Hybrid synchronization of hyperchaotic n-scroll Chua circuit using adaptive backstepping control

Abstract

In this paper, hybrid synchronization is investigated for n-scroll hyperchaotic Chua circuit using adaptive backstepping control. The theorem on hybrid synchronization for n-scroll hyperchaotic Chua circuit is established using Lyapunov stability theory. The backstepping scheme is recursive procedure that links the choice of Lyapunov function with the design of a controller and guarantees global stability performance of strict-feedback nonlinear systems. The backstepping control method is effective and convenient to hybrid synchronize the hyperchaotic systems which are mainly in this technique that gives the flexibility to construct a control law. Numerical simulations are also given to illustrate and validate the hybrid synchronization results derived in this paper.

Keywords:

• synchronization
• Chaos
• adaptive backstepping control
• n-scroll hyperchaotic Chua circuit

Public Interest Statement

Chaos synchronization can be applied in the areas of physics, engineering and biological science. Synchronization has been widely explored in a variety of fields including physical chemical, and ecological systems, secure communications etc. Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator. Because the butterfly effect which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a challenging problem. In most synchronization approaches, the master-slave or drive-response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the slave system so that the output of the response system tracks the output of the master system asymptotically.

1. Introduction

Synchronization in chaos refers to the tendency of two or more systems which are coupled together to undergo closely related motion, even when the motions are chaotic.

The synchronization for chaotic systems has been widespread to the scope (Alligood, Sauer, & Yorke, 1997; Fujisaka & Yamada, 1983; Pecora & Carroll, 1990), such as generalized synchronization (Harmov, Koronovskii, & Moskalenko, 2005a; b; Wang & Zhu, 2006), anti-synchronization, phase synchronization (Ge & Chen, 2006; Tokuda, Kurths, Kiss, & Hudson, 2008; Zhao, Lai, Wang, & Gao, 2004), lag synchronization, projective synchronization (Qiang, 2007), and generalized projective synchronization (Jian-Ping & Chang-Pin, 2006; Li, Xu & Li, 2007).

The property of anti-synchronization establishes a predominating phenomenon in symmetrical oscillators, in which the state vectors have the same absolute values but opposite signs.

When synchronization and anti-synchronization coexist, simultaneously, in chaotic systems, then that synchronization is called hybrid synchronization.

A variety of schemes to ensure the control and synchronization of such systems have been demonstrated based on their potential applications in various fields including chaos generator design, secure communication (Chen, 1996; Kanter, Kopelowitz, Kestler, & Kinzel, 2008; Yang & Chua, 1999), physical systems (Chern & Otsuka, 2012; Lakshmanan & Murali, 1996; Moreno & Pacheco, 2004), chemical reaction (Coffman, McCormick, Noszticzius, & Simoyi, 1987; Han, Kerrer, & Kuramoto, 1995), ecological systems (Blasius & Huppert, 1999), information science (Bauer, Atay, & Jost, 2010; Ghosh, Banerjee, & Chowdhury, 2007; Kocarev & Parlitz, 1995), energy resource systems, ghostburster neurons (Wang, Chen & Deng, 2009), biaxial magnet models (Moukam Kakmeni, Nguenang, & Kofane, 2006), neuronal models (Che, Wang, Tsang, & Chen, 2010; Hindmarsh & Rose, 1984; Qi, Huang, Chen, Wang, & Shen, 2008), IR epidemic models with impulsive vaccination (Zeng, Sun, Li, & Sun, 2005), and predicting the influence of solar wind to celestial bodies (Junxa, Dianchen, & Tian, 2006; Suresh & Sundarapandian, 2012a).

So far a variety of impressive approaches have been proposed for the synchronization of the chaotic systems such as OGY method (Ott, Grebogi, & Yorke, 1990), sampled feedback synchronization method (Murali & Lakshmaman, 2003), time delay feedback method (Park & Kwon, 2003), adaptive design method (Lu, Wu & Han, 2004; Park, 2008; Park, Lee, & Kwon, 2007), sliding mode control method (Ya, 2004), active control method (Sundarapandian & Suresh, 2010). and backstepping control design (Suresh & Sundarapandian, 2012b; Wu & Lu, 2003; Yu & Zhang, 2006).

Recently, backstepping method has been developed and designed to control the chaotic systems. A common concept in the method is to synchronize the chaotic system. The backstepping method is based on the mathematical model of the examined system, introducing new variables into a form depending on the state variables, controlling parameters, and stabilizing functions. The difficult work of synchronizing the chaotic system is to remove nonlinearities which were done in the system and influencing the stability of state operation. The use of backstepping method creates an additional nonlinearity and eliminates undesirable nonlinearities from the system (Suresh & Sundarapandian, sundarhybrid12; 2013; Wang, Zhang, & Guo, 2010; Wang 2011a; b).

The uncertainties are commonly in chaos synchronization and other control system problems. The uncertainties are one of the main factors in leading the adaptive-based synchronization. Adaptive control design is a direct aggregation of control methodology with some form of recursive system which identifies the system to determine the control of linear or nonlinear systems.

Adaptive control design is studied and analyzed in theory of unknown, but fixed parameter systems. The controller feedback gain could be depending on the system parameter.

2. Problem statement

Consider the chaotic system described by the dynamics(1) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{x}}_1& =F_1(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)\\ {\dot{x}}_2& =F_2(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)\\ {\dot{x}}_3& =F_3(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)\\ ⋮& ⋮& ⋮\\ {\dot{x}}_n& =F_n(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)\end{array}\end{array}$$(1)

where $x\in {\mathrm{R}}^n$ is the state of the system, in which the system (1) is considered as the master system; and ${\mathit{\alpha }}_i$ is the unknown parameter, $\widehat{{\mathit{\alpha }}_i}$ is the estimates as the parameter ${\mathit{\alpha }}_i$.

