Hidden Multistability in a Memristor-Based Cellular Neural Network

In this paper, we report a novel memristor-based cellular neural network (CNN) without equilibrium points. Dynamical behaviors of the memristor-based CNN are investigated by simulation analysis. The results indicate that the system owns complicated nonlinear phenomena, such as hidden attractors, coexisting attractors, and initial boosting behaviors of position and amplitude. Furthermore, both heterogeneous multistability and homogenous multistability are found in the CNN. Finally, Multisim circuit simulations are performed to prove the chaotic characteristics and multistability of the system.


Introduction
The cellular neural network (CNN) was proposed by Chua and Yang in 1988 for processing signals in real time, which is constituted of an array of the basic circuit units called cells [1]. The CNN has received widespread attention in academia because of its extensive applications, such as image processing, parallel computation, and complicated optimization problems [2,3]. Chua and Yang used CNN to realize image processing and pattern recognition [2]. For solving the constrained optimization problem, Shen et al. designed the utility-based radio resource scheduler by employing CNN [3]. Ref. [4] proposed a random weight change (RWC) algorithm to construct CNN weight and to make the hardware-based learning on CNN templates feasible. Based on a fuzzy cellular neural network (FCNN), an image encryption method was put forward in Ref. [5].
Ref. [6] further presented a state-controlled cellular neural network (SC-CNN), and a generalized Chua's circuit is constructed by three SC-CNN cells. A Colpitts-like oscillator also is implemented by SC-CNN [7]. An SC-CNN-based circuit could produce strange nonchaotic attractors in Ref. [8].
To explore multistability, initial boosting behaviors and attractor coexistence are investigated. Since the memory devices depend on its history, initial boosting behaviors were found in some memristor-based or meminductor-based systems [21][22][23]. Ref. [21] employed a memristor with sine memductance to construct a memristive jerk system. This novel memristive jerk system had four line equilibrium sets and periodical initial boosting. In Ref. [22], a memristormeminductor system was presented, which produced the amplitude, frequency, and position boosting. These systems have infinitely many equilibrium points. The initial boosting behaviors of a memristor-based system without equilibrium points are not put forward. Hence, we study the initial boosting behaviors in this memristor-based CNN. Moreover, multistability is divided into heterogeneous multistability and homogenous multistability [22,24]. This multistability is heterogeneous multistability if there exist coexisting attractors with different shapes. In contrast, the system owns homogenous multistability if it generates the same shape coexisting attractors with different positions and amplitudes or even frequencies.
This paper is organized as follows: Section 2 proposes the model of a novel memristive neural network. Its nonlinear dynamics is presented in Section 3, while its circuital implementation is completed in Section 4. Finally, conclusions are drawn in the last section.

A Memristor-Based Cellular Neural Network
The standard SC-CNN cell is defined as [6] where x i and x k are the state variables and always are the voltage of capacitors, y k is the output variable, i k is the independent current source, a ik and s ik are the feedback coefficient and state coefficient, respectively. A memristor-based cellular neural network is presented, which consists of one memristor-based CNN cell and two standard SC-CNN cells.
The memristor-based CNN cell is shown as Figure 1, from which the state equation is obtained as where a memristor is select as [25] M The memristor-based CNN cell is selected as the first cell, while the second and third ones are the standard SC-CNN cells. The state variables of two standard SC-CNN cells are independent of the output variable, and the state variable of the second cell does not relate to the independent current source. Therefore, the proposed CNN can be expressed as In order to better explore the feature of the memristorbased CNN, we let three cells own different numbers of state variables. Set s 11 = s 12 = s 21 = 0, Equation (4) can be simplified as Since the memristor-based CNN in this case has no equilibrium points, this chaotic attractor is a hidden attractor.

