Complexity
Volume 2021 (2021), Article ID 8857075, 13 pages
https://doi.org/10.1155/2021/8857075

Research Article

A Simple Chaotic Wien Bridge Oscillator with a Fractional-Order Memristor and Its Combination Synchronization for Efficient Antiattack Capability

Anitha Karthikeyan,¹ Karthikeyan Rajagopal,² Victor Kamdoum Tamba,³ Girma Adam,⁴ and Ashokkumar Srinivasan²

¹Ton Duc Thang University, Ho Chi Minh City, Vietnam
²Center for Nonlinear Systems, Chennai Institute of Technology, Malayambakkam, India
³Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P. O. Box: 134, Bandjoun, Cameroon
⁴Center for Nonlinear Dynamics, Defence University, Bishoftu, Ethiopia

Correspondence should be addressed to Karthikeyan Rajagopal; rkarthiekeyan@gmail.com

Received 1 September 2020; Revised 7 October 2020; Accepted 12 February 2021; Published 1 March 2021

Academic Editor: Chun-Biao Li

Copyright © 2021 Anitha Karthikeyan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Memristor-based oscillators are of recent interest, and hence, in this paper, we introduce a new Wien bridge oscillator with a fractional-order memristor. The novelty of the proposed oscillator is the multistability feature and the wide range of fractional orders for which the system shows chaos. We have investigated the various dynamical properties of the proposed oscillator and have presented them in detail. The oscillator is then realized using off-the-shelf components, and the results are compared with that of the numerical results. A combination synchronization scheme is proposed which uses more than one driver systems to synchronize with one response system. Indeed, we use two different techniques where the first one consists of splitting the transmitted signals into two parts where each part is loaded in different drive systems, while the second one consists of dividing time into different intervals and loading the signals in different drive systems. Such techniques improve the antiattack capability of the systems when used for secure communication.

1. Introduction

Versatile dynamical behaviors observed from memristor-based chaotic circuits obviously challenge the scientific community. After the physical realization of the memristor by HP Labs, many real-time circuits were designed for different kinds of applications. On the contrary, complex systems which hold intricate properties, much sensitive to initial conditions and influenced by history of variable states, demand memristor-based mimic models. Sufficient literature studies were identified for extracting and investigating special properties such as coexistence of multiple attractors [1–5], antimonotonicity [6, 7], hidden attractors [3, 8–11], and time delays [12, 13]. The challenges are handled effectively, while implementing those models in real-time circuits [1, 3, 8, 10, 11, 14].

A horde of physical phenomena holds fractional-order description; therefore, the differential equations formulated to analyze its dynamical nature need to be treated with the fractional-order form. Noninteger order formulation provides a more accurate model of physical systems than integer calculus do. A variety of applications [15–21], both in one dimension and in multidimensional space, especially nonlinear behaviors are modelled with fractional-order differential equations in order to reveal the characteristics in the complex scenario. In [22], fractional time derivatives-based oscillators are formulated using the Riemann–Liouville method. A comparative study was carried out on linearly damped oscillator equations, and the fractional-order treatment showed interesting phenomena over integer-order treatment [23].

The characteristics of nonlinearity present in the memristor element leads the memristor-based circuits to generate a chaotic signal easily. The presence of a “pinched hysteresis loop” in the current-voltage characteristics of memristive system shows its nonlinear behavior. The principle of fractional calculus is based on the memory property of the fractional-order integral or derivative. Hence, the relation between fractional calculus and memristor is clear enough to understand, and the memristor can be extended to the fractional order as well. In [24], a fractional-order memristor is modelled for Chua’s oscillator, and it successfully showed the chaotic behaviors. In [25], various fractional-order circuit elements were modelled and analyzed. Fractional-order generalized memristor-based chaotic systems are modelled and analyzed with circuit implementation [26]. Detailed analysis on fractional memristors and their intricate behaviors and influence on circuits are analyzed, and simple fractional-order circuits are also discussed in [27, 28]. In the last decade, many chaotic oscillators were formulated using memelements especially the memristor [29–32].

