1 Introduction

In 2016, researchers at North Carolina State University have developed nonlinear chaos-based integrated circuits, which enable computer chips to perform multiple functions with fewer transistors [1]. As the researchers say: “The potential of 100 morphable nonlinear chaos-based circuits doing work equivalent to 100 thousand circuits, or of 100 million transistors doing work equivalent to three billion transistors holds promise for extending Moore’s law-not through doubling the number of transistors every 2 years but through increasing what transistors are capable of when combined in nonlinear and chaotic circuit” [2]. Consequently, constructing chaotic circuits is still a research hotspot.

Fig. 1
figure 1

The SBT memristor-based Wien-bridge chaotic circuit

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It is well known that a nonlinear two-terminal electronic element is easily used to construct chaotic circuits. Coincidentally, the memristor is a nonlinear two-terminal electronic element revealing the relationship between magnetic flux \(\varphi \) and charge q [3, 4], which is very suitable for designing chaotic circuits in theory. Therefore, memristor-based chaotic circuits and their generating complex dynamical behaviors have been studied extensively [5,6,7,8,9,10,11,12,13,14,15,16,17]. In the published studies, those existing memristors in the memristor-based chaotic circuits were mainly memristor emulators. In 2008, the first physical TiO\(_{2}\)-based nanostructured memristor was prepared by researchers of Hewlett–Packard Laboratory [18], which rekindled the attention of researchers to the memristor because of its potential applications in nonvolatile memory and artificial neural network [19,20,21,22,23,24]. From then on, many new material systems have been reported toward the physical memristor [25,26,27,28,29,30,31]. However, the physical memristor is not applied into the chaotic circuit design and realization, because it is unavailable as a commercial element now.

In order to apply the physical memristor into nonlinear circuit designs, firstly, a \(\hbox {Sr}_{0.95}\hbox {Ba}_{0.05}\hbox {TiO}_{3}\) (SBT) nanometer film was prepared in our laboratory [32], and then a flux-controlled mathematical model with definite parameters was established [33]. In this paper, a physical SBT memristor-based Wien-bridge chaotic circuit is proposed, and its dynamic behaviors are analyzed by means of Lyapunov exponents [34,35,36], bifurcation diagrams and phase portraits. It can guide the research on the realization of physical SBT memristor-based chaotic circuit in the future.

This paper is organized as follows: Sect. 2 gives a flux-controlled mathematical model of the physical SBT memristor and the physical SBT memristor-based Wien-bridge circuit. In Sect. 3, the circuit system is modeled by fourth-order state equations, the system’s stability is analyzed, and the dynamics of dependence on the initial states are studied by means of numerical simulations. In Sect. 4, the impacts of circuit parameters on the dynamic behaviors of the circuit system are investigated. Finally, the conclusions are given in Sect. 5.

2 The physical SBT memristor-based Wien-bridge chaotic circuit

In our previous work, a \(\hbox {Sr}_{0.95}\hbox {Ba}_{0.05}\hbox {TiO}_{3}\) (SBT) nanometer film was prepared, which can be used as a physical memristive element [32]. And the SBT memristor’s flux-controlled mathematical model with deterministic parameters was obtained as follows:

$$\begin{aligned} \left\{ \begin{array}{l} i(t)=(A+B|\varphi (t)|)u(t)\\ \frac{\text {d}\varphi (t)}{\text {d}t}=u(t)\\ \end{array}\right. \end{aligned}$$

where \(A=0.0676\) S, and \(B=0.3682\) S/Wb [33]. Herein, the physical SBT memristor can be used to design chaotic circuit.

The physical SBT memristor-based Wien-bridge chaotic circuit is shown in Fig. 1. The chaotic circuit consists of an operational amplifier, three linear capacitors \(C_{1}\), \(C_{2}\), and \(C_{3}\), five linear resistors \(R_{1}\), \(R_{2}\), \(R_{3}\), \(R_{4}\), and \(R_{5}\), a linear negative conductance G, and a nonlinear physical SBT memristor. The physical SBT memristor is a fundamental circuit element, along with the resistor, capacitor and inductor, which is not composed of simulated circuit.