Dynamics of the Memristor-Based
Neural Network 3.1. Influence of the Parameter of the System. In a survey of the dynamical behaviors of the memristor-based CNN, with the parameter a increasing from 9 to 50, and the other parameters and the initial condition set as in Section 2, the Lyapunov exponent spectrum is shown in Figure 4(a), where LE1-3 represent the first three Lyapunov exponents, and the fourth Lyapunov exponent is neglected owing to its large negative value; the corresponding bifurcation diagram is depicted in Figure 4(b). From Figure 4(a), it is easy to observe that the memristor-based CNN keeps a chaotic state. Figure 4(b) shows that the amplitude of x 1 reduces nonlinearly with the increase of a.       Coexisting attractors illustrate that the memristor-based CNN has multistability. To explore its multistability nature, the attractive basins are drawn in Figure 6.     Advances in Mathematical Physics variables xð1Þ and φ irregularly change, whereas the mean values of the variables x 2 and x 3 still hardly change. Obviously, the route differs from Refs. [21][22][23]. Moreover, the initial variable φð0Þ is not only the booster of position but also of amplitude. The initial boosting behaviors of amplitude are shown in Figure 7(b). We can divide the figure into two parts. In the first part ð−0:25, 0:44, the mean absolute value of the variable φ almost keeps unchanged, but the other values nonlinearly decrease; in the second part ð0:44, 2:5, all the mean absolute values increase. When the initial value φð0Þ is 0.44, all the mean absolute values have a jump, which is the same as the mean values.

Attractor
For better illustrating the offset boosting, several coexisting attractors are plotted in Figure 8, whose positions and amplitudes are related to the initial variable φð0Þ. From Figure 8(a), three attractors with different shapes are observed, including two chaotic attractors and one periodic attractor, and thus, this system has heterogeneous x 1    Figure 7. Furthermore, if the ODE45 method with the time span ½0,500 is used to solve Equation (5), the time-domain waveforms of φðtÞ are shown in Figure 9(a), illustrating different frequencies with different initial variable φð0Þ. The corresponding frequency spectra of the chaotic signals are depicted in Figure 9(b). Therefore, the multistability is homogenous multistability. This memristor-based CNN owns not only heterogeneous multistability but also homogenous multistability.
The initial variable φð0Þ is an offset impact factor, but not only. Fixing the parameters, when the initial conditions are set as ð0, x 2 ð0Þ, 0, 0Þ, the initial boosting behaviors of position   Advances in Mathematical Physics and amplitude are shown in Figure 10. From Figure 10, it is clear to see that the offset adjuster x 2 ð0Þ also can control the attractor's position and amplitude, but this change process is different from the offset adjuster φð0Þ.

Circuit Design and Experiment Result
The memristive CNN can be implemented by the circuit. When the parameters are chosen as s 11 = s 12 = s 21 = 0, s 13 = 7, s 22 = 1:75, s 23 = −1:1, s 31 = −1:3, s 32 = 1:1, s 33 = 0:85, a = 20, b = −10, c = −6, and i 3 = −0:0001, and as we introduce the time scale factor K = 100, the circuit is established as Figure 11. Let R = 100 kΩ, and the state equations are yielded as Employing Multisim to simulate the circuit, the experimental results show that the circuit is in chaos as Figure 12. By giving different initial values and changing the value of the resistor R 2 , the phenomenon of coexisting attractors is obtained as shown in Figure 13. With the resistor R 2 = 14:286 kΩ, the coexisting attractors in Figure 13(a) are caught by the oscilloscope of Multisim. The red orbit comes from the initial value of ð0:1 V, 0, 0:1 V, 0:5 VÞ, while the blue one comes from the initial value of ð0, 0, 0:1 V, 0Þ. When the resistor R 2 is selected as 5 kΩ, Figure 13(b) demonstrates the other phenomenon of attractor coexistence.

Conclusions
In this paper, we introduce a memristor-based CNN without equilibrium points, which contains a memristor-based CNN cell and two standard SC-CNN cells. By analyzing its dynamical behaviors, the coexisting hidden attractors are found. More interestingly, heterogeneous multistability and homogenous multistability are observed in the CNN. The presented system owns initial boosting behaviors of position and amplitude. Then, the equivalent circuit of the memristor-based CNN is designed, with which its chaotic and multistable characteristic is verified. Owing to its rich dynamical characteristics, the memristor-based CNN can be utilized in the information encryption field.

Data Availability
The data used to support the findings of this study are included within the article.