Four-Scroll Hyperchaotic Attractor in a Five-Dimensional Memristive Wien Bridge Oscillator: Analysis and Digital Electronic Implementation

Research Unit of Condensed Matter, Electronics and Signal Processing (RU-MACETS), Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon Research Unit of Automation and Applied Computer (RU-AIA), Electrical Engineering Department of IUT-FV of Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Dschang, Cameroon Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi, B.P. 613, Porto Novo, Benin Center for Nonlinear Dynamics, Defence University, Bishoftu 6020, Ethiopia Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon


Introduction
An evidence fact in the research community is that the electronic circuits containing nonlinear elements exhibit rich dynamic behavior and it has been described in numerous books [1][2][3][4]. e research of chaotic memristive circuits is a hot topic of academic research in these recent years [5][6][7][8] due to their tremendous engineering applications. We can cite the field applications of communication systems, neural networks, image security, and so on. Memristor-based circuits are famous for displaying a rich variety of behaviors, including multiperiodic, quasiperiodic, and chaotic oscillations as well as self-pulsing and the coexistence of multiple attractors and hidden attractors [9][10][11][12][13][14]. e Wien bridge oscillator among many types of memristor-based oscillators appears to be one of the most studied recently with good standing papers published [15][16][17]. Memristor-based circuits are famous for displaying a rich variety of behaviors.
ese striking scenarios are defined as follows.
(i) Multistability is a critical property of nonlinear dynamical systems, where a variety of behaviors such as coexisting attractors can appear for the same parameters, but different initial conditions. e flexibility in the system's performance can be achieved without changing parameters. is striking scenario has been witnessed in numerous fields of engineering ranging across physics [18], biology [19], chemistry [20], electronics [21][22][23], and mechanics, as well as reported applications in oscillators and secure communications.
(ii) Quasiperiodicity is the property of a system that displays irregular periodicity [24]. Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It has been witnessed in rare systems such as the acoustic field [25], laser [26], and neural network [27].
(iii) Hyperchaotic scenario in the dynamical system is defined as a chaotic system with more than one positive Lyapunov exponent; this implies that its chaotic dynamics extend in several different directions simultaneously [24]. erefore, comparing with the traditional chaotic system, the hyperchaotic system has more complex dynamical behaviors which can be used to enhance the security of the chaotic communication system [28]. Consequently, the topic of theoretical design and circuitry realization of various hyperchaotic systems has recently become a hotspot in the nonlinear research field. Hyperchaos has been found numerically and experimentally such as Chua's circuit [29], Chen system [30], or Lorenz equation [31].
(iv) Bursting oscillations are defined as complex oscillations consisting of spiking (cluster of spikes or rapid oscillations) separated by periods of relative quiescence [32]. ey have been observed in many practical systems and found a multitude of applications in areas such as electromechanics [33], electronics [34], biology [35], and bioengineering of artificial organs [36]. ey have been discovered in many fields, magnetohydrodynamics [37], plasma confinement [25], and X-ray pulsar emission [26].
In biological neurons and cells electrophysiology, bursting oscillations play an important role in information processing. Moreover, in biological neurons, bursting oscillations are important for motor pattern generation and synchronization.
(v) Transient chaos is a dynamical behavior that displays the existence of chaotic behavior on finite time [38]. Generally, the phenomenon of transient chaos can be observed in dynamical system with boundary crisis [39] and also in families of the logistic and Hénon maps. Zhijun and Yicheng [40] employed a piecewise linear memristor to construct a fourth-order memristor-based Wien bridge circuit with hyperchaotic dynamics. Wu et al. [41] constructed an active generalized memristor, in which a fourthorder Wien bridge chaotic oscillator was designed further. In recent years, the electronic research team focused on the infinitely differentiable characteristic equation of the diode component in electronic chaotic circuits replacing nonsmooth ones. e synthesis are well presented in [42,43] just to name some well-standing papers. Some rare and interesting dynamics are found, namely, coexisting hidden attractors, quasi periodicity with antimonotonicity, and hyperchaos. As we recall, finding chaotic circuits, (i) which modeled some important unsolved problems in nature, (ii) shed insight on that problems, and (iii) exhibited some behavior previously unobserved [44], is still a major interest. For this purpose, we explore the 5D Wien bridge memristive oscillator with antiparallel diodes with smooth (i-v) characteristics not yet explored in this circuit with interesting dynamics discover: (i) Intermittency route to chaos (ii) Transient chaos (iii) Hyperchaos with offset boosting and partial amplitude control (iv) Multistability (v) Bursting oscillations (vi) e successful microcontroller implementation e (i-v) characteristic model without approximations of the behavior of the nonlinear element diodes connected in antiparallel direction, therefore, constitutes an advance in the field of research for this Wien bridge oscillator. e rest of this paper is organized as follows. In Section 2, the model and analysis of a memristive Wien bridge oscillator are presented. It is followed in Section 3 by the numerical analyses highlighting transitions to chaos. en, in Section 4, some complex dynamics are discovered in this oscillator. We then continue with the microcontroller implementation in Section 5 to verify the numerical findings. e paper ends with some concluding remarks.

