A novel memcapacitor and its application in a chaotic circuit

In this paper, a novel memcapacitor is designed by the SBT (Sr0.95Ba0.05TiO3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Sr}_{0.95}\hbox {Ba}_{0.05}\hbox {TiO}_{{3}}$$\end{document}) memristor and two capacitors. A fifth-order memcapacitor and memristor chaotic circuit is proposed. The stability of the equilibrium point of the system is analyzed theoretically. Lyapunov exponents spectra, bifurcation diagrams, poincaré maps and phase diagrams are used to analyze the dynamic behaviors of the system. The results show that under different initial values and parameters, the system produces rich dynamic behaviors such as stable points, limit cycles, chaos and so on. Specially, coexisting attractors, transient chaos, and steady-state chaos accompanied by burst period phenomena are also produced in the system. The proposed memcapacitor-based circuit expands the research methods of memcapacitor for application in chaotic circuits.

The behaviors of different memcapacitor models [20][21][22][23][24][25][26] were explored through previous researches on memcapacitor. Reference [20] described a synthesis of mutators which can transform the emulated memristor into memcapacitor and meminductor. Reference [21] converted a digital memristor into memcapacitor by virtue of the voltage following characteristics of operational amplifiers, which complicated the memcapacitor model. Reference [22] presented a memcapacitor simulator based on an light-dependent resistor (LDR) memristor and constructed a memcapacitor model. Refer-ence [23] designed a floating memcapacitor emulator, the circuit structure was simple and can be widely used in circuit design. Recently, researches based on memcapacitor have become a focus [27][28][29]. In Reference [27], a fractional-order memcapacitor model was proposed and a chaotic oscillator based on the model was investigated. Reference [28] proposed a logarithmic charge-controlled memcapacitor model and verified it non-volatility and switching features. These simulating memcapacitors were mostly based on complex conversion circuits, which were prone to errors, and it is not easy to the analysis of the characteristics of the memcapacitor. Therefore, it is necessary to find a new way to study the application of memcapacitor.
The memcapacitor device can be implemented by appending a memristor with a MIM (metal-insulatormetal) capacitor in Reference [30]. As the resistance of memristor changes, the capacitance of memcapacitor changes under external excitation. This structure has a potential for the artificial neural networks and chaotic circuits. The memcapacitor device also has nonlinear characteristics and low power consumption, and its structure is simpler than some simulating memcapacitors, so it has potential application in the integrability and scalability of CMOS technology. Based on the method of constructing physical memcapacitor proposed by Reference [30], this paper uses the physical SBT memristor [31,32] to design a memcapacitor, which can be directly used as a circuit element to generates chaos in the circuit. This paper constructs a fifth-order chaotic circuit composed of a memcapacitor and a memristor.
The paper is organized as follows. In Sect. 2, the dynamical modeling of the fifth-order memcapacitor and memristor chaotic circuit is introduced and its corresponding plane equilibrium and stability are analyzed. In Sect. 3, the influences of initial states and circuit parameters on system dynamic behaviors are studied. The conclusions are drawn in Sect. 4. In the previous researches, the mathematical model of the memristor had been proposed [31,32]. According to the experimental measurement of the memristor, the flux-controlled model of the memristor was obtained as follows: where A = 0.0676, and B = 0.3682. The fifth-order chaotic circuit based on the novel memcapacitor and the SBT memristor is shown in Fig. 1. The fifth-order chaotic circuit consists of a novel memcapacitor, a SBT memristor W 2 , a resistor R, a negative conductance-G , and an inductance L. According to Reference [30], the novel memcapacitor is composed of a memristor W 1 and two capacitors (C 1 and C 2 ), as shown in the dotted line. Its structure is simpler than previous memcapacitors [20][21][22], and it can be directly applied to the design of chaotic circuits without establishing a model of memcapacitor.
According to Kirchhoff's circuit laws, the current i L of inductor L, the voltage u 1 of capacitor C 1 , the voltage u 2 of capacitor C 2 , the magnetic flux ϕ 1 of memristor W 1 , and the magnetic flux ϕ 2 of memristor W 2 are selected as state variables, the state equations of the system are as follows:

Fig. 2
The typical chaotic attractor of fifth-order chaotic circuit R, and g = G, the dimensionless equations of the system are as follows:

Typical chaotic attractors
When the parameters are set as shown in Table 1, the complex dynamic behaviors occur in the system. The system generates a double-scroll attractor (see Fig. 2). By Jacobi matrix method, the five Lyapunov exponents are calculated as LE 1 = 0.1446, LE 2 = 0.01265, LE 3 = 0.0091, LE 4 = −0.2864, and LE 5 = −6.917. The sum of the Lyapunov exponents is negative, which mean that the system is chaotic.

Plane equilibrium and stability distribution
Letẋ =ẏ =ż =ẇ =v = 0 in Eq. (3), and the circuit parameters are set as shown in Table 1, the equilibrium point of the system can be obtained as: (4) where m and n are real constants. That is, the points on the w − v plane are the equilibrium point of the system (3). The Jacobian matrix J of Eq. (3) can be expressed as: The characteristic equation of the equilibrium point A can be given as: where a 1 , a 2 , a 3 are as follows: According the Routh-Hurwitz stability criterion, if all the nonzero eigenvalues of Eq. (6) are negative, the system is stable: If n = 0, m is a variable parameter, the conditions a 1 > 0, and a 3 > 0 in Eq. (8) cannot be satisfied simultaneously. If m = 0, n is a variable parameter, Eq. (8) has no solution. Obviously, the equilibrium point A is always unstable. No matter where the system (3) starts, the system tends to limit cycles, chaos, or infinite divergence.  The circuit parameters are taken as shown in Table 1, and the initial values of the system are assigned as (0.001, 0, 0, 0, 0). The Lyapunov exponents spectra and bifurcation diagrams varying with the initial states

Multistability depending on the initial condition z(0)
Multistability is a common characteristic in chaotic systems. Figure 8 shows the coexisting chaotic attractors and the corresponding Poincaré maps. Figure 8a, b shows the coexisting single-scroll chaotic attractors in detail and the corresponding Poincaré map, where the blue attractor starts from the initial conditions of (0.001, 0, 7, 0, 0) and the red one starts from (0.001, 0, − 7, 0, 0). Figure 8c, d shows the coexisting doublescroll chaotic attractors in detail and the corresponding Poincaré map, where the blue attractor starts from the initial conditions of (0.001, 0, 1.5, 0, 0) and the red one starts from (0.001, 0, − 1.5, 0, 0). Figure 8e, f shows the coexisting single-scroll chaotic attractor and double-scroll chaotic attractor in detail and the corresponding Poincaré map, where the blue attractor starts from the initial conditions of (0.001, 0, 3, 0, 0) and the red one starts from (0.001, 0, − 3, 0, 0).

Multistability depending on the initial conditions x(0), y(0), w(0), and v(0)
Other initial conditions can also produce coexisting chaotic attractors. The circuit parameters are taken as shown in Table 1, and the phase diagrams and the Poincaré maps of the multistability dependent on the initial conditions x(0), y(0), w(0), and v(0) are shown in Figs. 9, 10, 11 and 12 , the results are summarized in Table 2.

