Current mode multi scroll chaotic oscillator based on CDTA

1 Introduction

In the past 40 years, due to the unique advantages of chaotic systems such as extreme sensitivity to initial values and parameters, ergodicity, and pseudorandomness, chaos has paid more attention to the combination of theory and practical applications, and has been widely used in fields such as secure communication [1–3], image encryption [4–8], random number generators [9–11], memristors [12–17], neural networks [18–25], and chaotic synchronization control [26–30].

The chaotic signal generated by chaotic oscillation circuits or chaotic systems is the core part of the entire chaotic communication system and has always been a research hotspot in the field of chaos [31–38]. How to generate multi scroll chaotic attractors with more complex topological structures has been widely concerned [39–41]. At present, most of the multi scroll chaotic oscillators are implemented by operational amplifiers [42–44], transconductance operational amplifiers (OTA) [45], current feedback operational amplifiers (CFOA) [46, 47] and second-generation current conveyers (CCII) [48–50]. The principles and methods for designing multi scroll chaotic oscillators based on operational amplifiers, CFOAs, and OTAs are summarized in [45]. In [46], a multi scroll chaotic oscillator was implemented using CFOA, and 3–10 scrolls from 1 kHz to 100 kHz were generated in experiments. In [49], the authors proposed a simple multi scroll chaotic oscillator implemented using a positive CCII and a negative CCII-. Circuit simulation shows that the chaotic electronic oscillator can generate more scroll chaotic attractors with higher frequencies. Because the operational amplifier belongs to the traditional voltage mode (voltage input, voltage output) circuit, the chaotic oscillator based on the operational amplifier has the problem of narrow bandwidth; OTA, CFOA and CCII belong to the voltage and current mixed mode devices, The following problems exist in the chaotic oscillator based on OTA, CFOA and CCII:

1) Since the chaotic oscillator composed of OTA, CFOA and CCII still works in voltage mode, the output impedance is very high and changes with frequency.

2) Due to the large parasitic parameters at the input terminals of OTA, CFOA and CCII, the frequency bandwidth of the chaotic oscillator based on OTA, CFOA and CCII is not large.

Currently, analog integrated circuit designs mostly use voltage mode circuit designs [51–53]. With the development and breakthrough of various new technologies represented by the PCB process, traditional voltage mode circuits are no longer suitable for low power supply voltage design requirements due to their high impedance, high voltage gain, and high signal swing characteristics, while current mode circuits have attracted widespread interest due to their low impedance, zero or even negative voltage gain, and broadband characteristics [54–59]. Current differential transconductance amplifier (CDTA) is a current input and output device, characterized by extremely low input impedance, high output impedance, and large dynamic range. Compared to OTA, CFOA, and CCII, it is a true current mode device [60–63]. Two new implementations of current mode quadrature oscillators using CDTA as active components are proposed in [64]. The proposed circuit uses two grounded capacitors to achieve current controllability of the oscillation frequency. In [65], a floating decreasing and increasing memristor simulator using OTA, CDTA, and two grounded capacitors is used. Then, the proposed memristor simulator is used in the design of chaotic oscillators and adaptive learning circuits. Simulation results of a chaotic oscillator and an adaptive learning circuit verify the effectiveness of the proposed design. When it is used to form a current mode chaotic circuit, the input and output impedances have nothing to do with frequency, which can generate a larger number of chaotic attractors at high frequencies. In addition, since the input end of the CDTA is a virtual ground, the frequency parasitic parameters are small and the bandwidth is large.

In this paper, a current mode chaotic circuit based on CDTA is proposed, which can generate multi scroll chaotic attractor current signals, promote the practical application of chaos communication, chaotic neural network and other fields.

This paper is organized as follows. In Section 2, the CDTA is studied by theoretical analyses and Pspice simulation. In Section 3, the proposed current mode basic operation modules of chaotic circuit based on CDTA is studied by theoretical analyses and Pspice simulation. In Section 4, we draw our conclusions.

2 The CDTA

The CDTA is a current-input, current-output current mode device with a large dynamic range. When forming a current mode circuit, it has both low input impedance and high output impedance characteristics. Figure 1 is the circuit symbol of CDTA.

