frac{1}{2} (e\_1^2 + e\_2^2 + e\_3^2 + e\_4^2 + (\bar{a}\_2 - a\_2)^2 + (\bar{b}\_2 - b\_2)^2 + (\bar{c}\_2 - c\_2)^2) \\ &+ \frac{1}{2} ((\bar{a}\_1 - a\_1)^2 + (\bar{b}\_1 - b\_1)^2 + (\bar{c}\_1 - c\_1)^2 + (\bar{d}\_1 - d\_1)^2 + (\bar{k}\_1 - k\_1)^2 + (\bar{r}\_1 - r\_1)^2) \end{split} \tag{13}$$

Taking the time derivative of V along the trajectory of the error dynamical system (10) yields

<sup>V</sup>˙ = ˙e1e<sup>1</sup> + ˙e2e<sup>2</sup> + ˙e3e<sup>3</sup> + ˙e4e<sup>4</sup> + (¯a<sup>2</sup> <sup>−</sup> <sup>a</sup>2)a¯˙ <sup>2</sup> + (¯b<sup>2</sup> <sup>−</sup> <sup>b</sup>2)¯˙ <sup>b</sup><sup>2</sup> + (¯c<sup>2</sup> <sup>−</sup> <sup>c</sup>2)c¯˙<sup>2</sup> + (¯a<sup>1</sup> <sup>−</sup> <sup>a</sup>1)a¯˙ <sup>1</sup> + (¯b<sup>1</sup> <sup>−</sup> <sup>b</sup>1)¯˙ <sup>b</sup><sup>1</sup> + (¯c<sup>1</sup> <sup>−</sup> <sup>c</sup>1)c¯˙<sup>1</sup> + ( ¯ <sup>d</sup><sup>1</sup> <sup>−</sup> <sup>d</sup>1) ¯˙ d<sup>1</sup> + (¯ <sup>k</sup><sup>1</sup> <sup>−</sup> <sup>k</sup>1)¯˙ <sup>k</sup><sup>1</sup> + (¯r<sup>1</sup> <sup>−</sup> <sup>r</sup>1)r¯˙<sup>1</sup> =e1(a2x<sup>2</sup> − y2z<sup>2</sup> + w<sup>2</sup> − m11a1(y<sup>1</sup> − x1)z<sup>1</sup> − φ1(x)(x1y<sup>1</sup> − d1z1) + u1) + e2(x2z<sup>2</sup> − b2y<sup>2</sup> − m21(b1x<sup>1</sup> − x1z<sup>1</sup> − c1y<sup>1</sup> + w1)w<sup>1</sup> − φ2(x)(−k1y<sup>1</sup> − r1w1) + u2) + e3(x2y<sup>2</sup> − c2z<sup>2</sup> + x2w<sup>2</sup> − m31(x1y<sup>1</sup> − d1z1)x<sup>1</sup> − φ3(x)a1(y<sup>1</sup> − x1) + u3) + e4(−y<sup>2</sup> − m41(−k1y<sup>1</sup> − r1w1)y<sup>1</sup> − φ4(x)(b1x<sup>1</sup> − x1z<sup>1</sup> − c1y<sup>1</sup> + w1) + u4) + (¯a<sup>2</sup> <sup>−</sup> <sup>a</sup>2)a¯˙ <sup>2</sup> + (¯b<sup>2</sup> <sup>−</sup> <sup>b</sup>2)¯˙ <sup>b</sup><sup>2</sup> + (¯c<sup>2</sup> <sup>−</sup> <sup>c</sup>2)c¯˙<sup>2</sup> + (¯a<sup>1</sup> <sup>−</sup> <sup>a</sup>1)a¯˙ <sup>1</sup> + (¯b<sup>1</sup> <sup>−</sup> <sup>b</sup>1)¯˙ <sup>b</sup><sup>1</sup> + (¯c<sup>1</sup> <sup>−</sup> <sup>c</sup>1)c¯˙<sup>1</sup> + ( ¯ <sup>d</sup><sup>1</sup> <sup>−</sup> <sup>d</sup>1) ¯˙ d<sup>1</sup> + (¯ <sup>k</sup><sup>1</sup> <sup>−</sup> <sup>k</sup>1)¯˙ <sup>k</sup><sup>1</sup> + (¯r<sup>1</sup> <sup>−</sup> <sup>r</sup>1)r¯˙<sup>1</sup> (14)