Influences of parameter a on system dynamic behaviors
When parameter a is in the range of [1.70, 3.80] and initial values are (0.001, 0, 0, 0, 0), the Lyapunov exponents spectrum and bifurcation diagram can be obtained, as shown in Fig. 13. In order to better observe Table 2 The multistability of the circuit and corresponding initial conditions in Figs the Lyapunov exponents of the chaos, the fifth Lyapunov exponential curve is not drawn . When a is in the range of [1.70, 1.81], the system is stable; when a is in the range of [1.82, 3.67], the system exhibits chaotic attractors; when a is in the range of [3.68, 3.80], the system exhibits limit cycles. The dynamical evolution process of the system with the change of parameter a is shown in Fig. 14, where the values of a are 3.22, 3.39, and 3.70, respectively. When parameter a is in the range of [1.70, 3.80], and others are set as shown in Table 1. Dynamical behaviors with coexisting bifurcation diagrams are pre-  Fig. 15, where the trajectories colored in the blue start form the initial conditions (0.001, 0, 0, 0, 0), and those colored in red correspond to (0, 0.001, 0, 0, 0). When parameter a = 3.22, the coexisting signalscroll chaotic attractors in detail and the corresponding Poincaré map are shown in Fig. 16, where the blue attractor starts from the initial conditions of (0.001, 0, 0, 0, 0) and the red one starts from (0, 0.001, 0, 0, 0).

Influences of parameter c on system dynamic behaviors
Parameter c is set in the range of [9.00, 14.00], and other parameters are set as shown in Table 1. When the initial values are (0.001, 0, 0, 0, 0), the Lyapunov exponents spectrum and bifurcation diagram can be obtained, as shown in Fig. 17. In order to better observe the Lyapunov exponents of the chaos, the fifth Lyapunov exponential curve is not drawn. When parameter c is in the range of [9.00, 13.37], the system is chaotic, while in the range of [13.38, 14.00], the system is stable. Dynamical behaviors with coexisting bifurcation diagrams are presented in Fig. 18, where the trajectories colored in the blue start from the initial conditions (0.001, 0, 0, 0, 0), and those colored in red correspond to (− 0.001, 0, 0, 0, 0). When parameter c = 9.76, the coexisting signal-scroll chaotic attractors in detail and the corresponding Poincaré map are shown in Fig. 19, where the blue attractor starts from the initial conditions of (0.001, 0, 0, 0, 0) and the red one starts from (− 0.001, 0, 0, 0, 0).

Influences of parameters b, r , and g on system dynamic behaviors
The initial values of the system are set as (0.001, 0, 0, 0, 0), the Lyapunov exponents spectra and bifurcation diagrams with the variation in system parameters b, r , and g are shown in Figs. 20, 21 and 22. The dynamics of the chaotic circuit are shown in Table 3.
(a) (b) Fig. 22 a Lyapunov exponents spectrum and b bifurcation diagram varying with the system parameter g

Transient chaos
The initial values of the system are set as (0.001, 0, 0, 0, 0), other parameters remain unchanged. When b = 1.97, the chaotic phenomenon of the system is transient chaos in the finite time range. The phenomenon of transient chaos accompanied by boundary crisis is often encountered in dynamic systems. As shown in Fig. 23  period 8 in different planes, that is, a period 8 orbits coexist in chaos. This phenomenon is sensitive to the initial values of the system. When b = 0.65, the coexisting phenomenon of steady-state chaos accompanied by burst period in detail and the corresponding Poincaré map are shown in Fig. 25, where the blue attractor starts from the initial conditions of (0.001, 0, 0, 0, 0) and the red one starts from (− 0.001, 0, 0, 0, 0).

Conclusion
In this paper, a novel memcapacitor based on the SBT memristor is proposed. It has a simple structure and can be directly used in the design of chaotic circuits. Then, a fifth-order chaotic circuit based on this memcapacitor is designed. The characteristics of the circuit are analyzed by using the dynamics methods. With the changing of initial conditions (x(0), y(0), z(0), w(0), v(0)), the system is always chaotic and produces a wealth of coexisting attractors. With the changing of system parameters, the system generates complex dynamic behaviors such as transient chaos, steady-state chaos accompanied by burst period. The proposed memcapacitor and memristor chaotic circuit in this paper enriches the application of memcapacitor and memristor in high-order circuits and expands the research methods of memcapacitor in chaotic circuits.