The slave system is a chaotic system with the controller $u={[u_1,u_2,u_3...u_n]}^T$ described by the dynamics(2) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{y}}_1& =G_1(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)+u_1(t)\\ {\dot{y}}_2& =G_2(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)+u_2(t)\\ {\dot{y}}_3& =G_3(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)+u_3(t)\\ ⋮& ⋮& ⋮\\ {\dot{y}}_n& =G_n(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)+u_n(t)\end{array}\end{array}$$(2)

where $u_i$ is the input to the system with parameter estimator ${\widehat{\mathit{\alpha }}}_i,i=1,2,3,\dots ,n$, and $y\in {\mathrm{R}}^n$ is the state of the slave system and $F_i,G_i(i=1,2,3...n)$ linear or nonlinear functions with input from systems (1) and (2).

If $F_i=G_i$ for all i, then the system (1) and (2) are called identical and otherwise they are nonidentical chaotic systems.

The hybrid synchronization error is defined as(3) $$\begin{array}{c}\hfill \begin{array}{c}{e}_i=\left\{\begin{array}{ccc}{y}_i-x_i& \text{if}& \text{if i is odd}\\ y_i+x_i& \text{if}& \text{if i is even}\end{array}\right.\end{array}\end{array}$$(3)

Then the synchronization error dynamics is obtained as(4) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =G_1(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & -F_1(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_1,\hfill \\ {\dot{e}}_2& =G_2(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & +F_2(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_2,\hfill \\ ⋮& ⋮& ⋮\hfill \\ {\dot{e}}_n& =G_n(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & +{(-1)}^n{F}_n(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_n\hfill \end{array}\end{array}$$(4)

The parameter estimation error is defined as$$e_{{\mathit{\alpha }}_i}={\mathit{\alpha }}_i-\widehat{{\mathit{\alpha }}_i}$$

The hybrid synchronization problem basically requires the global asymptotically stability of the error dynamics (4), i.e.(5) $$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\Vert e(t)\Vert =0\end{array}$$(5)

for all initial conditions $e(0)\in {\mathrm{R}}^n.$

Backstepping design procedure is recursive and guarantee global stability performance of strict-feedback chaotic systems. By using the backstepping design, at the $i$th step, the $i$th order subsystem is stabilized with respect to a Lyapunov function $V_i,$ by the virtual control ${\mathit{\alpha }}_i$, and a control input function $u_i.$

Consider the global asymptotic stability of the system(6) $$\begin{array}{c}\hfill {\dot{e}}_1=G_1(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)-F_1(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_1\end{array}$$(6)

where $u_1$ is control input, which is the function of the error vector $e_i,$ and the state variables $x(t)\in {\mathrm{R}}^n,y(t)\in {\mathrm{R}}^n.$ As long as this feedback stabilizes, the system (6) will converge to zero as $t\to \infty$, where $e_2={\mathit{\alpha }}_1(e_1)$ is regarded as a virtual controller.

For the design of ${\mathit{\alpha }}_1(e_1)$ is to stabilize the subsystem (6), the Lyapunov function is defined by(7) $$\begin{array}{c}\hfill V_1(e)=e_1^T{P}_1{e}_1+\sum _{i=1}^k{e}_{{\mathit{\alpha }}_i}^T{R}_1{e}_{{\mathit{\alpha }}_i}\end{array}$$(7)

where $P_1,and{R}_1$ are positive definite matrices.

The derivative of $e_{{\mathit{\alpha }}_i}$ is(8) $$\begin{array}{c}\hfill {\dot{e}}_{{\mathit{\alpha }}_i}=-{\dot{\widehat{\mathit{\alpha }}}}_i\end{array}$$(8)

Suppose the derivative of $V_1$ is(9) $$\begin{array}{c}\hfill {\dot{V}}_1=-e_1^T{Q}_1{e}_1-\sum _{i=1}^k{e}_{{\mathit{\alpha }}_i}^T{S}_1{e}_{{\mathit{\alpha }}_i}\end{array}$$(9)

where $Q_1,and{S}_1$ are positive definite matrices.

Then ${\dot{V}}_1$ is a negative definite function.

Thus by Lyapunov stability theory, the error dynamics (6) is globally asymptotically stable.

The function ${\mathit{\alpha }}_1(e_1)$ is an estimative function when $e_2$ is considered as a controller.

The error between $e_2$ and ${\mathit{\alpha }}_1(e_1)$ is(10) $$\begin{array}{c}\hfill w_2=e_2-{\mathit{\alpha }}_1(e_1)\end{array}$$(10)

Consider the $(e_1,w_2)$ subsystem given by(11) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =G_1(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & -F_1(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_1\hfill \\ {\dot{w}}_2& =G_2(y_1,y_2,\dots ,y_n)\\ & & +F_2(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)-{\dot{\mathit{\alpha }}}_1(e_1)+u_2\hfill \end{array}\end{array}$$(11)

Let $e_3$ as a virtual controller in system (11).

Assume that when(12) $$\begin{array}{c}\hfill e_3={\mathit{\alpha }}_2(e_1,w_2)\end{array}$$(12)

the system (11) is made globally asymptotically stable.

Consider the Lyapunov function defined by(13) $$\begin{array}{c}\hfill V_2(e_2,w_2)=V_1(e_1)+w_2^T{P}_2{w}_2+\sum _{i=k+1}^m{e}_{{\mathit{\alpha }}_i}^T{R}_1{e}_{{\mathit{\alpha }}_i}\end{array}$$(13)

where $P_2,and{R}_2$ are positive definite matrices.

Suppose the derivative of $V_2(e_1,w_2)$ is(14) $$\begin{array}{c}\hfill {\dot{V}}_2=-e_1^T{Q}_1{e}_1-w_2^T{Q}_2{w}_2-\sum _{i=k+1}^m{e}_{{\mathit{\alpha }}_i}^T{S}_2{e}_{{\mathit{\alpha }}_i}\end{array}$$(14)

where $Q_1,Q_2,and{S}_2$ are positive definite matrices.

