Multiswitching Synchronization of a Driven Hyperchaotic Circuit Using Active Backstepping

1 Department of Physics, University of Agriculture Abeokuta, PMB 2240, Abeokuta, Nigeria 2 Department of Physics, Federal College of Education (Technical) Akoka, P.O. Box 269, Yaba, Lagos 101212, Nigeria 3 Department of Physics, University of Lagos, Akoka, Yaba, Lagos 101212, Nigeria 4Department of Physical Sciences, Redeemer’s University, Mowe, Ogun State, Redemption City 110115, Nigeria 5 Department of Physics, Lancaster University, Lancaster LA1 4YB, UK


Introduction
Nonlinear deterministic dynamical systems exhibiting sensitive dependence to initial conditions (chaos) have been found to exist and not unusual.Different methods used to describe their existence are found in the fields of sciences (physical and natural), medicine, and engineering [1][2][3].Various attributes of nonlinear dynamical systems such as chaos, bifurcation, multistability, pattern formation, control, and synchronization have been found useful or having potential applications in many disciplines.One of the most important attributes of nonlinear dynamical systems is that of synchronization of chaotic systems which has become more fascinating and has generated interest from researchers in recent time due to its applications in information processing, secure communication, chemical reactions, and modelling brain activity [4][5][6][7].Increasing interest in the study of synchronization of chaotic systems has led to the discovery of various types of synchronization which include complete synchronization [8], lag synchronization [9], phase synchronization [10], generalized synchronization [11][12][13], measure synchronization [14][15][16][17], projective synchronization [18][19][20], anticipated synchronization [21,22], and reduced-order synchronization [23][24][25][26][27][28][29][30][31].Recently multiswitching synchronization of chaotic systems was proposed by Uc ¸ar et al. [32].The method was successfully implemented with Lorenz system using active control technique.
Several synchronization methods have evolved to achieve stable synchronization between two or more chaotic systems.These methods include adaptive control [33], active control [34], sliding mode control [35], impulsive control [36], linear feedback control [37], backstepping control [38], open plus close loop control [39], adaptive fuzzy feedback [40], and passive control [41].Notable among this method is the backstepping control technique which has outstanding performance in the synchronization of identical and nonidentical chaotic systems [42,43].It has also been identified as a reliable design technique for stabilization and tracking [44].In recent time, backstepping technique has been applied for controlling and 2 Journal of Nonlinear Dynamics tracking hyperchaotic systems [45].In spite of the excellent performance of backstepping control technique, it has not been applied to multiswitching synchronization of chaotic system to the best of our knowledge.Motivated by the above discussion, we are reporting multiswitching synchronization of unified and periodically forced hyperchaotic Van der Pol-Duffing oscillator using active backstepping technique.
Van der Pol-Duffing oscillator is a popularly known and very significant classical model circuit that has been studied and even modified in some studies as reported by King et al. [46,47], Fotsin et al. [48], and Fodjouong et al. [49].The Van der Pol-Duffing oscillator in most reported cases is autonomous (unforced) whereas the periodically driven nonautonomous Van der Pol-Duffing oscillator considered in this paper displays more complex and richer dynamics as the amplitude and the frequency of the forcing signal are varied, thereby exhibiting chaos-hyperchaos transitions, and coexisting attractors as well as Hopf bifurcations in which quasiperiodic orbits are born [50].Recently, there has been a renewed interest and several attempts to understand the dynamics of nonautonomous (forced) oscillators excited from equilibrium by different external forcing techniques.This is because deterministic influences often arise in practice as in cellular dynamics, blood circulation, and brain dynamics [51,52].
In the present paper, we introduced the periodically forced Van der Pol-Duffing oscillator and study its synchronization.Specifically, we consider a more general form of synchronization, namely, multiswitching synchronization, in which a variable of a dynamical system synchronizes with another variable of the systems or another system.This type of synchronization was proposed by Uc ¸ar et al. [32].Despite its relevance, it has received less attention.Here, we propose an active backstepping technique to realize the multiswitching synchronization.The rest of the paper is organized as follows: the basic dynamical properties of our model system are described in Section 2. In Section 3, the multiswitching synchronization of UVDP using active backstepping is discussed.Numerical simulations are provided in Section 4, while the paper is concluded in Section 5. oscillator considered here can be modeled by the circuit shown in Figure 1, in which a dc series resistance is added to the  2 branch of the circuit, while another dc resistance is placed parallel to it.In addition, a periodic signal generator, acting as periodic driving, is connected to the left end of the circuit as shown.By applying Kirchhoff 's laws to the various branches of the circuit of Figure 1, and noting that the (V) characteristics of the nonlinear resistor () are approximated by the cubic polynomial (V) = V + V + V 3 , ( < 0,  > 0), we obtain the following set of equations:

Model Description
Making appropriate rescaling of (1) by setting and  = ( +   )/  , we obtain the following dimensionless equation: System ( 2) is a three-dimensional nonautonomous system.By letting  =  0 sin Ω, the system of (2) can be rewritten as a system of four-dimensional autonomous dynamical system (including  as a dynamical variable) as follows: where the new parameters  and  0 are the frequency and amplitude of the periodic driving force, respectively.Thus, our model system (2) or its equivalent system (3) has four Lyapunov exponents, allowing for hyperchaotic behaviour, that is, with the possibility of having two positive Lyapunov exponents along with one zero and one negative.Furthermore, under periodic driving, the unified Van der Pol-Duffing oscillator exhibits richer dynamical complexities, including the existence of hyperchaos-chaos transient and regime of periodic behaviour could be observed with the aid of bifurcations and Lyapunov exponents [50].In Figure 2, we present an illustration of a typical dynamics of our model from [50].The numerical results were performed using the standard fourth-order Runge-Kutta routine with step-size ℎ = 2/(), where  is the number of iterations.The complete Lyapunov exponent spectrum was computed using the Wolf et al. 's algorithm [53].For clarity and brevity, we will present only the first two exponents, namely,  1 and  2 , which determine exclusively the system's behaviours.Unless otherwise stated, the following parameters were fixed:  = 0.35,  = 300, and  = 100, while the other parameters, namely, , ,  0 , and , were varied for the different cases considered.In the absence of the forcing, we obtain the Fotsin and Woafo model when  = 1.0, and  = 0.2 [48], whereas, when  > 1.0 and  = 0, we have the Matouk and Agiza model [54].In Figures 2(a) and 2(b), we see clearly that  1 and  2 are positive with increasing amplitude,  0 , confirming the existence of hyperchaos in this driving amplitude regime.

Stability and Equilibria.
The changes discussed above maybe understood by examining the stability of system (3).First, we obtain the fixed points by solving the general equation ( u ) = 0, where  is the vector space containing , , and  at  0 = 0.The equilibrium points are found by equating the right-hand sides of (2) to zero such that from which the following fixed points were obtained: , +, +) ,  > 0, , −, −) ,  < 0,  0 = (0, 0, 0) , where  = ( ( + 1) + ) The characteristic equation of the Jacobian matrix of system (5) about the equilibrium points  = (, , ) is given as where Before analyzing the stability properties of each points, it is important to consider the practical process of modeling system (4).It is clear that only  can take on negative or positive values.The other parameters are always positive.In order to study the stability conditions of the equilibrium point, , we apply Routh-Hurwitz criterion [55], which states that all real eigenvalues and all real parts of complex conjugate eigenvalues are negative if and only if the following equation holds: At values of  = 100,  = 0.35,  = 300,  = 0.2, and  = 1.0.

