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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.

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 [

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 [

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 [

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 [

The memristor is believed to be the essential for mimicking the neuron network of the artificial brain [

The word memfractor denotes the fractional-order memristor and was proposed in [^{th} order voltage-controlled fractional-order memristor (memfractor) can be described as

(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 (

We use memfractor model (

Upper: absolute memfractor Wien bridge oscillator; lower: the I-V characteristic of the memfractor for

Applying Kirchoff’s law to the circuit shown in Figure

We have used the predictor-corrector method [

Equation (

The error estimate is

2D phase portraits of 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.

The AMWO system has three equilibrium points as follows:

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

In parameter values

The AMWO is asymptotically stable if

The eigenvalues of the AMWO for the equilibrium points

Condition for asymptotic stability of AMWO for the equilibriums

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

Using Corollary

Lyapunov exponents (LEs) of the AMWO are derived using Wolf’s algorithm [

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

(a) The bifurcation of the AMWO system with

Coexisting attractors of the AMWO for

In the second scenario, we have considered the fractional order of the AMWO system and have derived the bifurcation plot as seen in Figure

(a) Bifurcation of the AMWO with fractional order

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 (

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

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

Coexisting attractors of the AMWO obtained from Pspice with the initial voltages

From Figures

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 [

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

Differentiating (

Replacing systems (

The control functions are derived from system (

For

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

For

For numerical verifications, we fix

Combination synchronization errors between drive (

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.

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

The authors declare that they have no conflicts of interest.