Then ${\dot{V}}_2(e_1,w_2)$ is a negative definite function.

Thus by Lyapunov stability theory, the error dynamics (11) is globally asymptotically stable. The virtual controller $e_3={\mathit{\alpha }}_2(e_1,w_2)$ and the state feedback input $u_2$ make the system (11) asymptotically stable.

For the $n$th state of the error dynamics, define the error variable $w_n$ as(15) $$\begin{array}{c}\hfill w_n=e_n-{\mathit{\alpha }}_{n-1}(e_1,w_2,\dots ,w_{n-1})\end{array}$$(15)

Considering the $(e_1,w_2,\dots ,w_n)$ subsystem given by(16) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =G_1(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & -F_1(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)+u_1\hfill \\ {\dot{w}}_2& =G_2(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & +F_2(x_1,x_2,\dots ,x_n)-{\dot{\mathit{\alpha }}}_1(e_1)+u_2\hfill \\ ⋮& ⋮& ⋮\hfill \\ {\dot{w}}_n& =G_n(y_1,y_2,\dots ,y_n,{\mathit{\alpha }}_i)\\ & & -F_n(x_1,x_2,\dots ,x_n,{\mathit{\alpha }}_i)\hfill \\ & & -{\dot{\mathit{\alpha }}}_{n-1}(e_1,w_2,\dots ,w_{n-1})+u_n\hfill \end{array}\end{array}$$(16)

Consider the Lyapunov function defined by(17) $$\begin{array}{c}\hfill \begin{array}{cc}{V}_n(e_2,w_2,\dots ,w_n)& =V_{n-1}(e_1,w_2,\dots ,w_{n-1})\\ & & +w_n^T{P}_n{w}_n\sum _{i=m+1}^n{e}_{{\mathit{\alpha }}_i}^T{R}_n{e}_{{\mathit{\alpha }}_i}\hfill \end{array}\end{array}$$(17)

where $P_n,and{R}_n$ are positive definite matrices.

Suppose the derivative of $V_n(e_1,w_2,w_3....w_n)$ is(18) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{V}}_n(e_1,w_2,\dots ,w_n)& =-e_1^T{Q}_1{e}_1-w_2^T{Q}_2{w}_2-\cdots \\ & & -w_n^T{Q}_n{w}_n-\sum _{i=k+1}^m{e}_{{\mathit{\alpha }}_i}^T{S}_n{e}_{{\mathit{\alpha }}_i}\hfill \end{array}\end{array}$$(18)

where $Q_1,Q_2,\dots ,Q_n,S_n$ are positive definite matrices.

Then ${\dot{V}}_n(e_1,w_2,\dots ,w_n)$ is a negative definite function on ${\mathrm{R}}^n.$

Thus by Lyapunov stability theory (Hahn, 1967), the error dynamics (16) is globally asymptotically stable.

The virtual controller is(19) $$\begin{array}{c}\hfill e_n={\mathit{\alpha }}_{n-1}(e_1,w_2,\dots ,w_{n-1})\end{array}$$(19)

and the state feedback input $u_n$ makes the system (11) globally asymptotically stable.

Hence, the state of master and slave systems are globally and asymptotically synchronized.

3. System description

Recently, theoretical design and hardware implementation of different kinds of chaotic oscillators have attracted increasing attention, aiming real-world applications of many chaos-based technologies and information systems.

The n-scroll hyperchaotic Chua circuit (Yu, Lu, & Chen, 2007) is given by the dynamics(20) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{x}}_1& =\mathit{\alpha }[g(x_2-x_1)-x_3]\\ {\dot{x}}_2& =\mathit{\beta }[-g(x_2-x_1)-x_4]\\ {\dot{x}}_3& ={\mathit{\gamma }}_0(x_1+x_3)\\ {\dot{x}}_4& =\mathit{\gamma }{x}_2,\end{array}\end{array}$$(20)

where $g(x_2-x_1)$ is given by(21) $$\begin{array}{c}\hfill \begin{array}{cc}g(x_2-x_1)& =m_{N-1}(x_2-x_1)\\ & & +\frac{1}{2}{\sum }_{i=1}^{N-1}(m_{i-1}-m_i)\hfill \\ & & \times (|x_2-x_1+z_i|-|x_2-x_1+z_i|)\hfill \end{array}\end{array}$$(21)

The recursive positive switching points $z_i(i=2,3,4,...,N-1)$ can be deduced as(22) $$\begin{array}{c}\hfill \begin{array}{cc}{z}_2& =\frac{(1+k_1){\sum }_{i=1}^1(m_{i-1}-m_i)x_i}{m_1-1}-k_1{x}_1\\ z_3& =\frac{(1+k_2){\sum }_{i=1}^2(m_{i-1}-m_i)x_i}{m_2-1}-k_2{x}_2\\ ⋮& ⋮& ⋮\hfill \\ z_{N-1}& =\frac{(1+k_{N-1}){\sum }_{i=1}^{N-1}(m_{i-1}-m_i)x_i}{m_{N-2}-1}-k_{N-2}{x}_{N-2}\end{array}\end{array}$$(22)

and the $k_i$ values are obtained as(23) $$\begin{array}{c}\hfill k_i=\frac{x_{i+1}-x_i^E}{x_i^E-x_i}(1\le i\le N-2)\end{array}$$(23)

in which $x_i^E$ are the positive equilibrium points of $g(x_2-x_1)$.