3 Dynamic analysis of the physical SBT memristor-based Wien-bridge circuit

Table 1 The element parameter values of the physical SBT memristor-based Wien-bridge circuit
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Fig. 2
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The double-scroll chaotic attractor of the physical SBT memristor-based Wien-bridge circuit

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3.1 Modeling of a physical SBT memristor-based Wien-bridge circuit

There are four state variables of \(u_{1}(t)\), \(u_{2}(t)\), \(u_{3}(t)\) and \(\varphi (t)\), which represent the voltage of the capacitor \(C_{1}\), the voltage of the capacitor \(C_{2}\), the voltage of the capacitor \(C_{3}\), and the magnetic flux of the physical SBT memristor, respectively. The dynamical equations of the physical SBT memristor-based Wien-bridge circuit are as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\text {d}u_{1}\left( t\right) }{\text {d}t}=\frac{1}{C_{1}} \left( \frac{u_{2}\left( t\right) -u_{1}\left( t\right) }{R_{5}} -\left( A+B|\varphi (t)|+G\right) u_{1}(t)\right) \\ \frac{\text {d}u_{2}\left( t\right) }{\text {d}t}=\frac{1}{C_{2}}(\frac{R_{4}}{R_{2}R_{3}}u_{2}(t) -\frac{1}{R_{2}}u_{3}(t)-\frac{1}{R_{1}}u_{2}(t)-\frac{u_{2}(t)-u_{1}(t)}{R_{5}})\\ \frac{\text {d}u_{3}\left( t\right) }{\text {d}t}=\frac{1}{C_{3}} (\frac{R_{4}}{R_{2}R_{3}}u_{2}(t) -\frac{1}{R_{2}}u_{3}(t))\\ \frac{\text {d}\varphi \left( t\right) }{\text {d}t}=u_{1}\left( t\right) \end{array}\right. \end{aligned}$$
(1)

With some suitable parameters, this circuit can exhibit chaotic oscillations. Table 1 gives the selected circuit element parameter values: the capacitance values (10 or 20 nF), the resistance values (4, 10, 25, or \(45\,\hbox {k}\Omega \)), and the negative conductance value (− 0.0677 S), which are easily gained in the laboratory. The initial values of four state variables are assigned as \(u_{1}(0)=0.001\) V, \(u_{2}(0)=0\) V, \(u_{3}(0)=0\) V, and \(\varphi (0)=0\) Wb. The phase locus of the physical SBT memristor-based Wien-bridge circuit is simulated numerically, and the projections of the phase portraits onto the two-dimensional planes are shown in Fig. 2. The finite-time local Lyapunov exponents on the time interval \(t\in [0, 2]\) are shown in Fig. 3, and they are calculated as \(\hbox {LE}_{1}=12.9955\), \(\hbox {LE}_{2}=-\,4.4780\), \(\hbox {LE}_{3}=-\,102.0071\) and \(\hbox {LE}_{4}=-\,2829.3563\), which indicate that the physical SBT memristor-based Wien-bridge circuit is chaotic. The phase locus is a double-scroll chaotic attractor (see Fig. 2).

Fig. 3
figure 3

The Lyapunov exponents on the time interval \(t\in [0, 2]\)

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3.2 Equilibrium point and stability analysis

The system is invariant when variables (\(u_{1}\),\(u_{2}\),\(u_{3}\),\(\varphi \)) are transformed into (\(-u_{1}\),\(-u_{2}\),\(-u_{3}\),\(-\varphi \)) for the state equations (1). Therefore, the SBT memristor-based Wien-bridge circuit system is symmetrical with respect to the origin. Let the right hand of state equations (1) be equal to 0. An equilibrium point is obtained as:

$$\begin{aligned} \begin{array}{l} E=\{(u_{1},u_{2},u_{3},\varphi )|u_{1}=u_{2}=u_{3}=0,\varphi =\varphi _{0}\} \end{array} \end{aligned}$$

where \(\varphi _{0}\) is an arbitrary constant. All the points in the \(\varphi \) axis are equilibrium points of the fourth-order nonlinear system.