In a general Wien bridge oscillator, an operational amplifier is connected parallel to RC and series RC networks. An attempt is made to replace the resistor in parallel configuration with a memristor in [33–35] and come up with a strange nonchaotic attractor. A fractional-order memristor-based hyperchaotic Wien bridge oscillator circuit is modelled using the Adomian Decomposition method, and analysis found that compared with the integer order form, the fractional-order form revealed many intricate properties like coexisting of multiple attractors [36]. In [37], analysis of dynamical behaviors and realization of a fractional-order memristor-based chaotic circuit is carried out.

The memristor is believed to be the essential for mimicking the neuron network of the artificial brain [38], and the configuration of the nervous system is physically closer to the fractal dimensions [39, 40]. Henceforth, the research studies on the fractional-order memristive systems have great impact for the realization of the artificial intelligence. It is also necessary to do a detailed dynamical analysis for the memristive chaotic circuit system. There was recently a memristor-based chaotic oscillator which comes from the classical Chua circuit and is devoted to weak signal detection application discussed in the literature [41], and a van der Pol–Duffing oscillator was proposed and analyzed its applications in weak signal detection [42]. It is to be noted that all these literatures are integer order models but a fractional-order memristor can perfectly model the hysteresis and memory effects of a memristor compared to their integer order counterparts. Hence, in this paper, we propose a new fractional-order memristor-based Wien bridge oscillator.

2. Absolute Memfractor Wien Bridge Oscillator (AMWO)

The word memfractor denotes the fractional-order memristor and was proposed in [43]. By the definition of memristor, the nth order voltage-controlled fractional-order memristor (memfractor) can be described as

where is the current flowing through the memristor, V is the voltage across the memristor, and is the state variable. The new voltage-controlled fractional-order absolute memristor can be modelled as

where is the absolute memristor. The equivalent circuit for the fractional-order absolute memristor is shown in Figure 1. We have used the fractional-order integrator proposed in [44, 45] to design the circuit. Figure 1(a) shows the fractional-order memristor emulator, and Figure 1(b) shows the fractional-order integrator used in Figure 1(a).

(a)

(b)

Figure 1: (a) Fractional-order absolute memristor emulator; (b) fractional-order integrator using first-order approximation.

Using the memfractor circuit, we can derive the following modified forms of (2) as follows:

We use memfractor model (3) to design a new absolute memfractor Wien bridge oscillator, as shown in Figure 2. In the circuit, we consider voltage across the capacitors and the current through the memfractor () as the three state variables. We have also shown the V-I characteristics of the memfractor using the simulation and experimental model in Figure 2 (lower).

Figure 2: Upper: absolute memfractor Wien bridge oscillator; lower: the I-V characteristic of the memfractor for using (a) simulation and (b) Pspice circuit.

Applying Kirchoff’s law to the circuit shown in Figure 2, the mathematical model can be represented as

where. Let us define , , , , , , , and , and using these definitions in (4), the dimensionless model can be derived as

We have used the predictor-corrector method [46, 47] to discretize the AMWO system (5), and the predict evaluate correct evaluate (PECE) method of ABM studied in [48, 49] used. In order to derive the general model of the PECE [46, 47] method, the third-state variable of the AMWO with order is considered as

where for .

Equation (6) is similar to the Volterra integral equation [23] as

which in equation (7), and as . The discrete form of equation (8) can be defined as

where

The error estimate is , where . For the numerical analysis of the AMWO system, we have used RK-4 to solve the first two state equations, while the third fractional-order state is solved using PECE derived in (8). System (5) shows the chaotic attractor for , , , , , , and fractional order . We have shown the 2D phase portraits, as shown in Figure 3, and the step size . It is to be noted that most of the fractional-order literatures showed that having larger step size will require greater memory size. Hence, as said in literatures [21–24], the step size is taken as h = 0.01.

(a)

(b)

Figure 3: 2D phase portraits of AMWO system (5) for the initial conditions .

3. Dynamical Properties of the AMWO System

In order to investigate the dynamical behavior of the AMWO system, we have used the tools like the equilibrium points and its stability, Lyapunov exponents, bifurcation plots, Lyapunov spectrum, Poincare map, basin of attractions, etc.

3.1. Stability of Equilibrium Points

The AMWO system has three equilibrium points as follows:

The generalized characteristic polynomial of the AMWO system is calculated using the relation which is derived as

In parameter values , , , , , and , the characteristic equation of the AMWO system for the equilibrium point is . The stability of the AMWO system can be defined by the following corollaries.