FIGURE 1

FIGURE 1. Circuit model of CDTA.

Among them, p and n are the differential current input terminals, z is the auxiliary terminal, the current at the z terminal is the difference between the input currents of p and n, the x terminal is the current output terminal, and I_B is the external bias current. The port characteristics of CDTA are as follows:

${\mathrm{v}}_{\mathrm{p}}={\mathrm{v}}_{\mathrm{n}}=0,{\mathrm{i}}_{\mathrm{z}}={\mathrm{i}}_{\mathrm{p}}-{\mathrm{i}}_{\mathrm{n}},{\mathrm{i}}_{\mathrm{x}}={\mathrm{g}}_{\mathrm{m}}{\mathrm{v}}_{\mathrm{z}}={\mathrm{g}}_{\mathrm{m}}{\mathrm{Z}}_{\mathrm{z}}{\mathrm{i}}_{\mathrm{z}}(1)$

where $g_m$ is the function of the external bias current I_B, there is $g_m=f\left(I_B\right)$.

A CMOS CDTA circuit is designed, and the circuit schematic diagram is shown in Figure 2. Multiple x+ and x-ports can be expanded as needed. The transistor constitutes the current differential part, so that the z current of the auxiliary terminal is equal to the current difference between the p and n terminals. After an impedance is connected to the auxiliary terminal, the voltage v_z of the z terminal is obtained to realize the transconductance. The amplifying unit converts v_z into the current output of the x terminal.

FIGURE 2

FIGURE 2. Circuit schematic diagram of CDTA.

The Pspice simulation results of CDTA are as follows: Power supply voltage V_DD = 2.5V, V_SS = −2.5 V, external control current I_b = 200 uA. If only I_p is scanned when I_n = 0A is given, the relationship curve between I_z and I_p can be obtained as shown in Figure 3; If only I_n is scanned when Ip = 0A is given, the relationship curve between I_z and I_n can be obtained as shown in Figure 4. According to these two curves, it is not difficult to see that I_z is a difference relationship with I_p and I_n. When I_p = 1A, I_n = −1A, I_b = 200 uA, the transconductance gain of CDTA gm = I_x/V_z = I_x/(I_p-I_n) can be obtained, and the simulation results are shown in Figure 5.

FIGURE 3

FIGURE 3. Relation curve of Iz and Ip.

FIGURE 4

FIGURE 4. Relation curve of I_z and I_n.

FIGURE 5

FIGURE 5. The transconductance gain of CDTA.

3 The proposed current mode basic operation modules of chaotic circuit based on CDTA

The basic operation modules of the chaotic system (such as addition and subtraction, integration, etc.) and non-linear function generating circuits (such as step function, saturation function, etc.) can be easily realized by CDTA.

3.1 Integrator module

It can be seen from Equation 1 and Figure 1 that the input voltage of CDTA is zero and the input impedance is zero (the input impedance of the actual circuit is very small). In addition, the output impedance is also very high. When CDTA is used to form a current mode integrator, the input and output impedance characteristics are not destroyed. Figure 6 shows the current mode integrator composed of CDTA and capacitor, the output current expression is: $I_o=\frac{g_m}{C}\int I_{i}dt$.

FIGURE 6

FIGURE 6. CDTA-C current mode integrator.

Since the capacitor is not connected to the input and output terminals of CDTA, but is connected to the auxiliary terminal z of CDTA, the CDTA-C current mode integrator has very low input impedance and high output impedance, and has nothing to do with frequency, The input impedance of the CDTA-C current mode integrator is ideally 0, and the output impedance is the output impedance of CDTA (usually MΩ level). When implementing a chaotic circuit, the system parameters are independent of frequency, so that it can output chaos signal with a large bandwidth.

3.2 Adder module

Figure 7 shows the current mode addition and subtraction operation module composed of CDTA. Since CDTA has p and n as differential current input terminals, current mode addition and subtraction can be easily realized. The output current expression is: $I_o={\displaystyle \sum _{j=1}^i}{I}_j-{\displaystyle \sum _{j=i+1}^n}{I}_j$.

FIGURE 7

FIGURE 7. CDTA-C CDTA current mode adder and subtracter.