The Jacobi matrix at the equilibrium point (0, 0, 0, \(\varphi _{0}\)) can be expressed as:

$$\begin{aligned} J={ \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} -\frac{1}{C_{1}}\left( A+B|\varphi _{0}|+G+\frac{1}{R_{5}}\right) &{} \frac{1}{C_{1}R_{5}} &{} 0 &{} 0\\ \frac{1}{C_{2}R_{5}} &{} \frac{1}{C_{2}}\left( \frac{R_{4}}{R_{2}R_{3}}- \frac{1}{R_{1}}-\frac{1}{R_{5}}\right) &{} -\frac{1}{C_{2}R_{2}} &{} 0\\ 0 &{} \frac{R_{4}}{C_{3}R_{2}R_{3}} &{} -\frac{1}{C_{3}R_{2}} &{} 0\\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right] } \end{aligned}$$

The characteristic equation of Jacobi matrix J is as follows:

$$\begin{aligned} \begin{array}{l} det(\lambda I-J)=\lambda (\lambda ^{3}+a_{2}\lambda ^{2}+a_{1}\lambda +a_{0})=0 \end{array}. \end{aligned}$$
(2)

Setting the circuit element parameter values as shown in Table 1, the coefficients of characteristic equation are shown as:

$$\begin{aligned} a_{2}= & {} 0.3682|\varphi _{0}|\times 10^{8}-0.7667\times 10^{4},\\ a_{1}= & {} 4.0911|\varphi _{0}|\times 10^{9}+0.2889\times 10^{7},\\ a_{0}= & {} 2.2910|\varphi _{0}|\times 10^{14}-5.3333\times 10^{10}. \end{aligned}$$

Equation (2) indicates that the characteristic equation of Jacobi matrix J has one zero eigenvalue and three nonzero eigenvalues. According to the Routh–Hurwitz criterion of stability, all the nonzero eigenvalues of Eq. (2) have negative real parts when \(a_{2}>0\), \(a_{0}>0\) and \(a_{2}a_{1}-a_{0}>0\), namely:

$$\begin{aligned} \left\{ \begin{array}{l} a_{2}=0.3682|\varphi _{0}|\times 10^{8}-0.7667\times 10^{4}>0\\ a_{1}a_{2}-a_{0}=1.5063|\varphi _{0}|^{2}\times 10^{17}-1.5410| \varphi _{0}|\times 10^{14}+3.1185\times 10^{10}>0\\ a_{0}=2.2910|\varphi _{0}|\times 10^{14}-5.3333\times 10^{10}>0 \end{array}\right. \end{aligned}$$
Table 2 The three nonzero eigenvalues of Jacobi matrix with different \(|\varphi _{0}|\) values
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Fig. 4
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Dynamic behaviors with the variation in initial state \(\varphi (0)\) of the SBT memristor: a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 5
figure 5

The phase portraits on the \(u_{2}{-}\varphi \) plane with different initial state \(\varphi (0)\) value: a \(\varphi (0)=-\, 0.55\,\hbox {mWb}\), b \(\varphi (0)=-\,0.1\,\hbox {mWb}\), and c \(\varphi (0)=0.45\,\hbox {mWb}\)

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Fig. 6
figure 6

The phase portraits on the \(u_{2}-\varphi \) plane with different initial state \(\varphi (0)\) value: a \(\varphi (0)=-\,0.335\,\hbox {mWb}\), b \(\varphi (0)=0.13\,\hbox {mWb}\), and c \(\varphi (0)=0.32\,\hbox {mWb}\)

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The solutions of inequality group are 0.2328 mWb \(<|\varphi _{0}|<\) 0.2778 mWb or \(|\varphi _{0}|\,>\) 0.7452 mWb. On the contrary, when 0 mWb \(<|\varphi _{0}|<\) 0.2328 mWb or 0.2778 mWb \(<|\varphi _{0}|<\) 0.7452 mWb, the equilibrium point is unstable. The system in the neighborhood of the equilibrium point may give rise to a variety of trajectories, such as stable point, period or chaos. The three nonzero eigenvalues of Jacobi matrix J are listed in Table 2 with different \(|\varphi _{0}|\) values. The results show that the type of equilibrium point with different \(|\varphi _{0}|\) values converts among unstable saddle-focus and stable focus.