Multiswitching Synchronization via Backstepping
Here, we develop multi-switching synchronization procedure based on active-backstepping and applied to the periodically forced Van der Pol-Duffing oscillator due to its richer dynamical complexities and relevance in the area of chaotic systems that can be physically built and analyzed.It is convenient to transform the driven Van der Pol-Duffing oscillator system (2) by redefining the variables as follows:  = ,  = , and  =  such that we have the following master system: and the corresponding slave system where  11 (),  12 (), and  13 () are the set of nonlinear controller for the first switch ( = 1).We define the error signals for this switch as  11 =  2 −  1 ,  12 =  2 −  1 , and  13 =  2 −  1 .By considering the time derivative of the error signals together with (10) and (11), we determine the controller that would cause the three terms of error signals to asymptotically approach zero as  → ∞.The controller was found from the following closed loop dynamics: With error dynamics represented by (12), the synchronization problem is equivalent to stabilizing the system at one of its equilibria.Since the equilibria  + and  − are stable and symmetric, so if appropriate  11 (),  12 (), and  13 () are chosen such that the equilibrium  + or  − is stable and unchanged, then asymptotic stabilization would be realized leading to globally stable synchronization of systems (10) and (11).To start with, let  1 =  11 , which implies that q 1 = ė 11 .Then, choose  1 to be a Lyapunov function for the q 1 subsystem given as and then If  12 =  1 is a virtual control and a function of  11 , making  11 = 0 and  1 = 0, thus we obtain which is negative definite, and this means that the subsystem,  11 , is fully stabilized.To stabilize the second subsystem in (12), let the error state between  12 and  1 be  2 Introduce the second Lyapunov function  2 given as Taking time derivative of ( 16), we have Again, if  13 =  2 = 0 is a virtual control and  12 =  1 (1 − ), then Equation ( 18) is negative definite, meaning that the subsystem,  12 , is also fully stabilized.To stabilize the third subsystem in (12), let the error state between  13 and  2 be We choose the third Lyapunov function and its time derivative expressed as If the controller  13 () is chosen such that then is negative definite, so that the two systems (10) and ( 11) are globally asymptotically synchronized.Summarily, the virtual controllers for the system state  = 1 are stated as follows: More switching states that were investigated will be presented with the parameter symbol  = 2, 3, 4, 5, and 6.The error signals for these various switching states can be defined as follows: Following the same procedure that is itemized from (13) to (24), the controller signals  1 (),  2 (), and  3 () for  = 2, 3, 4, 5, and 6 switching, designed to cause the respective error dynamics to approach zero as time approaches infinity, are as follows:

Numerical Simulations
Numerical solutions are now presented to verify the effectiveness of controllers ( 24) and ( 30)- (34).In all six cases presented, the periodically driven oscillator parameters selected remain constant at  = 100,  = 300,  = 0.35,  = 0.2,  = 1.0,  0 = 1.51, and  = 10.This step is very important such that the hyperchaotic state obtained in [50] and shown in Figure 2 is retained and also to disable the multistability property of the system.The synchronization of the slave (13) with the master system (12) for different cases represented by  = 1, 2, 3, 4, 5, and 6 is presented in Figures 3, 4 when each of the controllers was activated at  ≥ 50, where the unified and periodically driven Van der Pol-Duffing oscillator was found to converge to zero as time tends to infinity which signifies that the multiswitching synchronization between systems ( 12) and ( 13) has been achieved.We remark, however, that quasi-synchronized (partial synchronization) state may also be achieved.Here, the oscillators suffer from achieving full synchronization but settle for a rather realistic form of complete synchronization wherein the state in which the defining limit of synchronization is bounded within a definite small region around zero.That is, lim  → ∞ ‖  ‖ =  ̸ = 0. We find this phenomenon when for  1 , the switching is between  1 and  2 as well as, for  3 , when, the switching is between  2 and  2 , denoted as  12 and  22 in Figures 3(b) and 4(b), respectively.This kind of synchronization phenomena are intriguing in nature as well as in practical situations and have been previously reported [56][57][58][59].

Conclusion
In this paper, we have presented in brief the dynamics of a unified and periodically driven Van der Pol-Duffing oscillator circuit and have shown that the model admits hyperchaotic behaviour when the amplitude of the driving force is increased with a good choice of forcing frequency.Using our model circuit, an approach for realizing multiswitching synchronization was proposed based on active backstepping technique.The numerical results obtained confirmed the effectiveness of the proposed analytical method.

Figure 3 :
Figure 3: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 1 with controller defined in (24) when the control was activated at  = 50 for (a)  11 , (b)  12 , and (c)  13 .

Figure 4 :
Figure 4: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 2 with controller defined in (30) when the control was activated at  = 50 for (a)  21 , (b)  22 , and (c)  23 .

Figure 5 :
Figure 5: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 3 with controller defined in (31) when the control was activated at  = 50 for (a)  31 , (b)  32 , and (c)  33 .

Figure 6 :
Figure 6: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 4 with controller defined in (32) when the control was activated at  = 50 for (a)  41 , (b)  42 , and (c)  43 .

Figure 7 :
Figure 7: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 5 with controller defined in (33) when the control was activated at  = 50 for (a)  51 , (b)  52 , and (c)  53 .

Figure 8 :
Figure 8: Error dynamics for the unified and periodically forced Van der Pol-Duffing oscillator of the system state for  = 6 with controller defined in (34) when the control was activated at  = 50 for (a)  61 , (b)  62 , and (c)  63 .