3.1. Case 1: 2-scroll hyperchaotic attractor

The parameters of the systems (20) are taken in the case of hyperchaotic case as $\mathit{\alpha }=2,\mathit{\beta }=20.$

When $N=2$, In Equation (20), the function $g(x_2-x_1)$ is given by(24) $$\begin{array}{c}\hfill \begin{array}{cc}g(x_2-x_1)& =m_1(x_2-x_1)+\frac{1}{2}(m_0-m_1)\\ & & \times (|x_2-x_1+z_1|-|x_2-x_1+z_1|)\hfill \end{array}\end{array}$$(24)

When $m_0=-0.2,m_1=3\text{and}{z}_1<1,$ the 2-scroll hyperchaotic attractor is generated. Figures 1–3 depict the 2-scroll hyperchaotic attractor.

Figure 1. 2-scroll hyperchaotic attractor.

Figure 2. 2-scroll hyperchaotic attractor.

Figure 3. 2-scroll hyperchaotic attractor.

3.2. Case 2: 3-scroll hyperchaotic attractor

When $N=3$, in Equation (20), the function $g(x_2-x_1)$ is given by(25) $$\begin{array}{c}\hfill \begin{array}{cc}g(x_2-x_1)& =m_2(x_2-x_1)+\frac{1}{2}(m_0-m_1)\\ & & \times (|x_2-x_1+z_1|-|x_2-x_1+z_1|)\hfill \\ & & +\frac{1}{2}(m_1-m_2)\hfill \\ & & \times (|x_2-x_1+z_2|-|x_2-x_1+z_2|)\hfill \end{array}\end{array}$$(25)

When $m_0=3,m_1=-0.8,m_2=3,z_2=1.8333\text{and}{z}_1<1,$ the 3-scroll hyperchaotic attractor is generated. Figures 4–6 depict the 3-scroll hyperchaotic attractor.

Figure 4. 3-scroll hyperchaotic attractor.

Figure 5. 3-scroll hyperchaotic attractor.

Figure 6. 3-scroll hyperchaotic attractor.

3.3. Case 3: 4-scroll hyperchaotic attractor

When $N=4$, in Equation (20), the function $g(x_2-x_1)$ is given by(26) $$\begin{array}{c}\hfill \begin{array}{cc}g(x_2-x_1)& =m_3(x_2-x_1)+\frac{1}{2}(m_0-m_1)\\ & & \times (|x_2-x_1+z_1|-|x_2-x_1+z_1|)\hfill \\ & & +\frac{1}{2}(m_1-m_2)\hfill \\ & & \times (|x_2-x_1+z_2|-|x_2-x_1+z_2|)\hfill \\ & & +\frac{1}{2}(m_2-m_3)\hfill \\ & & \times (|x_2-x_1+z_3|-|x_2-x_1+z_3|)\hfill \end{array}\end{array}$$(26)

When $m_0=m_2=-0.7,m_1=m_3=2.9,m_2=3,z_2=1.5289,z_3=3.0239\text{and}{z}_1<1,$ the 4-scroll hyperchaotic attractor is generated.

Figures 7–9 depict the 4-scroll hyperchaotic attractor.

Figure 7. 4-scroll hyperchaotic attractor

Figure 8. 4-scroll hyperchaotic attractor.

Figure 9. 4-scroll hyperchaotic attractor.

4. Hybrid Synchronization of n-scroll hyperchaotic Chua circuits via backstepping control with recursive feedback

In this section, the backstepping method with recursive feedback function is applied for the hybrid synchronization of identical hyperchaotic n-scroll Chua circuits (Yu et al., 2007).

The n-scroll hyperchaotic Chua circuit is taken as the master system, which is described by(27) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{x}}_1& =\mathit{\alpha }[g(x_2-x_1)-x_3]\\ {\dot{x}}_2& =\mathit{\beta }[-g(x_2-x_1)-x_4]\\ {\dot{x}}_3& ={\mathit{\gamma }}_0(x_1+x_3)\\ {\dot{x}}_4& =\mathit{\gamma }{x}_2,\end{array}\end{array}$$(27)

where $g(x_2-x_1)$ is given by(28) $$\begin{array}{c}\hfill \begin{array}{cc}g(x_2-x_1)& =m_{N-1}(x_2-x_1)+\\ & & \frac{1}{2}{\sum }_{i=1}^{N-1}(m_{i-1}-m_i)\hfill \\ & & \times (|x_2-x_1+z_i|-|x_2-x_1+z_i|)\hfill \end{array}\end{array}$$(28)

where $x(t)(i=1,2,3,4)\in {\mathsf{R}}^4$ are state variables.

The n-scroll hyperchaotic Chua circuit is also taken as the slave system, which is described by(29) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{y}}_1& =\mathit{\alpha }[g(y_2-y_1)-y_3]+u_1\\ {\dot{y}}_2& =\mathit{\beta }[-g(y_2-y_1)-y_4]+u_2\\ {\dot{y}}_3& ={\mathit{\gamma }}_0(y_1+y_3)+u_3\\ {\dot{y}}_4& =\mathit{\gamma }{y}_2+u_4\end{array}\end{array}$$(29)

where $g(y_2-y_1)$ is given by(30) $$\begin{array}{c}\hfill \begin{array}{cc}g(y_2-y_1)& =m_{N-1}(y_2-y_1)+\frac{1}{2}{\sum }_{i=1}^{N-1}(m_{i-1}-m_i)\\ & & \times (|y_2-y_1+z_i|-|y_2-y_1+z_i|)\hfill \end{array}\end{array}$$(30)

where $y(t)(i=1,2,3,4)\in {\mathsf{R}}^4$ are state variables.