3.3 Dynamic analysis of dependence on the initial state \(\varphi (0)\) of the SBT memristor

The selected circuit element parameters are shown as in Table 1, the initial state values except \(\varphi (0)\) are set as \(u_{1}(0)=0.001\) V, \(u_{2}(0)=0\) V, and \(u_{3}(0)=0\) V. The variation range of \(\varphi (0)\) is from − 1 to 1 mWb. When the initial state \(\varphi (0)\) gradually increases, the Lyapunov exponents spectrum and the bifurcation diagram of the state variable \(u_{1}\) are displayed in Fig. 4a, b, respectively. A part of the minimum Lyapunov exponent is depicted in Fig. 4a for clarity. The bifurcation diagram coincides with Lyapunov exponents spectrum well. Figure 4 shows that the physical SBT memristor-based Wien-bridge circuit system can exhibit multiple dynamical behaviors with the variation in initial state \(\varphi (0)\).

Table 3 The Lyapunov exponents and dynamical behavior with some typical initial state \(\varphi _{0}\) values
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Table 4 The dynamics of the circuit system with the variation in the initial state (\(u_{1}(0), u_{2}(0), u_{3}(0)\))
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Fig. 7
figure 7

Dynamic behaviors with the variation in initial state \(u_{1}(0)\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 8
figure 8

Dynamic behaviors with the variation in initial state \(u_{2}(0)\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 9
figure 9

Dynamic behaviors with the variation in initial state \(u_{3}(0)\): a Lyapunov exponents spectrum, and b bifurcation diagram

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If the initial state \(\varphi (0)\) is in the range of [− 1, − 0.75 mWb], [− 0.28, − 0.25 mWb], [0.19, 0.28 mWb] or [0.75, 1 mWb], the four Lyapunov exponents are less than zero and the dynamic behaviors of system can be stabilized finally (see Fig. 4).

If \(\varphi (0)\) is in the range of [− 0.74, − 0.35 mWb], [− 0.24, 0.11 mWb] or [0.35, 0.74 mWb], there are positive Lynpunov exponents, and the sum of four Lyapunov exponents is negative, so the physical SBT memristor-based Wien-bridge circuit system is chaotic (see Fig. 4). For \(\varphi (0)=-\) 0.55, − 0.1 and 0.45 mWb, the phase portraits on the \(u_{2}\)-\(\varphi \) plane are depicted in Fig. 5. The circuit system exhibits chaotic behaviors, including two single-scroll attractors (Fig. 5a, b) and one double-scroll attractor (Fig. 5c).

Fig. 10
figure 10

A single-scroll hidden chaotic attractor at \(\varphi (0)=-\,0.235\,\hbox {mWb}\)

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If \(\varphi (0)\) is in the range of [− 0.34, − 0.29 mWb], [0.12, 0.18 mWb] and [0.29, 0.34 mWb], the maximum Lyapunov exponent is zero and the system is periodic (see Fig. 4). The phase portraits on the \(u_{2}\)-\(\varphi \) plane for \(\varphi (0)=-\,0.335\), 0.13, and 0.32 mWb are shown in Fig. 6. The circuit system is 4-periodic for \(\varphi (0)=-\,0.335\) mWb (Fig. 6a) or 0.13 mWb (Fig. 6b) and is 1-periodic for \(\varphi (0)= 0.32\) mWb (Fig. 6c).

The four finite-time local Lyapunov exponents on the time interval t\(\in \)[0, 2] and dynamical behavior of the circuit system with some typical initial state \(\varphi (0)\) are listed in Table 3.

3.4 Dynamic analysis of dependence on other initial states

The circuit element parameters are selected as shown in Table 1. The dynamics of the circuit system with the variation in the initial state (\(u_{1}(0)\), \(u_{2}(0)\), \(u_{3}(0)\)) are depicted in Table 4. The corresponding Lyapunov exponents spectrums and bifurcation diagrams are displayed in Figs. 7, 8 and 9; the minimum Lyapunov exponents are not depicted for clarity. In a word, the circuit system is very sensitive to the initial states.