3.3 Step function module

The step function can be approximated as a saturation function with a sufficiently large slope. The basic unit circuit of the step function using CDTA is shown in Figure 8. The saturation current that the CDTA can achieve for a given supply voltage is denoted by $\pm \left|I_{\mathrm{s}at}\right|$. Then the output current can be approximately expressed as: $I_o=\left|I_{sat}\right|\text{sign}\left({\mathrm{I}}_i-I_j\right)$. By connecting several basic units in parallel, the step function sequence can be obtained, and the expression is: $I_o={\displaystyle \sum _{j=1}^Q}\left|I_{sat}\right|sign\left(I_i-I_j\right)$.

FIGURE 8

FIGURE 8. Current step function generation circuit composed of CDTA.

It can be seen from the above that the non-linear function generation circuit composed of CDTA compares the state variable current with the comparison current, and outputs the current saturation function. It can be seen that when CDTA is used to form a chaotic circuit, whether it is a linear circuit part or a non-linear circuit, all of them work in the current mode, which can realize the real current mode chaotic circuit.

In addition, due to the grounding of the input terminal (generally virtual grounding when the circuit is implemented), the parasitic parameters are small, the frequency characteristics are good, and the operating frequency bandwidth is wide.

Therefore, compared with operational amplifiers, OTA, CFOA and CCII, CDTA is more suitable for implementing current mode chaotic oscillator circuits. However, so far, there is no report on the use of CDTA to form a chaotic oscillator circuit.

3.4 The proposed CDTA-based current mode multi scroll Jerk chaotic oscillator circuit

Due to the simplicity and good recursion characteristics of the Jerk system, it has become a typical example for the study of multi scroll chaotic systems. This design adopts the classic Jerk system, and its dimensionless state equation is:

$\left\{\begin{array}{l}\dot{x}=y-f\left(y\right)\\ \dot{y}=z-f\left(z\right)\\ \dot{z}=-a\left(x+y+z\right)\end{array}\right.(2)$

The proposed CDTA-based current mode multi scroll Jerk chaotic oscillator circuit is shown in Figure 9.

FIGURE 9

FIGURE 9. The proposed current mode multi scroll Jerk chaotic oscillation circuit.

The main circuit of the chaotic oscillation circuit is composed of three CDTAs and three programmable equivalent capacitances CEQ, and the circuit structure is very simple. Its non-linear function adopts a step function, and the step function generating circuit is shown in Figure 8. By connecting several basic units in parallel, the step function sequence can be obtained.

$\left\{\begin{array}{l}{\dot{I}}_1=\frac{g_{m1}}{C_1}\left(I_2\right)\\ {\dot{I}}_2=\frac{g_{m2}}{C_2}\left(I_3\right)\\ {\dot{I}}_3=-\frac{g_{m3}}{C_3}\left(I_1+I_2+I_3-f\left(I_1\right)\right)\end{array}\right.(3)$

It can be seen from Figures 8, 9 that the non-linear function generating circuit and the main circuit of the chaotic circuit are both current mode circuits implemented by CDTA, with good high frequency characteristics and large dynamic range, and can be designed to generate more scrolls chaotic system. And because the capacitor is connected to the auxiliary z terminal, the input and output impedances are independent of the frequency, so the chaotic system equation will not change with the frequency.

Derive the dynamic equation of the multi scroll chaotic oscillator circuit corresponding to the circuit diagram shown in Figure 9.

This is a set of third-order non-linear autonomous ordinary differential equations with the currents at the x + ports of the output terminals of CDTA1, CDTA2, and CDTA3 respectively, and as the three state variables, and the non-linear functions $f\left(I_2\right)$ and $f\left(I_3\right)$ as the current step function. Through reasonable design of transconductance and selection of capacitance, the dimensioned current signals $I_1$, $I_2$, $I_3$, time t are converted into signals x, y, z and dimensionless time. It can be seen that the proposed current mode multi scroll chaotic oscillator circuit shown in Figure 9 can realize the multi scroll Jerk system.