Corollary 1. The AMWO is asymptotically stable if

for any of the equilibrium points. In this case, the components of the state decay towards 0 like t.

The eigenvalues of the AMWO for the equilibrium points are plotted against the fractional order , as shown in Figure 4, and it could be easily seen that, for , the system is asymptotically stable.

(a)

(b)

Figure 4: Condition for asymptotic stability of AMWO for the equilibriums ((a), (b)).

Corollary 2. For the chaotic attractor to exist in the AMWO, the equilibrium points corresponding to the oscillations should exhibit instability. So, the necessary condition for the existence of the unstable equilibrium is

where are the roots of for each

Using Corollary 1 and Corollary 2, the AMWO has chaotic dynamics in for .

3.2. Lyapunov Exponents

Lyapunov exponents (LEs) of the AMWO are derived using Wolf’s algorithm [50] and the fractional-order predictor-corrector [49] solver fde12 [51] as the ode solvers [52]. The LEs for the AMWO system are calculated as for fractional order . The Kaplan–Yorke dimension of the AMWO system is .

3.3. Bifurcation

To investigate the parameter impact on the AMWO system, we derive the bifurcation plots. We considered two different bifurcation scenarios with the first one taking the parameter as the control value and in the second one taking the fractional order as the control variable. We have used forward and backward continuation in which either the parameter is increased or decreased with reinitializing of the initial conditions in every iteration to the end values of state trajectories, as shown in Figure 5(a). We have also plotted the corresponding finite-time Lyapunov exponents (LEs) calculated using Wolf’s algorithm [50] and the fractional-order predictor-corrector [49] solver fde12 [51] as the ode solvers [52], as shown in Figures 5(b) and 5(c). We could clearly see coexisting chaotic attractors for , coexisting period-4 limit cycles for , coexisting period-2 limit cycles for , and coexisting period-1 limit cycle for . The various coexisting attractors for different values of at are shown in Figures 6(a)–6(c). The basin of attraction of the system in x-z plane is shown in Figure 6(d).

(a)

(b)

(c)

Figure 5: (a) The bifurcation of the AMWO system with (red dots showing backward continuation and blue dots showing forward continuation); (b) LEs of forward continuation; (c) LEs of backward continuation.

(a)

(b)

(c)

(d)

Figure 6: Coexisting attractors of the AMWO for and initial conditions [0.1; 0.1; 0] shown in green colour and [−0.1; −0.1; 0] shown in purple for (a) ; (b) ; (c) . (d) The basin of attraction of the AMWO in the x-z plane having the cross-section at y = 0.1.

In the second scenario, we have considered the fractional order of the AMWO system and have derived the bifurcation plot as seen in Figure 7(a). We could see a very narrow band of chaos for and a wide band of chaos for . The LEs are computed and shown in Figure 7(a).

(a)

(b)

Figure 7: (a) Bifurcation of the AMWO with fractional order ; (b) the corresponding LEs.

4. Circuit Implementation

In this section, we perform some Pspice-based circuit simulations in order to compare the results with those obtained numerically. The analogue circuit diagram modelling of the AMWO system (5) is provided in Figure 8. By using Kirchhoff’s electric laws on the analogue circuit diagram of Figure 8, the corresponding circuit state equations can be expressed as

where , , and are the output voltages of the operational amplifiers OP1, OP2, and OP3, respectively. The capacitor denotes the fractional-order unit circuit consisting of , , , and . For the value of system (5) selected as , , , , , and and fractional order , the corresponding values of circuit elements can be calculated as , , , , , , , , , , , and .

(a)

(b)

(c)

Figure 8: Analogue circuit diagram modelling of the AMWO system (a). The fractional-order unit circuit (b). Circuit realization of the absolute value function (c).

The 2D phase portraits of the AMWO system implemented in Pspice are shown in Figure 9.

Figure 9: 2D phase portraits of the absolute memfractor Wien bridge oscillator (AMWO) obtained from Pspice with initial voltages .

The coexistence of attractors in the AMWO system obtained from the Pspice is presented in Figure 10.

(a)

(b)

(c)

Figure 10: Coexisting attractors of the AMWO obtained from Pspice with the initial voltages and for (a) , (b) , and (c) . The other values of circuit elements are defined as in the text.