The hybrid synchronization error is defined by(31) $$\begin{array}{c}\hfill e_1=y_1-x_1,e_2=y_1+x_2,e_3=y_3-x_3,e_4=y_4+x_4\end{array}$$(31)

The error dynamics is obtained as(32) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]-\mathit{\alpha }{e}_3+u_1\\ {\dot{e}}_2& =-\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]-\mathit{\beta }{e}_4+u_2\\ {\dot{e}}_3& ={\mathit{\gamma }}_0(e_1+e_3)+u_3\\ {\dot{e}}_4& =\mathit{\gamma }{e}_2+u_4\end{array}\end{array}$$(32)

The modified error dynamics is defined by(33) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]-\mathit{\alpha }{e}_3+u_1\\ {\dot{e}}_2& =-\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\\ & & -\mathit{\beta }{e}_4+e_1-y_1+x_1+u_2\hfill \\ {\dot{e}}_3& ={\mathit{\gamma }}_0(e_1+e_3)+e_2-y_2-x_2+u_3\\ {\dot{e}}_4& =\mathit{\gamma }{e}_2+e_3-y_3+x_3+u_4\end{array}\end{array}$$(33)

Now the objective is to find control law $u_i,i=1,2,3,4$ and for the parameter update law $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},{\widehat{\mathit{\gamma }}}_0$ for stabilizing the system (32) at the origin.

First consider the stability of the system(34) $$\begin{array}{c}\hfill {\dot{e}}_4=\mathit{\gamma }{e}_2+e_3-y_3+x_3+u_4\end{array}$$(34)

where $e_3$ is regarded as virtual controller.

Consider the Lyapunov function defined by(35) $$\begin{array}{c}\hfill V_1(e_4)=\frac{1}{2}{e}_4^2+\frac{1}{2}{e}_{\mathit{\gamma }}^2\end{array}$$(35)

Let define the parameter estimation error as(36) $$\begin{array}{c}\hfill e_{\mathit{\gamma }}=\mathit{\gamma }-\widehat{\mathit{\gamma }}\end{array}$$(36)

Differentiating the Equation (36)(37) $$\begin{array}{c}\hfill {\dot{e}}_{\mathit{\gamma }}=-\dot{\widehat{\mathit{\gamma }}}\end{array}$$(37)

Differentiate $V_1$ along with the Equation (37)(38) $$\begin{array}{c}\hfill \dot{V_1}=e_4(\mathit{\gamma }{e}_2+e_3-y_3+x_3+u_4)+e_{\mathit{\gamma }}(-\dot{\widehat{\mathit{\gamma }}})\end{array}$$(38)

Assume the controller $e_3={\mathit{\alpha }}_1(e_4)$.

If(39) $$\begin{array}{c}\hfill {\mathit{\alpha }}_1(e_4)=-k_1{e}_4,\text{and}{u}_4=y_3-x_3-\widehat{\mathit{\gamma }}{e}_2\end{array}$$(39)

and the parameter update law $\dot{\widehat{\mathit{\gamma }}}$ is taken as(40) $$\begin{array}{c}\hfill \dot{\widehat{\mathit{\gamma }}}=e_2{e}_4+k_2{e}_{\mathit{\gamma }}\end{array}$$(40)

then(41) $$\begin{array}{c}\hfill \dot{V_1}=-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2\end{array}$$(41)

which is a negative definite function. Hence, the system (34) is globally asymptotically stable.

The function ${\mathit{\alpha }}_1(e_4)$ is an estimative function when $e_3$ is considered as a controller.

The error between $e_3$ and ${\mathit{\alpha }}_1(e_4)$ is(42) $$\begin{array}{c}\hfill w_2=e_3-{\mathit{\alpha }}_1(e_4)=e_3+k_1{e}_4\end{array}$$(42)

Consider the $(e_1,w_2)$ subsystem given by(43) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =e_{\mathit{\gamma }}{e}_2+e_3\\ \dot{w_2}& ={\mathit{\gamma }}_0(e_1+w_2-k_1{e}_4)+(k_1{e}_{\mathit{\gamma }}+1)e_2\\ & & +k_1(w_2-k_1{e}_4)-y_2-x_2+u_3\hfill \end{array}\end{array}$$(43)

Let $e_2$ be a virtual controller in system (43).

Assume that when $e_2={\mathit{\alpha }}_2(e_4,w_2)$ and the system (43) is made globally asymptotically stable.

Consider the Lyapunov function defined by(44) $$\begin{array}{c}\hfill V_2(e_4,w_2)=V_1(e_4)+\frac{1}{2}{w}_2^2+\frac{1}{2}{e}_{{\mathit{\gamma }}_0}^2\end{array}$$(44)

Let us define the parameter estimation error as(45) $$\begin{array}{c}\hfill e_{{\mathit{\gamma }}_0}={\mathit{\gamma }}_0-\widehat{{\mathit{\gamma }}_0}\end{array}$$(45)

Differentiating the Equation (45), we get(46) $$\begin{array}{c}\hfill \dot{e_{{\mathit{\gamma }}_0}}=-\dot{\widehat{{\mathit{\gamma }}_0}}\end{array}$$(46)

The derivative of $V_2(e_4,w_2)$ is(47) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{V_2}& ={\dot{V}}_1+w_2{\dot{w}}_2+e_{{\mathit{\gamma }}_0}\dot{e_{{\mathit{\gamma }}_0}}\\ & =e_4(e_{\mathit{\gamma }}{e}_2+w_2-k_1{e}_4)+e_{\mathit{\gamma }}(\dot{\widehat{\mathit{\gamma }}})\\ & & +w_2({\mathit{\gamma }}_0(e_1+w_2-k_1{e}_4)+(k_1{e}_{\mathit{\gamma }}+1)e_2\hfill \\ & & +k_1(w_2-k_1{e}_4)-y_2-x_2+u_3)+e_{{\mathit{\gamma }}_0}(-\dot{\widehat{{\mathit{\gamma }}_0}})\hfill \end{array}\end{array}$$(47)

Substituting for $e_3$ from (42) into (47) and simplifying, we get(48) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{V_2}& =-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2\\ & & +w_2(e_4+{\mathit{\gamma }}_0(e_1+w_2-k_1{e}_4)+(k_1{e}_{\mathit{\gamma }}+1)e_2\hfill \\ & & +k_1(w_2-k_1{e}_4)-y_2-x_2+u_3)+e_{{\mathit{\gamma }}_0}(-\dot{\widehat{{\mathit{\gamma }}_0}})\hfill \end{array}\end{array}$$(48)