Fig. 11
figure 11

The Lyapunov exponents at \(\varphi (0)=-\,0.235\,\hbox {mWb}\)

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Fig. 12
figure 12

Dynamic behaviors with variation in the circuit parameter \(C_{1}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 13
figure 13

The phase portraits on the \(u_{2}\)-\(\varphi \) plane with different circuit parameter \(C_{1}\): a \(C_{1}=8\) nF, and b \(C_{1}=13\) nF

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Fig. 14
figure 14

The phase portraits on the \(u_{2}\)-\(\varphi \) plane for \(C_{1}=18\) nF

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Table 5 The Lyapunov exponents and dynamical behavior with the variation in circuit parameter \(C_{1}\)
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3.5 A hidden chaotic attractor

The basin of a hidden attractor is not connected with equilibrium point. For example, the hidden attractors are the attractor in the system with no equilibrium point or with only one stable equilibrium point [37]. When − 0.2400 mWb \(<\varphi (0)\,<\,-\) 0.2328 mWb, the theoretical analysis in Sect. 3.2 shows that the equilibrium point is stable, the numerical simulation in Sect. 3.3 shows that the system is chaotic (see Fig. 4b), so the hidden attractors may exist in the system. For example, when \(\varphi (0)=-\,\)0.235 mWb, the equilibrium point corresponding to three nonzero eigenvalues \(\lambda _{1}=-\,135.3871\) and \(\lambda _{2,3}=-\,425.3231\pm 1885.2675i\) is stable. By numerical simulation at \(\varphi (0) = -0.235\,\hbox {mWb}\), the single-scroll chaotic attractor is obtained (see Fig. 10). The finite-time local Lyapunov exponents on the time interval t\(\in \)[0, 2] at \(\varphi (0)=-\,0.235\) mWb are shown in Fig. 11, and they are calculated as \(\hbox {LE}_{1}\)=121.5404, \(\hbox {LE}_{2}\)=16.5271, \(\hbox {LE}_{3}=-\,38.2758\) and \(\hbox {LE}_{4}=-\,2667.7087\), which indicates that the physical SBT memristor-based Wien-bridge circuit system is chaotic. Consequently, a hidden chaotic attractor exists in the system.

Fig. 15
figure 15

Bifurcation diagram with the variation in circuit parameter \(C_{1}\)

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Fig. 16
figure 16

The phase portraits on the \(\varphi \)-\(u_{2}\) plane: a coexisting chaotic attractors at \(C_{1}=13.5\) nF, b coexisting periods at \(C_{1}=17.0\) nF

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4 Dynamic analysis of dependence on circuit element parameters

4.1 Multiple dynamics with the variation in capacitance \(C_{1}\)

The circuit element parameters except \(C_{1}\) are selected as shown in Table 1. The initial state values are set as \(u_{1}(0)=0.001\) V, \(u_{2}(0)=0\) V, \(u_{3}(0)=0\) V and \(\varphi (0)=0\) Wb. The Lyapunov exponents spectrum and bifurcation diagram with the variation in \(C_{1}\) are displayed in Fig. 12a, b, respectively. A part of the third Lyapunov exponent and the minimum Lyapunov exponent are not depicted in Fig. 12a for clarity. The bifurcation diagram coincides with Lyapunov exponents spectrum well.

Table 6 The dynamics of the system with the variation in circuit element parameters (\(C_{2}\), \(C_{3}\), \(R_{1}\), \(R_{2}\), \(\frac{R_{4}}{R_{3}}\), \(R_{5}\))
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Fig. 17
figure 17

Dynamic behaviors with the variation in circuit parameter \(C_{2}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 18
figure 18

Dynamic behaviors with the variation in circuit parameter \(C_{3}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 19
figure 19

Dynamic behaviors with the variation in circuit parameter \(R_{1}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 20
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Dynamic behaviors with the variation in circuit parameter \(R_{2}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 21
figure 21

Dynamic behaviors with the variation in circuit parameter \(\frac{R_{4}}{R_{3}}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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Fig. 22
figure 22

Dynamic behaviors with the variation in circuit parameter \(R_{5}\): a Lyapunov exponents spectrum, and b bifurcation diagram

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With the increase in the capacitance \(C_{1}\), the system exhibits various dynamic behaviors (see Fig. 12). If the circuit parameter \(C_{1}\) is in the range of [5.0, 14.8 nF], there are positive Lynpunov exponents and the sum of four Lyapunov exponents is negative, so the physical SBT memristor-based Wien-bridge circuit system is chaotic. For \(C_{1}=8\) and 13 nF, the phase portraits on the \(u_{2}\)-\(\varphi \) plane are depicted in Fig. 13. The circuit system exhibits chaotic behaviors, including a double-scroll attractor and a single-scroll attractor (see Fig. 13).

