1 Introduction

Chaos synchronization is one the critical issues in nonlinear dynamical science because of its various applications in physics, secure communication, chemical reactors, control theory, biological networks, artificial neural networks, etc. Particularly, in neuroscience research, synchronization plays a very important role in the analysis of migraine and epilepsy [1]. In recent decades, the synchronization of chaotic systems has been extensively investigated both theoretically and practically [2–7]. The study of chaos synchronization has led to the discovery of various types of synchronization such as complete synchronization [2], phase synchronization [3], lag synchronization [4], generalized synchronization [5], anticipated synchronization [6], projective synchronization [7], etc.

Projective synchronization was reported by Mainnieri and Rehacek [7] in partially linear coupled three-dimensional systems in which the response of the dynamical state variables synchronized up to a scaling constant with the drive dynamical variables. Projective synchronization was later extended to general class of chaotic systems and not only to partially linear systems and this is called generalized projective synchronization [8, 9]. Projective synchronization is one of the most interesting synchronization scheme because of the proportional relation that exists between the dynamical state variables of the drive and response systems, a feature that can be used to extend binary digit to M-nary digit for fast communication [10]. Following the seminal work of Pecora and Carroll [2], various synchronization techniques have been developed in search of improved and effective methods of achieving stable synchronous state between identical and nonidentical chaotic systems. These methods include feedback control, adaptive control, sliding mode, backstepping control, active control [11–19] to list a few. Notable among these methods are the active control and backstepping techniques which have excellent performance in the synchronization of identical and nonidentical chaotic systems [20, 21].

The ubiquitous application of active control techniques has encouraged researchers to introduce active control based on different stability criteria. For instance, Lei et al [22] introduced active control based on Lyapunov stability theory and Routh Hurwitz criteria which has the advantage of possible implementation and has been used to synchronize a few chaotic systems [22, 23]. However, the active control technique always gives controllers which are as many as the dimensions of the system been synchronized. In this article, active control technique is used to design a controller for projective synchronization of three-dimensional autonomous chaotic system in such a way that the number of controllers is reduced from three to one thereby, reducing significantly the controller complexity and cost. Hence, this makes it feasible for practical implementation.

Meanwhile, the backstepping technique has been recognized as a powerful design technique for stabilization, tracking and synchronization of chaotic systems. It has been reported in [24] that backstepping can guarantee global stability, tracking and transient performance of a broad class of strict-feedback nonlinear systems. Recently, it has been used for controlling and tracking hyperchaotic systems [25]. According to ref. [26], some of the advantages of the method include: applicability to a variety of chaotic systems irrespective of whether they contain external excitation or not; needs only one controller to realize synchronization of chaotic systems and finally there is no derivative in the controller. Zhang et al [25] states that the controller is singularity-free from nonlinear term of quadratic forms, gives flexibility to construct a control law which can be extended to higher-dimensional hyperchaotic systems, while ref. [27] adds that it requires less control effort compared to differential geometric methods.

The Bonhöffer–van der Pol (or Fizhugh–Nagumo) oscillator model was derived from van der Pol oscillator to give more accurate and reliable description of nonlinear dynamical systems which can show a stable threshold phenomenon as well as stable oscillations. The Bonhöffer–van der Pol oscillator is closely related to Fizhugh–Hulley (FH) model of the squid giant axon, to the cat’s carotid sinus nerve [28]. Indeed, many studies on Bonhöffer–van der Pol oscillator have shown its various applications in medicine (see, for example, refs [28–31] and references therein). Recently, there has been a resurgent interest in the theoretical and experimental generation of multiscroll chaotic attractors [32]. This is partly due to many practical applications foreseen in such fields as digital and secure communication, synchronous prediction, random number generation, information systems, etc. [31]. An efficient secure communication model should exhibit appreciable synchronous performance even in chaotic state. Thus, exploring projective synchronization of multiscroll chaotic system will be a stimulating subject of research. However, it has received inadequate attention. A few reports on synchronization of multiscroll chaotic systems are limited to complete synchronization [33–35].

The goal of this paper is to design a single control function via active control and backstepping techniques and compare their performance in projective synchronization of two identical double-scroll chaotic attractors generated from an extended Bonhöffer–van der Pol oscillator. To the best of our knowledge, this problem has not been considered till now. The rest of this paper is organized as follows: In §2, a brief description of the system of the extended Bonhöffer–van der Pol oscillator is given. Sections 3 and 4 are devoted to projective synchronization of extended Bonhöffer–van der Pol oscillators via active control and backstepping techniques with numerical simulations. Section 5 deals with the comparison of active control and backstepping techniques while, §6 concludes the paper.