Assume the virtual controller $e_2={\mathit{\alpha }}_2(e_4,w_2)$(49) $$\begin{array}{c}\hfill \begin{array}{cc}{\mathit{\alpha }}_2(e_1,w_2)& =0\\ u_3& =y_2+x_2-e_4-k_1(w_2\\ & & -k_1{e}_4)-k_3{w}_2-\widehat{{\mathit{\gamma }}_0}(e_1+w_2-k_1{e}_4)\hfill \end{array}\end{array}$$(49)

The parameter update law $\dot{\widehat{{\mathit{\gamma }}_0}}$ is(50) $$\begin{array}{c}\hfill \dot{\widehat{{\mathit{\gamma }}_0}}=w_2(e_1+w_2-k_1{e}_4)+k_4{e}_{{\mathit{\gamma }}_0}\end{array}$$(50)

Then it follows that(51) $$\begin{array}{c}\hfill {\dot{V}}_2=-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2-k_3{w}_2^2-k_4{e}_{{\mathit{\gamma }}_0}^2\end{array}$$(51)

Thus, ${\dot{V}}_2$ is a negative definite function and hence the system (43) is globally asymptotically stable.

Define the error variable $e_2$ and ${\mathit{\alpha }}_2(e_4,w_2)$ as(52) $$\begin{array}{c}\hfill w_3=e_2-{\mathit{\alpha }}_2(e_4,w_2)\end{array}$$(52)

Consider the $(e_4,w_2,w_3)$ subsystem given by(53) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =e_{\mathit{\gamma }}{e}_2+e_3\\ {\dot{w}}_2& =(k_1{e}_{\mathit{\gamma }}+1)e_2-e_4-k_3{w}_2\\ & & +e_{{\mathit{\gamma }}_0}(e_1+w_2-k_1{e}_4)\hfill \\ {\dot{w}}_3& =-\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\\ & & -\mathit{\beta }{e}_4+e_1-y_1+x_1+u_2\hfill \end{array}\end{array}$$(53)

Let $e_1$ be a virtual controller in system (53).

Assume when it is equal to $e_1={\mathit{\alpha }}_3(e_4,w_2,w_3),$ the system (53) is made globally asymptotically stable.

Consider the Lyapunov function defined by(54) $$\begin{array}{c}\hfill V_3(e_1,w_2,w_3)=V_2(e_1,w_2)+\frac{1}{2}{w}_3^2+\frac{1}{2}{e}_{\mathit{\beta }}^2\end{array}$$(54)

Let us define the parameter estimation error as(55) $$\begin{array}{c}\hfill e_{\mathit{\beta }}=\mathit{\beta }-\widehat{\mathit{\beta }}\end{array}$$(55)

The derivative of (55) is(56) $$\begin{array}{c}\hfill \dot{e_{\mathit{\beta }}}=-\dot{\widehat{\mathit{\beta }}}\end{array}$$(56)

The derivative of $V_3(e_3,w_2,w_3)$ is(57) $$\begin{array}{c}\hfill \dot{V_3}={\dot{V}}_2(e_1,w_2)+w_3{\dot{w}}_3+e_{\mathit{\beta }}\dot{e_{\mathit{\beta }}}\end{array}$$(57)

i.e.(58) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{V_3}& =e_4(e_{\mathit{\gamma }}{e}_2+w_2-k_1{e}_4)+e_{\mathit{\gamma }}(-e_2{e}_4-k_2{e}_{\mathit{\gamma }})\\ & & +w_2[(k_1{e}_{\mathit{\gamma }}+1)w_3-e_4-k_3{w}_2\hfill \\ & & +e_{{\mathit{\gamma }}_0}(e_1+w_2-k_1{e}_4)]\hfill \\ & & +e_{{\mathit{\gamma }}_0}[-w_2(e_1+w_2-k_1{e}_4)-k_4{e}_{{\mathit{\gamma }}_0}]\hfill \\ & & +w_3[-\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\hfill \\ & & -\mathit{\beta }{e}_4+e_1-y_1+x_1+u_2]+e_{\mathit{\beta }}(-\dot{\widehat{\mathit{\beta }}})\hfill \end{array}\end{array}$$(58)

Substituting for $e_2$ from (52) into (58) and simplifying, we get(59) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{V_3}& =-k_1{e}_1^2-k_2{e}_{\mathit{\gamma }}^2-k_3{w}_2^2-k_4{e}_{{\mathit{\gamma }}_0}^2\\ & & +w_3[w_2(k_1{e}_{\mathit{\gamma }}+1)\hfill \\ & & -\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\hfill \\ & & -\mathit{\beta }{e}_4+e_1-y_1+x_1+u_2]+e_{\mathit{\beta }}(-\dot{\widehat{\mathit{\beta }}})]\hfill \end{array}\end{array}$$(59)

Assume the virtual controller $e_1={\mathit{\alpha }}_3(e_4,w_2,w_3).$

choose(60) $$\begin{array}{c}\hfill \begin{array}{cc}{\mathit{\alpha }}_3(e_4,w_2,w_3)& =0\\ u_2& =y_1-x_1+\widehat{\mathit{\beta }}{e}_4+\\ & & \mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\hfill \\ & & -k_5{w}_3-w_2(k_1{e}_{\mathit{\gamma }}+1)\hfill \end{array}\end{array}$$(60)

The parameter update law $\dot{\widehat{\mathit{\beta }}}$ is(61) $$\begin{array}{c}\hfill \dot{\widehat{\mathit{\beta }}}=-w_3{w}_4+k_6{e}_{\mathit{\beta }}\end{array}$$(61)

Then it follows that(62) $$\begin{array}{c}\hfill {\dot{V}}_3=-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2-k_3{w}_2^2-k_4{e}_{{\mathit{\gamma }}_0}^2-k_5{w}_3^2-k_6{e}_{\mathit{\beta }}^2\end{array}$$(62)

Thus, ${\dot{V}}_3$ is a negative definite function and hence the system (53) is globally asymptotically stable.