Modelling and Analysis of a Memristive Wien
Bridge Oscillator e schematic diagram of a memristive circuit based on the 5D Wien bridge oscillator is presented in Figure 1.

e Model of the Circuit.
e circuit of Figure 1 consists of three capacitors C 1 , C 2 , C 3 ; an inductor L 1 with its internal resistor R 1 ; an operational amplifier; two antiparallel diodes D 1 , D 2 ; three resistors R 2 , R 3 , R 4 ; and a flux-controlled memristor ω(φ). e authors in [16,45] used a piecewise linear function to describe the voltage-current characteristics of two antiparallel diodes. e exponent of the internal state of the fluxcontrolled memristor was set to first order as in [46]. In this work, the current-voltage characteristic of two antiparallel diodes D 1 and D 2 is described without any approximation: , the parameters i s � 2.682 nA, and V T � 26 mV stand for the reverse saturation current, ideality factor, and the thermal voltage of the diodes, respectively. e exponent of the internal state of the memristor here is set to second order as in [47]. e five dynamic elements C 1 , C 2 , C 3 , L, and memristor correspond to each state variable u 1 , u 2 , u 3 , i 1 , and ϕ, respectively. For analysis of the circuit, Kirchhoff's law is applied to the circuit of Figure 1 to reveal five sets of first-order differential equations: where the parameter α is associated with the memristor and w(ϕ) � (a + 3b ′ ϕ 2 ) is the memristance. Let us define is symmetry about the origin. e origin of the state space is a trivial equilibrium point E(0, 0, 0, 0, 0) meaning that the solution shows twin symmetric around the origin. e other equilibrium points of system (2a)-(2e) are obtained by solving dx/dt � 0, dy/dt � 0, dz/dt � 0, dw/dt � 0, dv/dt � 0, which gives

Mathematical Problems in Engineering
Equation (3d) cannot be solved analytically. en, the Newton-Raphson method [48] is used to find the value of x * for the chosen value a � 0.05, b � 0.03, f � 1, g � 1, α � 2, delta � 1, e � 2, and the method yield x * � − 3.08e − 18 resulting in the trivial E.
where Jac is the Jacobian matrix. us, the stability of E can be determined by solving the characteristic equation det(M j − λI 5 ) � 0, where I 5 represents the 5 × 5 identity matrix. Table 1 illustrates the eigenvalues obtained by the Newton-Raphson method. e screening parameter α is kept in the range 1.4 < α < 2 while the other parameters of the model are the ones defined previously.
We can conclude that, for α belonging to the interval, the system can develop self-excited attractors. e overview of the stability of system (2a)-(2e) is performed by plotting the stability diagram versus parameters a and α.
In Figure 2, we can notice that the unstable area is limited to the stable area at a critical value of a around 0.75; this helps to choose the value of the parameter for numerical analyses.

2D Bifurcation Diagrams.
Numerical analyses of the dynamical system can be obtained by plotting the 2-D MLE when varying simultaneously two parameters to provide global information about the dynamic behavior of the system under investigation [49]. e colors on these diagrams of the model of our oscillator vary according to the value of the MLE computed using the well-known method of Wolf et al. [50]. In these figures, the light green, cyan, and magenta characterize a chaotic motion while the dark-green yellow and dark red represent periodic or quasiperiodic motion. It is therefore visible that parameters c, d, and g provide many diverse dynamics in contrary to the parameter e that is monotone. For these reasons, we choose them in their interesting interval to study the scenario toward chaos ( Figure 3).