2 Description of the Bonhöffer–van der Pol oscillator model

The normalized equations for the extended Bonhöffer–van der Pol oscillator are

$$ \begin{array}{rll}\label{eq1} \dot x &=& -z+Ax+\tanh Bx, \nonumber\\ \dot y &=& z-\delta y \nonumber\\ \dot z &=& x-y \end{array} $$
(1)

where x and y are the state variables corresponding to the voltage across the capacitors and z corresponds to the value of fixed resistor. A and B are the control parameters of the oscillator. Extensive experimental and numerical study of the bifurcation and chaotic phenomenon of the oscillator described in (1) was recently carried out by Nshiuchi et al [36]. Besides, various dynamical behaviours observed in double-scroll chaotic attractors of Bonhöffer–van der Pol oscillator were reported for (A, B, δ) = (1.0, 1.0, 1.2). This attractor is shown in figure 1.

Figure 1
figure 1

Two-dimensional view of the double-scroll chaotic attractor of the extended Bonhöffer–van der Pol oscillator with A = 1.0, B = 1.0 and δ = 1.2.

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3 Projective synchronization of chaos in an extended Bonhöffer–van der Pol oscillators via active control technique

3.1 Design of a single active controller

Let the system of the Bonhöffer–van der Pol oscillator be written as

$$ \begin{array}{rll}\label{eq2} \dot x_1 &=& -z_1+Ax_1+\tanh Bx_1 , \nonumber\\ \dot y_1 &=& z_1-\delta y_1 , \nonumber\\ \dot z_1 &=& x_1-y_1. \end{array} $$
(2)

Then the response system is

$$ \begin{array}{rll}\label{eq3} \dot x_2 &=& -z_2+Ax_1+\tanh Bx_2+u_1(t) , \nonumber\\[2.5pt] \dot y_2 &=& z_2-\delta y_2+u_2(t) , \nonumber\\[2.5pt] \dot z_2 &=& x_2-y_2+u_3(t), \end{array} $$
(3)

where u i (t), i = 1,2,3 are the control functions to be determined. Subtracting (1) from (1) we obtain the error dynamics as

$$ \begin{array}{rll}\label{eq4} \dot e_1 &=& -e_3+A e_1+\tanh Bx_2-\alpha \tanh Bx_1+u_1(t), \nonumber\\[2.5pt] \dot e_2 &=& e_3-\delta e_2+u_2(t), \nonumber\\[2.5pt] \dot e_3 &=& e_1-e_2+u_3(t), \end{array} $$
(4)

where e 1 = x 2 − αx 1, e 2 = y 2 − αy 1 and e 3 = z 2 − αz 1. The projective synchronization between the drive (1) and the response (1) reduces to asymptotic stability of the error system (1) at equilibrium. To achieve this, control functions are re-defined to eliminate terms in (1) which cannot be expressed as a linear term in e 1, e 2 and e 3 as follows:

$$ \begin{array}{rll}\label{eq5} u_1(t) &=& \alpha \tanh Bx_1-\tanh Bx_2+v_1(t) \nonumber\\[2.5pt] u_2(t) &=& v_2(t) \nonumber\\[2.5pt] u_3(t) &=& v_2(t). \end{array} $$
(5)

Substituting (1) into (1) yields

$$ \begin{array}{rll}\label{eq6} \left(\begin{array}{ccc} \dot e_1 \\ \dot e_2 \\ \dot e_3 \end{array}\right)= \left(\begin{array}{rrr} A & 0 & -1 \\ 0 & -\delta & 1 \\ 1 & -1 & 0 \\ \end{array}\right) \left(\begin{array}{ccc} e_1 \\ e_2 \\ e_3 \end{array}\right)+ \left(\begin{array}{ccc} v_1(t) \\ v_2(t) \\ v_3(t) \end{array}\right). \end{array} $$
(6)

When eq. (1) is stabilized by the feedback v 1(t), v 2(t) and v 3(t), the errors will converge to zero as t→ ∞ which implies that projective synchronization of systems (1) and (1) is achieved. To achieve this goal using the active control technique, a constant matrix C is chosen to control the error dynamics (1) such that the feedback v 1(t), v 2(t) and v 3(t) are

$$ \begin{array}{rll}\label{eq7} \left(\begin{array}{ccc} v_1(t) \\ v_2(t) \\ v_3(t) \end{array}\right)=\mathbf{C} \left(\begin{array}{ccc} e_1 \\ e_2 \\ e_3 \end{array}\right), \end{array} $$
(7)

where C is a 3×3 matrix. There are various choices of the feedback C that would control the error dynamics in (1). We optimize the way this choice is made so that the problem of controller complexity is significantly reduced and the choice is

$$ \begin{array}{rll}\label{eq8} \mathbf{C}= \left(\begin{array}{ccc} -(\lambda+A) & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right). \end{array} $$
(8)

With this choice the control functions are reduced from three to one which is

$$\label{eq9} u_1(t) = \alpha \tanh Bx_1-\tanh Bx_2-e_1(\lambda+A)+e_3, $$
(9)

where λ is a positive constant control parameter.