If \(C_{1}\) is in the range of [14.9, 21.2 nF], the maximum Lyapunov exponent is zero, so the system is periodic. For \(C_{1}=18\) nF, the phase portrait on the \(u_{2}\)-\(\varphi \) plane is 1-periodic, as shown in Fig. 14. If \(C_{1}\) is in the range of [21.3, 25.0 nF], the four Lyapunov exponents are less than zero and the locus curves converge into a stable equilibrium point.

The four finite-time local Lyapunov exponents on the time interval t\(\in \)[0, 2] and dynamical behavior of the circuit system with the variation in circuit parameter \(C_{1}\) are listed in Table 5.

4.2 Coexisting chaotic attractors and coexisting periods

Selecting the circuit parameters as Table 1 except \(\hbox {C}_1\), and setting the initial states as (\(0, 0, 0.001\hbox {V}, 0\hbox {Wb}\)) and (\(0.001, 0, 0\hbox {V}, 0\hbox {Wb}\)) separately, the bifurcation diagram of the state variable \(\varphi (t)\) with variation of the capacitance \(\hbox {C}_1\) is shown in Fig. 15. The red line is corresponding to (0, 0, 0.001 V, 0 Wb), and the blue line is corresponding to (0.001, 0, 0 V, 0 Wb) (see Fig. 15). With the two initial states, the system has coexisting chaotic attractors when \(C_{1}\) is at the range of [12.5, 14.8 nF] and has coexisting periods when \(C_{1}\) is at the range of [14.9, 20.0 nF]. For \(C_{1}=13.5\) and 17.0 nF, the phase portraits on the \(\varphi \)-\(u_{2}\) plane are coexisting chaotic attractors and coexisting periods with the two initial states of (0, 0, 0.001 V, 0 Wb) and (0.001, 0, 0 V, 0 Wb), as shown in Fig. 16a, b.

4.3 Dynamic analysis of dependence on other circuit element parameters

The circuit element parameters are selected as shown in Table 1. The dynamics of the circuit system with the variation in other circuit element parameters (\(C_{2}\), \(C_{3}\), \(R_{1}\), \(R_{2}\), \(\frac{R_{4}}{R_{3}}\), \(R_{5}\)) are depicted in Table 6. The corresponding Lyapunov exponents spectrum and bifurcation diagram are displayed in Figs. 17, 18, 19, 20, 21 and 22; the minimum Lyapunov exponents are not depicted for clarity. In a word, the circuit system exhibits multiple dynamics with the variation in circuit element parameters.

5 Conclusion

In the paper, the physical SBT memristor-based Wien-bridge circuit is proposed, and its mathematical model is established using fourth-order state equations. The system generates typical chaotic attractors by choosing suitable circuit element parameters. By means of theoretical analysis, when 0.2328 mWb \(<|\varphi _{0}|<\) 0.2778 mWb or \(|\varphi _{0}|>0.7452\) mWb, the equilibrium point is stable; when 0 mWb \(<|\varphi _{0}|<\) 0.2328 mWb and 0.2778 mWb \(<|\varphi _{0}|<\) 0.7452 mWb, the equilibrium point is unstable. Moreover, the numerical simulation results indicate that this circuit system exhibits various dynamic behaviors with the variation in the initial states and the circuit element parameters. Specifically, some interesting dynamic behaviors have been found. When the initial state \(\varphi (0)\) is in the range of [− 0.2400, − 0.2328 mWb], the system can generate hidden chaotic attractors. When the capacitance \(C_{1}\) is in the range of [12.5, 20.0 nF], the system can generate coexisting chaotic attractors and coexisting periods. All the results provide an important theoretical basis for the next physical implementation of the chaotic circuit.