Substituting Equation (11) into Equation (14) yields

$$\begin{aligned} \dot{V} &= -l\_1 e\_1^2 - l\_2 e\_2^2 - l\_3 e\_3^2 - l\_4 e\_4^2 \\ &- (\bar{a}\_1 - a\_1)^2 - (\bar{b}\_1 - b\_1)^2 - (\bar{c}\_1 - c\_1)^2 - (\bar{d}\_1 - d\_1)^2 - (\bar{k}\_1 - k\_1)^2 - (\bar{r}\_1 - r\_1)^2 \\ &- (\bar{a}\_2 - a\_2)^2 - (\bar{b}\_2 - b\_2)^2 - (\bar{c}\_2 - c\_2)^2 \\ &< 0 \end{aligned} \tag{15}$$

Since the Lyapunov function V is positive definite and its derivative V˙ is negative definite in the neighborhood of the zero solution for system (10). According to the Lyapunov stability theory, the error dynamical system (10) can converge to the origin asymptotically. Therefore, the SGFPS between the two hyperchaotic systems (6) and (7) is achieved with the adaptive controllers (11) and the parameter update laws (12).

This completes the proof.

#### 4. Numerical Simulation

In this section, to verify and demonstrate the effectiveness of the proposed method we consider a numerical example. In the numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. The true values of the "unknown" parameters of systems (6) and (7) are chosen as a<sup>1</sup> = 12, b<sup>1</sup> = 23, c<sup>1</sup> = 1, d<sup>1</sup> = 2.1, k<sup>1</sup> = 6, r<sup>1</sup> = 0.2, a<sup>2</sup> = 8, b<sup>2</sup> = 40, c<sup>2</sup> = 14.9, so that the two systems exhibit hyperchaotic behavior, respectively. The initial values for the drive and response systems are x1(0) = 8.3, y1(0) = 10.8, z1(0) = 17.4, w1(0) = −11.1, x2(0) = −0.2, y2(0) = −0.1, z2(0) = 16.9 and w2(0) = −0.7, and the estimated parameters have initial conditions 0.1. Given that the function factors are φ1(x)=2x<sup>1</sup> − 0.3, φ2(x)=2y<sup>1</sup> + 0.5, φ3(x)=0.5z<sup>1</sup> + 0.03, φ4(x) = −0.5w<sup>1</sup> + 0.03, and the gain matrix L is given as diag{10, 10, 10, 10}. The simulation results are shown in Figures 3–5. Figure 3 demonstrates the SGFPS errors of the drive system (6) and response system (7). From this figure, it can be seen that the SGFPS errors converge to zero, *i.e.*, these two systems achieved SGFPS. And Figures 4 and 5 show that the unknown system parameters approach the true values.

Figure 3. The time evolution of SGFPS errors for the drive system (6) and response system (7) with controllers (11) and parameter update laws (12), where e<sup>1</sup> = x<sup>2</sup> −(2x<sup>1</sup> − 0.3)z1, e<sup>2</sup> = y2−(2y1+0.5)w1, e<sup>3</sup> = z2−(0.5z1+0.03)x1, e<sup>4</sup> = w2−(−0.5w1+0.03)y1.

Figure 4. The time evolution of the estimated unknown parameters of system (6).

Figure 5. The time evolution of the estimated unknown parameters of system (7).

#### 5. Conclusions

In this paper, we have investigated switched generalized function projective synchronization between two new different hyperchaotic systems with fully unknown parameters, which extended the switched modified function projective synchronization scheme. In this synchronization scheme, a state variable of the drive system synchronizes with a different state variable of the response system up to a generalized scaling function matrix. Due to the unpredictability of the switched states and scaling function matrix, this synchronization scheme can provide additional security in secure communication. By applying the adaptive control theory and Lyapunov stability theory, the appropriate adaptive controllers with parameter update laws are proposed to achieve SGFPS between two different hyperchaotic systems. A numerical simulation was conducted to illustrate the validity and feasibility of the proposed synchronization scheme.

#### Acknowledgments

This work was supported by the Youth Foundation of Yunnan University of Nationalities under grant No.11QN07, the Natural Science Foundation of Yunnan Province under grants No.2009CD019 and No.2011FZ172, the Natural Science Foundation of China under grant No.61263042.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### References


Reprinted from *Entropy*. Cite as: Zhou, X.; Jiang, M.; Cai, X. Synchronization of a Novel Hyperchaotic Complex-Variable System Based on Finite-Time Stability Theory. *Entropy* 2013, *15*, 4334–4344.

*Article*