From Figures 9 and 10, we can observe that the Pspice-based circuit simulations’ results are consistent with those obtained from numerical simulations (see Figures 3 and 6). This means that the proposed analogue circuit of Figure 8 is able to mimic the dynamics of the AMWO system 5.

5. Combination Synchronization of the Wien Bridge Oscillator with the Fractional-Order Memristor

Chaos synchronization plays a very important role in secure communication systems. In recent decades, many secure communication systems based on the synchronization of the one drive system with a response system have been intensively investigated both theoretically and practically using many different methods. However, this type of secure communication systems is relatively vulnerable to the attacks due to the fact that they use only one transmitter. To overcome this problem of security and ensure safer communication, some interesting methods of synchronization have been developed including combination-combination synchronization, multiswitching combination synchronization, combined projective synchronization, equal combination synchronization, and combination synchronization [53–58]. The latter synchronization method consists of using more than one driver systems to synchronize with one response system. This synchronization method has great advantages (e.g., stronger antiattack ability and antitranslated capability) over the traditional drive-response synchronization systems. The combination synchronization method is described as follows: (a) the drive systems are divided into at least two groups, and each group exists as one of the parallel branches in the system combination; (b) the drive systems on the same branch jointly drive the response system; (c) the drive systems on different branches synchronize with the response system separately; (d) the response system takes merely one controller in the combination synchronization.

According to the advantages of such synchronization methods, in this section, we design and perform numerically the synchronization of two drives and one-response absolute memfractor Wien bridge oscillators (AMWO) based on the combination method. The drive-response systems are constructed as

where and

where are the control functions to be determined such that the synchronization between the three systems can be realized. For the combination synchronization, the error is defined as

where , , , and and , , . The main objective is to design the control functions for the response system [15], such that the error defined by (17) can be asymptotically stable at the zero equilibrium, i.e., . Thus, the combination synchronization is achieved between the three drive-response systems. To achieve this, we assume that , , and . Considering these assumptions, system (17) is rewritten as

Differentiating (18), we obtain the error dynamical system as

Replacing systems (15), (16), and (18) into system (19) yields

The control functions are derived from system (20) as follows:

in which () are linear functions selected such that the error dynamical system becomes stable. Lets choose these linear functions in the following form:

where

is a real matrix.

For , , , , , , , , , , , , and , the error dynamical system becomes

Note that these values of parameters are chosen such that the error dynamical system becomes stable.

For , , and , the eigenvalues of error dynamical system (24) are , , and . The condition for the stability of the error dynamical system (24) is . Using the above eigenvalues, we obtain which satisfy the above condition for stability. This means that the error states converge asymptotically to zero as , and thus, the combination synchronization between the three drive-response systems is achieved.

For numerical verifications, we fix , , , , , , and such that the AMWO system is still chaotic. The initial states of drive and response systems (15) and (16) are chosen, respectively, as  =  =  = 0.1, , , , , , and . The combination synchronization results depicting the errors between the drive and response systems (15) and (16) are presented in Figure 8. From Figure 11, one can observe that the error states converge to zero which implies that the combination synchronization is realized between the three drive-response systems (15) and (16).

(a)

(b)

(c)

Figure 11: Combination synchronization errors between drive (1) and response (2) absolute memfactor Wien bridge oscillators for , , , , , and , and the control parameters are selected as .

6. Conclusion

In this paper, we proposed a Wien bridge oscillator with the fractional-order memristor. Most of the fractional-order Wien bridge oscillators consider all the states as noninteger orders, but in this paper, we considered only the internal state of the memristor as the fractional order which enables us to explore a wide variety of unexplored properties like the multistability and coexisting attractors. Pspice-based circuit simulations are shown to prove the realizability of the proposed oscillator. The results of the Pspice simulations are compared with theoretical results. A combination synchronization scheme is proposed in which we have used two slave systems to synchronize with the master system. Such techniques enable us to implement much complex antiattack features by using two different methods. The first one consists of splitting the transmitted signals into two parts where each part is loaded in different drive systems, while the second one consists of dividing time into different intervals and loading the signals in different drive systems. The synchronization errors are numerically calculated and are shown to prove the effectiveness of the proposed technique.

Data Availability

All the numerical simulation parameters are included within the article, and there were no additional data requirements for the simulation results.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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