The error between $e_1$ and ${\mathit{\alpha }}_3(e_4,w_2,w_3)$ is(63) $$\begin{array}{c}\hfill w_4=e_1-{\mathit{\alpha }}_3(e_1,w_2,w_3)\end{array}$$(63)

Consider $(e_4,w_2,w_3,w_4)$ subsystem given by(64) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{e}}_1& =e_{\mathit{\gamma }}{e}_2+w_2-k_1{e}_4\\ {\dot{w}}_2& =(k_1{e}_{\mathit{\gamma }}+1)e_2-e_4-k_3{w}_2+\\ & & e_{{\mathit{\gamma }}_0}(e_1+w_2-k_1{e}_4)\hfill \\ {\dot{w}}_3& =-e_{\mathit{\beta }}{e}_4+e_1-k_5{w}_3-w_2(k_1{e}_{\mathit{\gamma }}+1)\\ {\dot{w}}_4& =\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]-\mathit{\alpha }{e}_3+u_1\end{array}\end{array}$$(64)

Consider the Lyapunov function defined by(65) $$\begin{array}{c}\hfill V_4(e_1,w_2,w_3,w_4)=V_3(e_1,w_2,w_3)+\frac{1}{2}{w}_4^2+\frac{1}{2}{e}_{\mathit{\alpha }}^2\end{array}$$(65)

Let define the parameter error as(66) $$\begin{array}{c}\hfill e_{\mathit{\alpha }}=\mathit{\alpha }-\widehat{\mathit{\alpha }}\end{array}$$(66)

The derivative of $e_{\mathit{\alpha }}$ is(67) $$\begin{array}{c}\hfill \dot{e_{\mathit{\alpha }}}=-\dot{\widehat{\mathit{\alpha }}}\end{array}$$(67)

The derivative of $V_4(e_3,w_2{w}_3,w_4)$ is(68) $$\begin{array}{c}\hfill \dot{V_4}={\dot{V}}_3(e_1,w_2,w_3)+{\dot{w}}_4{w}_4+e_{\mathit{\alpha }}\dot{e_{\mathit{\alpha }}}\end{array}$$(68)

i.e.(69) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{V_4}& =-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2-k_3{w}_2^2-k_4{e}_{{\mathit{\gamma }}_0}^2-k_5{w}_3^2-k_6{e}_{\mathit{\beta }}^2\\ & & +w_4(w_3+\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]\hfill \\ & & -\mathit{\alpha }{e}_3+u_1)+e_{\mathit{\alpha }}(-\dot{\widehat{\mathit{\alpha }}})\hfill \end{array}\end{array}$$(69)

Choose the controller(70) $$\begin{array}{c}\hfill u_1=-w_3-\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]+\widehat{\mathit{\alpha }}{e}_3-k_7{w}_4\end{array}$$(70)

and the parameter update law $\dot{\widehat{\mathit{\alpha }}}$ is(71) $$\begin{array}{c}\hfill \dot{\widehat{\mathit{\alpha }}}=-e_3{w}_4+k_8{e}_{\mathit{\alpha }}\end{array}$$(71)

Then(72) $$\begin{array}{c}\hfill \begin{array}{cc}{\dot{V}}_4& =-k_1{e}_4^2-k_2{e}_{\mathit{\gamma }}^2-k_3{w}_2^2-k_4{e}_{{\mathit{\gamma }}_0}^2-k_5{w}_3^2-k_6{e}_{\mathit{\beta }}^2\\ & & -k_7{w}_4^2-k_8{e}_{\mathit{\alpha }}^2\hfill \end{array}\end{array}$$(72)

Thus, ${\dot{V}}_4$ is a negative definite function.

Thus, by Lyapunov stability theory (Hahn, 1967), the error dynamics (53) is globally asymptotically stable for all initial condition.

Thus, the states of master and slave systems are globally and asymptotically hybrid synchronized.

5. Theorem

The identical n-scroll hyperchaotic Chua’s circuit (27) and (29) are globally and asymptotically hybrid synchronized with the adaptive backstepping controls(73) $$\begin{array}{c}\hfill \begin{array}{cc}{u}_1& =-w_3-\mathit{\alpha }[g(y_2-y_1)-g(x_2-x_1)]+\\ & & \widehat{\mathit{\alpha }}{e}_3-k_7{w}_4\hfill \\ u_2& =y_1-x_1+\widehat{\mathit{\beta }}{e}_4+\mathit{\beta }[g(x_2-x_1)+g(y_2-y_1)]\\ & & -k_5{w}_3-w_2(k_1{e}_{\mathit{\gamma }}+1)\hfill \\ u_3& =-w_2-{\mathit{\gamma }}_0{e}_1-2{\mathit{\gamma }}_0{w}_3+e_4\\ u_4& =y_2+x_2-e_4-k_1(w_2-k_1{e}_4)-k_3{w}_2\\ & & -\widehat{{\mathit{\gamma }}_0}(e_1+w_2-k_1{e}_4)\hfill \end{array}\end{array}$$(73)

and with the parameter update laws(74) $$\begin{array}{c}\hfill \begin{array}{cc}\dot{\widehat{\mathit{\alpha }}}& =-e_3{w}_4+k_8{e}_{\mathit{\alpha }}\\ \dot{\widehat{\mathit{\beta }}}& =-w_3{w}_4+k_6{e}_{\mathit{\beta }}\\ \dot{\widehat{{\mathit{\gamma }}_0}}& =w_2(e_1+w_2-k_1{e}_4)+k_4{e}_{{\mathit{\gamma }}_0}\\ \dot{\widehat{\mathit{\gamma }}}& =e_2{e}_4+k_2{e}_{\mathit{\gamma }}\end{array}\end{array}$$(74)

6. Numerical simulation

For the numerical simulations, the fourth order Runge–Kutta method is used to solve the differential Equations (27) and (29) with the backstepping controls $u_1,u_2,u_3$, and $u_4$ given by (29).