Transitions to Chaos.
In this section, intensive numerical analyses are performed by monitoring the bifurcation parameter α and initial states. We plot the local maxima of the coordinate x (x max ) and record the Lyapunov spectra. We noticed sparse chaotic windows alternating with periodic ones while increasing or decreasing the bifurcation parameter α. e red curve is obtained during the increasing path of α while the red one is during the decreasing path.
In this manner, the hysteresis firstly discovered in dynamical systems by Berglund [51] occurs in Figure 4(a) while the red and black curve does not overlap. It is used here to discover symmetric or multiple attractors. e chaos behavior is obtained as we can see by the intermittency route. We note that this phenomenon is rarely found in dynamical system interring chaotic dynamics [52]. It is described as a scenario involving several frequencies and spontaneously becomes chaotic while varying the bifurcation parameter. e sudden changes in the quality behavior (competition between several frequencies and chaos) are revealed in detail in Figure 5 where some phase portraits with corresponding power density spectra are plotted.
According to Figure 5, we can reveal to the reader the transition to chaos (in forward and in the reversed directions of the bifurcation parameter α) from Figure 5(j) € Figure 5(k). We can observe a periodic behavior suddenly followed by chaotic dynamics by a tiny variation of α.
In this numerical research, it is usually during the intermittency route to encounter the same scenario while time elapsed. In this situation, chaos can appear and disappear to become periodic: this scenario is called transient chaos and it is very rare in dynamical systems.

Transient Chaos Behavior.
e findings of transients' chaos are of great interest since they are believed to be the culprit for disastrous such as voltage collapse in electric power systems [53] and species extinction in ecology. We choose the set of system parameters a � 0.
3, delta � 1 and plot the time trace of the dynamical system in Figure 6.
In Figure 6, one can notice that starting the system at t � 0, the behavior of the system is chaotic until t � 600 s. is description is revealed by the chaotic attractor in (b) with the corresponding power density spectra. Passing the critical time 600 s, the system becomes regular as shown in Figure 6(d). is phenomenon is also reported in memristive systems [54] including Chua's [55] and Duffing oscillators [56] and deserves to be shared.

Intermittency Route to Chaos during Symmetric
Interior Crisis. Another interesting dynamic found in this system is the interior crisis. It describes the bifurcation events in which a chaotic attractor suddenly expands in size. It was initially observed by Grebori et al. [57].
In Figure 7, when the control parameter is varied, we noticed the expansion in size of the red and black attractors until they merged to form a unique huge attractor in Figure 7(f). is striking phenomenon is shown with the time traces of the black and red attractors in the right column of Figure 7. e description of the multiple routes to chaos observed in system (2a)-(2e) rise to a very interesting and complex behavior as we revealed in the next section.

Multistability.
e famous and interesting behavior of multistability in the dynamical system was shown in the optical chain by [58] and in the isolation of a new defect in n-type silicon [59,60]. It was recently encountered in well-known systems such as Chua ( [61]) and Sprott [62], jerk system [63]. A simple technique to detect this property is to scan the bifurcation diagram upward and downward using the same control parameter (see Figure 4). In the light of this technique, one can obverse in Figure 4(a) some windows where the black curve and the red curve do not overlap showing the multistability phenomenon. We draw a zoom in the interesting interval of the control parameter to share this scenario, Figure 8(a).
In Figures 8(b1) and (b2), one can see that the 4 attractors coexist for the same set of system parameters. e red ones are obtained with positive initial conditions while black attractors are obtained with negative initial conditions. e basin of attraction showing the space of initial conditions resulting from each coexisting steady state is plotted in Figure 9. Recall that the basin of attraction associated with attractors red or black is the closure of the set of initial points that, taken as initial conditions, converge to red or black attractors when time increases to infinity. is subset-plane of initial points are determined using the computation of maximum Lyapunov Exponent (MLE) using the well-known algorithm by Wolf et al. [50]. For the 5D system under study, we fixed the initial points x 2 (0) � x 3 (0) � x 4 (0) � 1. e system parameters are set in the caption of Figure 9 and remain unchanged during the computation. For any couple of starting points.
− 4 < x 1 (0) < 4 and − 10 < x 5 (0) < 10, the long-term behavior of system (1a)-(1e) is computed using the Rung-Kutta algorithm, and the MLE is determined using the Wolf et al.'s method. en, it is saved. If: (a) MLE > 0, we plot on the substate space the point with magenta color for positive chaotic attractors while the blue color area is for negative ones. (b) MLE ≤ 0, we plot on the substate space the point with black color (for negative limit cycles) and red color (for positive limit cycles) for initial conditions that led to periodic attractors.
In Figure 9, the reader can discover the fractal form of the substate space resulting in the complexity of system (1a)-(1e). Note that other planes are not plotted for simplicity purposes.