3.2 Numerical simulation results

To verify the effectiveness of the designed controller we used the fourth-order Runge–Kutta algorithm with initial conditions (x 1,y 1,z 1) = (1.0, − 1.0, 1.0) and (x 2,y 2,z 2) = (0.5, − 0.5,0.5), a time step of 0.005, and fixing the parameter values of the system such that the systems exhibits double-scroll chaotic attractors as in figure 1 to ensure chaotic dynamics of the state variables. We solved systems (1) and (1) with the control function defined in (4) for λ = 1. The result displayed in figure 2a shows the projection of the response attractor on the drive system for α = 2.0. The two systems achieved projective synchronization as indicated by the convergence of the error state variables as soon as the controller is switched on for t ≥ 100 (figure 3a). The results obtained confirm the effectiveness of the single control function designed for the synchronization of a third-order double-scroll chaotic Bonhöffer–van der Pol oscillators via the active control technique.

Figure 2
figure 2

Projection of the attractor of the response Bonhöffer–van der Pol oscillator (green colour) on the drive Bonhöffer–van der Pol oscillator (red colour) with scaling factor of α = 2 via (a) active control; (b) the backstepping technique.

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Figure 3
figure 3

Error dynamics between two Bonhöffer–van der Pol oscillators with controller deactivated for 0 < t < 100 and activated for t ≥ 100 for a scaling factor α = 2 via (a) active control; (b) backstepping technique.

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4 Projective synchronization of chaos in an extended Bonhöffer–van der Pol oscillator via backstepping technique

4.1 Design of a single active controller

Let the drive system of the Bonhöffer–van der Pol oscillator be as in (1) and the response Bonhöffer–van der Pol oscillator as

$$ \begin{array}{rll}\label{eq10} \dot x_2 &=& -z_2+Ax_2+\tanh Bx_2+u(t) , \nonumber\\[3pt] \dot y_2 &=& z_2-\delta y_2 , \nonumber\\[3pt] \dot z_2 &=& x_2-y_2, \end{array} $$
(10)

where u(t) is the control function to be determined. Subtracting (1) from (5) and re-arrange the error dynamics, the error system can be written as

$$ \begin{array}{rll}\label{eq11} \dot e_2 &=& e_3-\delta e_2 , \nonumber\\[2.5pt] \dot e_3 &=& e_1-e_2 , \nonumber\\[2.5pt] \dot e_1 &=& -e_3+Ae_1+\tanh Bx_2-\alpha \tanh Bx_1+u(t), \end{array} $$
(11)

where e 1 = x 2 − αx 1, e 2 = y 2 − αy 1 and e 3 = z 2 − αz 1. The objective of this paper is to find a control function u(t) that can stabilize the error state in (5) at the origin. Firstly, we stabilize the first equation in (5) by regarding e 3 as a controller, choosing a Lyapunov function \(V_1(e_2)=\frac {1}{2}e_2^2\) and differentiating it with respect to time we have

$$\label{eq12} \dot V_1 = e_2e_3-\delta e_2^2. $$
(12)

We estimate that the controller e 3 = α 1 (e 2). Then, eq. (5) can be written as \(\dot V_1=-\delta e_2^2+e_2\alpha_1 (e_2)\). \(\dot V_1\) is negative definite if the estimated function α 1 (e 2) = 0. The error W 2 between e 3 and α 1 (e 2) is

$$\label{eq13} W_2 = e_3-\alpha_1 (e_2). $$
(13)

Differentiating (6) with respect to time yields

$$\label{eq14} \dot W_2 = e_1-e_2. $$
(14)

Choose a Lyapunov function \(V_2(e_2,W_2)=V_1(e_3)+\frac {1}{2}W_2^2\), regarding e 1 as the controller in (7) and assume that when e 1 = α 2 (e 2,W 2) then, the time derivative of the Lyapunov function is

$$\label{eq15} \dot V_2 = -\delta e_2^2+W_2\alpha_2 (e_2,W_2) $$
(15)

if α 2 (e 2,W 2) = − kW 2 then \(\dot V_2=-\delta e_2^2-kW_2^2<0\) is negative definite. Therefore, system (7) is asymptotically stable. The error W 3 between e 1 and α 2 (e 2,W 2) is

$$\label{eq16} W_3=e_1-\alpha_2 (e_2,W_2)=e_1+k W_2 $$
(16)

or

$$\label{eq17} e_1 = W_3-k W_2. $$
(17)

Substitution of (10) into the time derivative of (9) yields

$$\label{eq18} \dot W_3 = -e_3+A e_1+\tanh Bx_2-\alpha\tanh Bx_1+k(e_1-e_2)+u(t). $$
(18)

