# 基于简化Lorenz混沌电路的频率特性分析Frequency Characteristics Analysis Based on Simplified Lorenz System

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In the hardware circuit of the chaotic system, the integral constant of the integrating circuits can be configured to change the size of the system time scale transformation. However, the different parameter configurations of the integrating circuits will change the frequency distribution range of the system signal. What effect does the change in the signal frequency distribution have on the nonlinear dynamic characteristics of the system? In this paper, the variation of the oscillation frequency range of the system signal is observed by changing the integral time constant in chaotic circuit. The circuit simulation of the simplified Lorenz system is used to analyze the oscillation frequency range and explore the potential influence for the chaotic system with different parameter configuration of the integrating circuits. The experimental results have showed that the different parameter configuration of integrating circuits can change the oscillation frequency range of the chaotic system signal, but it does not change the chaotic characteristics of the system. In the chaotic systems based on memristors whose volt-ampere characteristics exhibited vibration with frequency, the parameter configuration of integrating circuits might cause potential effect on nonlinear dynamic characteristics of the system.
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1. 引言

2. 简化Lorenz系统

$\left\{\begin{array}{l}\stackrel{˙}{x}=10\left(y-x\right)\\ \stackrel{˙}{y}=\left(24-4c\right)x-xz+cy\\ \stackrel{˙}{z}=xy-8z/3\end{array}$ (1)

$\left\{\begin{array}{l}\text{d}x/\text{d}t=\frac{1}{{R}_{5}{C}_{1}}\left(\frac{{R}_{3}}{{R}_{1}}y-\frac{{R}_{4}\left({R}_{1}+{R}_{3}\right)}{{R}_{1}\left({R}_{2}+{R}_{4}\right)}x\right)\\ \text{d}y/\text{d}t=\frac{1}{{R}_{17}{C}_{2}}\left(\frac{{\text{R}}_{16}}{{R}_{8}}x-\frac{{R}_{16}{R}_{7}}{10{R}_{9}{R}_{6}}xz+\frac{{R}_{16}{R}_{14}}{{R}_{15}}\left(\frac{{R}_{11}}{{R}_{12}{R}_{10}}y-\frac{1}{{R}_{13}}x\right)\right)\\ \text{d}z/\text{d}t=\frac{1}{{R}_{22}{C}_{3}}\left(\frac{{R}_{20}}{10{R}_{18}}xy-\frac{{R}_{21}\left({R}_{18}+{R}_{20}\right)}{{R}_{18}\left({R}_{21}+{R}_{19}\right)}z\right)\end{array}$ (2)

Figure 1. Phase portraits of chaotic attractor (x-y plane, x-z plane)

Figure 2. Circuit of simplified Lorenz system

Figure 3. The signals x(t), y(t), z(t) of simplified Lorenz circuit (R = 1 kΩ, R14 = 3.4 kΩ). (a) C = 1 nF; (b) C = 100 nF; (c) C = 4.7 uF

(a) (b) (c)

Figure 4. Spectrum of signals x(t), y(t), z(t) of simplified Lorenz circuit (R = 1 kΩ, R14 = 3.4 kΩ). (a) C = 1 nF; (b) C = 100 nF; (c) C = 4.7 uF

Figure 5. Phase portraits of attractors with different parameters. (a) R14 = 7.38 kΩ, C = 4.7 uF; (b) R14 = 7.38 kΩ, C = 100 nF; (c) R14 = 5.55 kΩ, C = 1 nF

(a) (b) (c)

Figure 6. The spectrogram of x(t), y(t), z(t) in non-chaotic state. (a) R14 = 7.38 kΩ, C = 4.7 uF; (b) R14 = 7.38 kΩ, C = 100 nF; (c) R14 = 5.55 kΩ, C = 1 nF

3. 结论

NOTES

*通讯作者。

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