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本文采用对数函数序列构造了一个新Chua多涡卷混沌系统, 分析了该系统的非线性动力学行为, 主要包括对称性、不变性、平衡点、最大李雅普诺夫指数等. 然后, 设计递归反步控制器控制Chua多涡卷混沌系统中的混沌行为. 最后, 利用Chua多涡卷混沌系统检测了多频微弱周期信号. 结果表明, 对数函数序列与新Chua双涡卷混沌系统相结合能够产生丰富的Chua多涡卷混沌吸引子. 产生机制为指标2的鞍焦平衡点用于产生涡卷, 指标1的鞍焦平衡点用于连接涡卷. 递归反步控制器能够将Chua多涡卷混沌系统控制到不动点或给定的正弦函数. Chua多涡卷混沌系统与递归反步控制器相结合的新微弱信号检测方法能够检测出淹没在高斯噪声背景下多频微弱周期信号的各频率.
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关键词:
- Chua多涡卷混沌系统/
- 对数函数序列/
- 递归反步控制器/
- 微弱信号检测
Chaos has great potential applications in engineering fields, such as secure communication and digital encryption. Since the double-scroll Chua’s circuit was developed first by Chua, it has quickly become a paradigm to study the double-scroll chaotic attractors. Compared with the conventional double-scroll chaotic attractors, the multi-scroll chaotic attractors have complex structures and rich nonlinear dynamical behaviors. The multi-scroll chaotic attractors have been applied to various chaos-based information technologies, such as secure communication and chaotic cryptanalysis. Hence, the generation of the multi-scroll chaotic attractors has become a hot topic in research field of chaos at present. In this paper, a novel Chua multi-scroll chaotic system is constructed by using a logarithmic function series. The nonlinear dynamical behaviors of the novel Chua multi-scroll chaotic system are analyzed, including symmetry, invariance, equilibrium points, the largest Lyapunov exponent, etc. The existence of chaos is confirmed by theoretical analyses and numerical simulations. The results show that the rich Chua multi-scroll chaotic attractors can be generated by combining the logarithmic function series with the novel Chua double-scroll chaotic system. The generation mechanism of the Chua multi-scroll chaotic attractors is that the saddle-focus equilibrium points of index 2 are used to generate the scrolls, and the saddle-focus equilibrium points of index 1 are used to connect these scrolls. Then, three recursive back-stepping controllers are designed to control the chaotic behavior in the novel Chua multi-scroll chaotic system. The recursive back-stepping controllers can control the novel Chua multi-scroll chaotic system to a fixed point or a given sinusoidal function. Finally, a new method of detecting a weak signal embedded in the Gaussian noise is proposed on the basis of the novel Chua multi-scroll chaotic system and the recursive back-stepping controllers. The immunity of the novel Chua multi-scroll chaotic system to the Gaussian noise with the zero mean is analyzed by using the stochastic differential equation theory. The results show that the proposed new method of detecting the weak signal can detect the frequencies of the multi-frequency weak periodic signal embedded in the Gaussian noise. In addition, the novel Chua multi-scroll chaotic system has strong immunity to any Gaussian noise with the zero mean. The proposed method provides a new thought for detecting the weak signal.-
Keywords:
- Chua multi-scroll chaotic system/
- logarithmic function series/
- recursive back-stepping controllers/
- weak signal detection
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平衡点 特征值 平衡点的类型 ${Q_0}\left( {0,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{1,2}}\left( { \pm 10,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{3,4}}\left( { \pm 20,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{5,6}}\left( { \pm 30,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{7,8}}\left( { \pm 40,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{9,10}}\left( { \pm 50,0,0} \right)$ $67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$ Ⅰ ${Q_{11,12}}\left( { \pm 5,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ ${Q_{13,14}}\left( { \pm 15,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ ${Q_{15,16}}\left( { \pm 25,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ ${Q_{17,18}}\left( { \pm 35,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ ${Q_{19,20}}\left( { \pm 45,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ ${Q_{2{\rm{1}},2{\rm{2}}}}\left( { \pm 55,0,0} \right)$ $ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$ Ⅱ 
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