Hyperchaos.
e new memristive Wien Bridge oscillator, as in Figure 1, generates hyperchaotic attractor with two positive Lyapunov exponents. In Table 2, Lyapunov exponents and dynamics of system (1a)-(1e) for different values of α are given. Figure 10 displays the phase portraits of the hyperchaotic attractors.

Offset Boosting.
e flexibility of the Wien Bridge oscillator can be used as a chaotic encoding circuit by means of varying parameters. For this purpose, we exploit the rescaling factor k to illustrate the displaced attractor on the w axis as follows: w⟶w + µ. System (2a)− (2e) is written as follows: In Figure 11, the attractor moved in the z-axis in accordance with the offset variable μ.

Amplitude Control.
In this section, we show that the amplitude of the attractor of the oscillator can be varied by means of the scaling factor m. In Figure 12, one can see that the dimension attractor can be shrunk or expanded in accordance with the amplitude control factor m.  0)) showing the initial conditions that lead to each coexisting steady state: the orange area is for positive chaotic attractor while the blue area is for negative ones. e black and red zones are for initial conditions that led to periodic attractors. System parameters are a � 0.05, b � 0.03, c � 6, d � 3, e � 2, f � 1, g � 1, k � 2, α � 1.3, δ � 1. e other initial conditions are X 2 (0) � 1, X 3 (0) � 1, X 4 (0) � 1 (color version online).  As we can see, the striking amplitude behavior can be exploited in engineering instrumentation.

Bursting
Oscillations. By selecting some discrete values of the parameter α � 0.01, the time series and phase portraits of y versus x and z versus y are displayed in Figures 13(iii) and (iv) illustrating the periodic bursting oscillations. According to these figures, one can observe that sometimes, the system is fast while other times, it is slow in regular space by a constant period, T � 50 s.

Electronic Implementation of a Memristive Wien Bridge Oscillator
In this section, the objective of the study is to verify some interesting behaviors found during the numerical simulation of the model of the Wien bridge oscillator with real antiparallel diodes.
It is important to note that recently, numerous advantages of digital components like FPGA, FPAA, DSP, and SOC have made them suitable for the implementation of chaotic systems just to name a few compared to the analog electronics component. Among them, the microcontrollers offer more flexibility for setting control parameters and initial conditions accurately, reducing the system to a portable source code, and realizing complicated mathematical operations or algorithms without needing special tools.
For this purpose, the experimental setup is drawn in Figure 14.
e experimental setup is composed of an Arduino MEGA board powered by a 9 V DC battery. e computer is connected to the USB port of the Arduino card built with an ATMEGA2560 microprocessor. It is connected to the computer to display data from the Arduino MEGA interface.   As the reader can see, the experimental design produces on the screen of the computer data that are captured and plotted in MATLAB software for comparison purposes and similar results are recovered.

Intermittency Route to Chaos with Symmetry Crisis Verification
4.2. Hyperchaotic Behavior. As we can see, Figure 15 reproduces the intermittency scenario and Figure 16 the hyperchaotic dynamic of the real component.

Conclusion
In this contribution, a new Wien bridge oscillator was introduced and analyzed. e investigations show that some additional behavior found apart from those already revealed in this oscillator was discovered, namely, intermittency route to chaos, transient chaos, hyperchaos with offset boosting, partial amplitude control, and bursting oscillations. e route to chaos is intermittent, transient chaos with some multistability characterized by the coexistence of up to 4 attractors for the same set of parameters. A basin of attraction is the plot to highlight this scenario. Experiment results based on an isolated Arduino card-built ATMEGA2560 processor producing a digital output of each state variable of the Wien bridge oscillator are consistent with theoretical and numerical predictions. With the research on fracmemristor [68,69] increasing with high interest, the outlook of this paper is to propose the fractional version of the circuit in Figure 1 with fracmemristor to increase the complexity of the nonlinear Wien bridge oscillator because it can increase the number of disconnected attractors of the Wien bridge. e dimension of the system under scrutinizing is 5D greater than the ones in the literature, but the presence of 4-wing hyperchaotic attractors is an important metric for the system to be used as an image encryption oscillator embedded on an Arduino microcontroller. erefore, we are planning in the nearest future to experiment with this fascinating application.

Data Availability
No data were used to support this study.