6.1. Case 1: 2-scroll hyperchaotic attractor

The parameters of the systems (27) are taken in the case of hyperchaotic case as $\mathit{\alpha }=2,and\mathit{\beta }=20$.

When $N=2,m_0=-0.2,m_1=3\text{and}{z}_1=0.5,$ the double scroll hyperchaotic attractor is generated.

The initial values of the master system (27) are chosen as $x_1(0)=0.947,x_2(0)=0.234,x_3(0)=0.472,x(4)=0.198$ and the initial values of the slave system (29) are chosen as $y_1(0)=0.157,y_2(0)=0.648,y_3(0)=0.810,y(4)=0.108.$

The initial values of the estimated parameters are $\widehat{\mathit{\alpha }}=2.3,\widehat{\mathit{\beta }}=4.9,\widehat{\mathit{\gamma }}=10,\widehat{{\mathit{\gamma }}_0}=5.5.$

Figure 10 depicts the hybrid synchronization of 2-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 10. Hybrid synchronization of 2-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 11. Hybrid synchronization error between 2-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 12. Parameter estimation of 2-scroll hyperchaotic Chua’s circuits (27) and (29). The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1$.

Figure 11 depicts the hybrid synchronization error between 2-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 12 depicts the parameter estimation of 2-scroll hyperchaotic Chua’s circuits (27) and (29).

The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1.$

6.2. Case 2: 3-scroll hyperchaotic attractor

When $N=3,m_0=3,m_1=-0.8,m_2=3,z_2=1.8333,\text{and}{z}_1=0.5$ the 3-scroll hyperchaotic attractor is generated.

The initial values of the master system (27) are chosen as $x_1(0)=0.431,x_2(0)=0.281,x_3(0)=0.983,x(4)=0.731$ and the initial values of the slave system (29) are chosen as $y_1(0)=1.012,y_2(0)=3.012,y_3(0)=2.018,y(4)=0.112.$

The initial values of the estimated parameters are $\widehat{\mathit{\alpha }}=10.318,\widehat{\mathit{\beta }}=3.121,\widehat{\mathit{\gamma }}=5.000,\widehat{{\mathit{\gamma }}_0}=3.$

Figure 13 depicts the hybrid synchronization of 3-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 13. Hybrid synchronization of 3-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 14. Hybrid synchronization error between 3-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 15. Parameter estimation of 3-scroll hyperchaotic Chua’s circuits (27) and (29). The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1$.

Figure 14 depicts the hybrid synchronization error between 3-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 15 depicts the parameter estimation of 3-scroll hyper chaotic Chua’s circuits (27) and (29).

The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1.$

6.3. Case 3: 4-scroll hyperchaotic attractor

When $N=4,m_0=m_2=-0.7,m_1=m_3=2.9,m_2=3,z_2=1.5289,z_3=3.0239\text{and}{z}_1=0.5,$ the 4-scroll hyperchaotic attractor is generated.

The initial values of the master system (27) are chosen as $x_1(0)=1.938,x_2(0)=2.138,x_3(0)=1.708,x(4)=3.325$ and the initial values of the slave system (29) are chosen as $y_1(0)=0.125,y_2(0)=0.986,y_3(0)=0.065,y(4)=1.363.$

The initial values of the estimated parameters are $\widehat{\mathit{\alpha }}=20.363,\widehat{\mathit{\beta }}=0.563,\widehat{\mathit{\gamma }}=10.613,\widehat{{\mathit{\gamma }}_0}=9.$

Figure 16 depicts the hybrid synchronization of 4-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 17 depicts the hybrid synchronization error between 4-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 18 depicts the parameter estimation of 4-scroll hyperchaotic Chua’s circuits (27) and (29).

The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1.$

Figure 16. Hybrid synchronization of 4-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 17. Hybrid synchronization error between 4-scroll hyperchaotic Chua’s circuits (27) and (29).

Figure 18. Parameter estimation of 4-scroll hyperchaotic Chua’s circuits (27) and (29). The estimated values of the parameters $\widehat{\mathit{\alpha }},\widehat{\mathit{\beta }},\widehat{\mathit{\gamma }},\text{and}\widehat{{\mathit{\gamma }}_0}$ converge to system parameters $\mathit{\alpha }=2,\mathit{\beta }=20,\mathit{\gamma }=1.5,\text{and}{\mathit{\gamma }}_0=1$.

7. Conclusion

In this paper, the adaptive backstepping control method has been applied to achieve global chaos hybrid synchronization for a family of n-scroll hyperchaotic Chua circuits. The backstepping control is a systematic procedure for hybrid synchronizing hyperchaotic systems and there is no derivative in controller. The adaptive backstepping control design has been demonstrated to class of n-scroll hyperchaotic Chua circuits. Numerical simulations have been given to illustrate and validate the effectiveness of the proposed hybrid synchronization schemes of the chaotic circuit. The adaptive backstepping method is very effective and convenient to achieve global chaos hybrid synchronization.

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Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Suresh Rasappan

Suresh Rasappan obtained his PhD degree in Mathematics from Vel Tech Rangarajan Dr Sakunthala R & D Institute of Science and Technology, Chennai, Tamil Nadu, India in 2013, and MPhil degree in Mathematics from Bharathiar University, Coimbatore, Tamil Nadu, India in 2008. He is currently working as an associate professor in the Department of Mathematics, Vel Tech Dr RR & Dr SR Technical University, Chennai, India. He has published over 28 papers in international journals and book chapters. His research interest is differential equations.