We choose Lyapunov function \(V_3(e_3,W_2,W_3)=V_2(e_2,W_2)+\frac {1}{2}W_3^2\) and obtain its time derivative as

$$ \begin{array}{rll}\label{eq19} \dot V_3 &=& -\delta e_2^2-k W_2^2+W_3[-e_3+(A+k)(W_3-W_2)-k e_2+\tanh Bx_2 \nonumber\\[3pt] && - \ \alpha \tanh Bx_1+ u(t)]. \end{array} $$
(19)

The control function is chosen as

$$ \begin{array}{rll}\label{eq20} u(t) &=& e_3-(A+k)(W_3-k W_2)-k(W_3-e_2)\nonumber\\[2.5pt] &&-\tanh Bx_2+\alpha \tanh Bx_1 \end{array} $$
(20)

so that \(\dot V_3=-\delta e_2^2-kW_2^2-kW_3^2<0\) which is negative definite, where k is a positive constant control parameter. It follows that all solutions of (5) converge to the manifold e 1 = e 2 = e 3 = 0 as t→ ∞ and hence systems (1) and (5) are globally synchronized. Thus, projective synchronization is achieved.

4.2 Numerical simulation results

To verify the effectiveness of the designed controller we used the fourth-order Runge–Kutta algorithm with initial conditions (x 1,y 1,z 1) = (1.0, − 1.0, 1.0) and (x 2,y 2,z 2) = (0.5, − 0.5,0.5), with a time step of 0.005, fixing the parameter values of the system such that the system exhibits double-scroll chaotic attractors as in figure 1 to ensure chaotic dynamics of the state variables. We solve systems (1) and (5) with the control function defined in (12) for k = 1. The result is displayed in figure 2b which shows the projection of the response attractor on the drive system for α = 2. The two systems achieved projective synchronization as indicated by the convergence of the error state variables as soon as the controller is switched on for t ≥ 100 (figure 3b). The results obtained confirm the effectiveness of the single control function designed for the synchronization of a third-order double-scroll chaotic Bonhöffer–van der Pol oscillators via the backstepping technique.

5 Comparison of active control and backstepping techniques

In order to make a detailed comparison between the two technique we obtained a relation between synchronization time and the control parameters. In our simulation, we varied the value of the control parameter k and λ between 0.1 and 10, and the result obtained for backstepping technique is shown in figure 4a, while, that for active control is shown in figure 4b. For clarity and effective comparison, the results in figures 4a and 4b are combined in figure 5. From the figures we notice that for active control, the least synchronization time is 111.8 which occurs at λ = 0.6 while, for backstepping technique the least synchronization time is 105.1 which occurs at k = 2.2 and 2.5. In order to obtain the fastest synchronization time (111.8) using the active control the choice of λ should be 0.6 while, for the backstepping technique k should be 2.2 or 2.5 in order to obtain fastest synchronization time (105.1). Moreover, it clear from figure 5 that the active control has a more stable synchronization time. Increase in the control parameters beyond the values quoted above amount to waste of energy. There is no significant difference in the time for the onset of synchronization. We also compared the result using the rate of error convergence e and the rate at which the ratio r/d tends to the predefined scaling factor (figures 6a and 6b) respectively, where

$$ e=\sqrt{e_1^2+e_2^2+e_3^2}, \quad r=\sqrt{x_2^2+y_2^2+z_2^2}, $$

and

$$ d=\sqrt{x_1^2+y_1^2+z_1^2}. $$

The results also show that the transient error dynamics via the backstepping technique converges faster than that of the active control technique.

Figure 4
figure 4

Comparison of synchronization time between active control and backstepping technique with controller deactivated for 0 < t < 100 and activated for t ≥ 100 for a scaling factor α = 2 via (a) backstepping technique; (b) active control.

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Figure 5
figure 5

Synchronization time against the variation of the positive constant parameters with controller deactivated for 0 < t < 100 and activated for t ≥ 100 for a scaling factor α = 2 via (a) backstepping technique; (b) active control.

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Figure 6
figure 6

(a) The time evolution of transient error dynamics; (b) the variation of the scaling factor via active control (red) and backstepping (green).

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6 Conclusion

We have investigated projective synchronization of double-scroll chaotic attractors of an extended Bonhöffer–van der Pol oscillator via backstepping and active control technique. In each synchronization scheme, we designed a single control function which achieved projective synchronization between two identical Bonhöffer–van der Pol oscillators evolving from different initial conditions. For the active control technique, the choice of the coefficient matrix of the error dynamics was chosen such that the number of control functions reduced from three to one, thereby, reducing the controller complexity in the design. The results show that the transient error dynamics convergence and synchronization time are achieved faster via the backstepping than via the active control technique. However, the control function obtained via the active control is simpler with a more stable synchronization time and hence, it is more suitable for practical implementation. Numerical simulations are presented to confirm the effectiveness of the analytical results.