Hyperchaotic Oscillation in the Deformed Rikitake Two-Disc Dynamo System Induced by Memory Effect

Collaborative Innovation Center of Memristive Computing Application (CICMCA), Qilu Institute of Technology, Jinan 250200, China Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology, Nanjing University of Information Science & Technology, Nanjing 210044, China Biomedical Engineering Faculty, Amirkabir University of Technology, 424 Hafez Ave, Tehran 15875-4413, Iran


Introduction
Memristor is a passive element with nonlinearity and nonvolatility. According to the completeness of basic circuit components, in 1971, Chua first predicted the existence of the fourth circuit component, which describes the relationship between charge and flux [1]. In 2008, Hewlett-Packard Laboratory successfully fabricated the nanoscale memristor based on metal and metal oxides [2], which aroused great interest in the scientific and technological community. Recently, great progress has been made in the research of memristors, and the application of memristoralso has become a hot focus. Because of the small size, low power consumption, nonlinearity, and non-volatility, the memristor can be applied in many areas such as nonlinear chaotic circuits [3][4][5][6], electronic engineering [7][8][9], artificial intelligence [10][11][12], and neural networks [13][14][15].
It is considered that the geomagnetic field is associated with the conductive outer core, which has been proved by some dynamical models. In 1958, T. Rikitake firstly proposed a Two-Disc Dynamo System (RTDDS) and observed this physical law [16]. RTDDS is a simple model to demonstrate the polarity reversal of the earth's magnetic field, in which the current from each disc excites the coil of the other [17]. In fact, it is a chaotic system [18] and exhibits abundant dynamical behavior [19][20][21].
Recently, much attention has been paid to the RTDDS. A new attractor synthesis algorithm was applied to model the attractors in the Rikitake system [22]. By applying synchronization technique based on control theory, an active controller was designed for the synchronization of two identical RTDDS [19]. A method to stabilize asymptotically the nontrivial Lyapunov stable states of Rikitake two-disk dynamo dynamics was given in [23]. It was proven that the two non-hyperbolic equilibrium points of the Rikitake system are all stable for all positive parameters [24]. A simple realization of the symmetric Rikitake system was given in [25]. A reduced-order projective synchronization system was designed for the Rikitake system without any equilibria or with two non-hyperbolic equilibrium points in [26]. In [27], a 4-D hyperchaotic Rikitake dynamo system without any equilibria was proposed. In [28], a 5-D hyperchaotic Rikitake dynamo system with three positive Lyapunov exponents was proposed, which has a hidden attractor without any equilibrium point. From aforementioned references, the hyperchaotic system of RTDDS was constructed by artificial numerical methods directly through feedback and other means.
With the continuous development of industrial automation, intelligent motors with control and learning capabilities have also been realized. In this paper, a deformed RTDDS is constructed by introducing an extra flux-controlled memristor. e motor can be controlled to avoid the chaotic region through the combination of memristors and memristor control parameters. After the memristor is added, the bifurcation point of the system is changed, and the original behavior state of the RTDDS is also changed. erefore, the research in this paper provides an important idea for the control of the double-disk motor. is system is proven with abundant dynamical behaviors, including a line of equilibria and hyperchaos [29][30][31][32][33]. In Section 2, the brief introduction of the flux-controlled memristor is given, thereafter a new lossy RTDDS is given, and the deformed RTDDS is constructed by adding one extra flux-controlled memristor. Rich dynamics of the presented system are analyzed in Section 3. In Section 4, the analog circuit of the new memristive hyperchaotic system is implemented based on Multisim simulation. Some conclusions are finally drawn in Section 5.

Memristor Model.
According to the relationship between voltage v and current of a flux-controlled memristor [1,34,35], choose a cubic nonlinearity to describe the q function [36][37][38], and then, where α and β are two positive constants. Figure 1 shows the voltage-current relationship of the memristor with α � 2 and β � 0.1 under the frequency f � 1 Hz. e hysteresis curve agrees the inherent characteristics of the memristor.
In the following, the above memristor is applied to study the memory effect of the deformed RTDDS.