3.5 Design of even-numbered scroll Jerk system

The non-linear function adopts step function:

$f\left(x\right)=\left(N-M\right)A1+s\left(x\right)+{\displaystyle \sum _{n=1}^N}s\left(x-2nA1\right)+{\displaystyle \sum _{m=1}^M}s\left(x+2mA1\right)(4)$

Where, N, M = 1,2,3,4, etc. Especially when N = M, there is

$f\left(x\right)=s\left(x\right)+{\displaystyle \sum _{n=1}^N}s\left(x-2nA1\right)+{\displaystyle \sum _{m=1}^N}s\left(x+2mA1\right)(5)$

Where A = A1 is the saturation value of the saturation function, and the number of scrolls can be generated is 2 *N+2. When N = 0, $f\left(x\right)=s\left(x\right)$, From this, 2-step waves can be obtained, and the simulation is shown in Figure 10. It can be seen that the 2-step function can be achieved, which can generate two saddle focal equilibrium points with two indicators, and can achieve two scrolls.

FIGURE 10

FIGURE 10. 2-step wave simulation.

From the above figures, we can get A = A1 = 30uA, take N = 1, then we can get 4-step waves, and the simulation is shown in Figure 11. It can be seen that the 4-step function can be achieved, which can generate four saddle focal equilibrium points with two indicators, and can achieve four scrolls.

FIGURE 11

FIGURE 11. 4-step wave simulation.

3.6 Design of odd-numbered scroll Jerk system

Using Eq. (4), an odd number of scrolls can be generated, and the number of scrolls is N + M+2. Now take the generation of three scroll numbers as an example. Let N = 1, M = 0, and the scroll number be N + M+2 = 1 + 0+2 = 3. Then Eq. (4) becomes Eq. (6)

$f\left(x\right)=A1+s\left(x\right)+s\left(x-2A1\right)(6)$

From this, 3-step waves can be obtained, and the simulation is shown in Figure 12. It can be seen that the 3-step function can be achieved, which can generate three saddle focal equilibrium points with two indicators, and can achieve three scrolls.

FIGURE 12

FIGURE 12. 3-step wave simulation.

3.7 Simulation of multi scroll chaotic circuit

According to the expression Eq. (4) of the non-linear function f(x), when N and M take different values, different step waves will be generated, which will affect the number of scroll generated. When N = M and N = 0, two scroll attractors are generated, as shown in Figure 13.

FIGURE 13

FIGURE 13. Two scroll chaotic attractor. (A) x-y phase diagram of chaotic attractor. (B) x-z phase diagram of chaotic attractor.

When N = M, N = 1, four scroll attractors are generated, as shown in Figure 14.

FIGURE 14

FIGURE 14. Four scroll chaotic attractor. (A) x-y phase diagram of chaotic attractor. (B) x-z phase diagram of chaotic attractor.

When N = 1, M = 0, three scroll attractors are generated, as shown in Figure 15.

FIGURE 15

FIGURE 15. Three scroll chaotic attractor. (A) x-y phase diagram of chaotic attractor. (B) x-z phase diagram of chaotic attractor.

As shown in these figures, the proposed CDTA-based current mode multi scroll Jerk chaotic oscillator circuit can display a theoretical number of scrolls in both the x-y and x-z directions. The experimental results are consistent with the theoretical results.

4 Conclusion

The circuit structure of the multi scroll chaotic oscillator based on CDTA proposed in this paper is simple, the main circuit does not contain passive resistance elements, has low input impedance, high output impedance, and the input and output impedance are independent of frequency, and the dynamic range is large, so that the chaotic oscillator can generate more scrolls, and the signal is not distorted.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

YL: Conceptualization, Methodology, Visualization, Project administration, Supervision, Writing–Review and Editing. JG: Methodology, Investigation, Formal analysis, Supervision. FY: Methodology, Investigation, Formal analysis, Writing–Original Draft. YH: Investigation, Formal analysis, Supervision. All authors contributed to the article and approved the submitted version.

Funding

This work is supported by the Scientific Research Fund of Hunan Provincial Education Department under Grants 20k036, 21B0345 and 19C0083, and by the Hunan Provincial Natural Science Foundation of China under Grant 2022JJ30624, and by the Postgraduate Training Innovation Base Construction Project of Hunan Province under Grant 2020-172-48.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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