Description of the Deformed RTDDS.
In 1958, Rikitake first proposed a two-disc generator, in which the dimensionless equations are [16,18] _ where x 1 and x 2 are dimensionless currents, x 3 is the angular velocity of the two discs, μ and σ are adjustable parameters.
In the chaotic double-disk generator, considering the wear rate of the double-disk generator, the model of the deformed double-disk coupled generator is constructed under electromechanical coupling.
e new deformed RTDDS is proposed as where a and b are the ratio of resistance to self-inductance of the two loops, which represent the dissipative performance of the generator, and are closely related to the working conditions of the generator. Here, c is the difference in angular velocity between two rotors of the coupled generators, d is the wear parameter, x and y represent the current across the two loops, and z is the angular velocity of the double discs. System (3)

Memristive Deformed RTDDS.
In RTDDS, assume that the loop current of the first disc depends on the change of the current of the other disc, and the memory effect can be indicated by the memristor. A flux-controlled memristor is applied in RTDDS with α = 2 and β = 0.1, and the memristive deformed RTDDS is where a, b, c, and d are positive parameters, and k is a positive parameter representing the strength of the memristor. System (4) with a = 2, b = 3, c = 5, d = 0.75, and k = 1 has a hyperchaotic attractor with Lyapunov exponents LE 1 = 0.3784, LE 2 = 0.0218, LE 3 = 0.0000, and LE 4 = − 4.6503, two of which are positive indicating hyperchaos, as shown in Figure 3. e Kaplan-York dimension is D KY = 3.0861. Poincaré map is a line representing chaos if the surface represents hyperchaos, and the Poincaré map of the system is shown in Figure 4; in this system, Poincaré map is surface, so it is hyperchaos. e invariance of system (4) under the 2 Complexity rotational symmetric structure. erefore, newly introduced memristor transforms the system to be a hyperchaotic one.

Line of Equilibria and Stability
Analysis. e equilibrium points of system (4) can be derived by solving the following equations: System (5) has a line of equilibria [0, 0, 0, φ], where φ is a real variable. By linearizing system (5) at the equilibria, the Jacobian matrix can be obtained:  Complexity According to equation (6), the characteristic equation can be obtained as So, the eigenvalues are erefore, the stability of the line equilibria is associated with the values of a, b, c, k, and W (φ). From λ 2 and λ 3,4 , it can be seen that when a, b, and d are positive, the line equilibria point is unstable, command Δ k � (a − b) 2 − 4ckW(φ), where if Δ k > 0, and the line equilibria are unstable nodes or else if Δ k < 0, the line equilibria are unstable saddle foci. Specifically, when a � 2, b � 3, c � 5, d � 0.75, and k � 1, the line equilibria are unstable saddle focus.

Dynamical Analysis.
e dynamic behavior of system (4) will be further investigated with Lyapunov exponent spectra and bifurcation diagram.
When b � 3, c � 5, d � 0.75, k � 1, and a varies in [2,10], set the initial condition IC � (0.1, 0.1, 0.1, 0.1), step size of a is 0.01, time step is 0.01 s, and running time is 300 s; the corresponding Lyapunov exponent spectra and bifurcation diagram are obtained as shown in Figure 5. As shown in Figure 5 Figure 5(b) agrees with the Lyapunov exponents. Specific periodic oscillations are shown Figure 6, and the detail information of Lyapunov exponents is shown in Table 1. We noticed that system (4) sometimes provides a symmetric oscillation and sometimes gives a symmetric pair of limit cycles. e memristor bridges the loop current of two disks. Different phase portraits can be obtained under different initial values, as shown in Figure 7. A symmetric oscillation or a symmetric pair of limit cycles is found simultaneously.

Circuit Simulation Based on Multisim
In order to further observe the specific hyperchaotic oscillation behavior induced from the memory effect, a circuit simulation is realized based on Multisim software [39]. In the circuit design, resistors, capacitors, operational amplifiers (OPA404AG), analog multipliers, and other elements are applied. e supply voltages for OPA404AG operational amplifiers (with saturated voltages V sat ≈ ±13.5 V) are ±15 V. In fact, the variable z is beyond the normal operating range of the device. erefore, here, the variables are transformed by proportional compression, that is, v , v z1 , and v φ1 are the voltages on the integral capacitor, respectively. By time rescaling of system (4), the equations can be obtained as follows:    Complexity e corresponding analog circuit is shown in Figure 11. e circuit in Figure 11(a) represents the quadratic nonlinear flux-controlled memristor, which consists of an integration circuit and a proportional circuit. Resistor R41, operational amplifier U2C, and capacitor C4 constitute an integral circuit, which integrates the voltage v across the memristor giving the magnetic flux through the memristor.
In Figure 11, the capacitances and resistances are  Figure 12 gives the phase trajectories shown in the oscilloscopes.

Conclusions and Discussion
e deformed Rikitake two-disc dynamo system possesses rich dynamics including chaos, hyperchaos, and different periodic oscillations. A memristive deformed RTDDS was constructed for observing the memory effect. Consequently, an analog circuit based on the flux-controlled memristor was designed for further verification. Circuit simulation agrees with the theoretical analysis and numerical simulation. By the memristor model built the control circuit, through memristor and matching parameters of the memristor, the RTDDS can be controlled to avoid the chaotic region and realize the smooth operation.
is research provides a meaningful reference for motor design and control.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.