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Study on Nonlinear Dynamics of a Class of Deformed Chua’s Circuit under Dual-Frequency Excitation
DOI: 10.12677/AAM.2021.102068, PDF, HTML, XML, 下载: 402  浏览: 562

Abstract: This paper considers a typical Chua’s circuit with piecewise nonlinear resistance and dual-frequency excitation, and studies the chaos and bifurcation behavior in piecewise smooth dynamic system. If the excitation parameters of the system are changed, different bifurcation and chaos phenomena can be produced. The bifurcation behavior of the fast subsystem is discussed by taking the excitation term with slowly varying period as bifurcation parameter, and the bursting phenomenon of the system are discussed when the two excitation frequencies are the same or different and are in integral ratio. The nonlinear dynamic behaviors of folding bifurcation, Hopf bifurcation and chaos are analyzed by theoretical method and numerical simulation.

1. 引言

2. 蔡氏电路数学模型

$\left\{\begin{array}{l}{{x}^{\prime }}_{1}=a\left(-{x}_{1}+{x}_{2}-f\left({x}_{1}\right)\right)+\alpha \\ {{x}^{\prime }}_{2}={x}_{1}-{x}_{2}+{x}_{3}\\ {{x}^{\prime }}_{3}=-b{x}_{2}+\beta \end{array}$，其中 $f\left({x}_{1}\right)=\left\{\begin{array}{l}m{x}_{1}+n,{x}_{1}<-{x}_{0}\\ c{x}_{1}^{3}+d{x}_{1},|{x}_{1}|\le {x}_{0}\\ m{x}_{1}-n,{x}_{1}>{x}_{0}\end{array}$ (1)

$\alpha =\mu \mathrm{sin}{\omega }_{1}t$$\beta =\mu \mathrm{sin}{\omega }_{2}t$$a,b,c,d,m,n$ 为蔡氏电路无量纲化对应参数。

3. 平衡点和分岔分析

${D}_{0}=\left\{{x}_{1}||{x}_{1}|\le {x}_{0}\right\},\text{}{D}_{+}=\left\{{x}_{1}|{x}_{1}>{x}_{0}\right\},\text{}{D}_{-}=\left\{{x}_{1}|{x}_{1}<-{x}_{0}\right\}$ (2)

$a\left[\left(1+d\right){x}_{0}+c{x}_{0}^{3}-\frac{\beta }{b}\right]-\alpha =0$，这里 ${p}_{+}$ ${p}_{-}$${p}_{0}$ 处相应的雅克比矩阵为：

${J}_{{p}_{+}}={J}_{{p}_{-}}=\left(\begin{array}{ccc}-a\left(m+1\right)& a& 0\\ 1& -1& 1\\ 0& -b& 0\end{array}\right),\text{}{J}_{{p}_{0}}=\left(\begin{array}{ccc}-a-ad-3ac{x}_{0}^{2}& a& 0\\ 1& -1& 1\\ 0& -b& 0\end{array}\right)$ (3)

$|\lambda E-{J}_{{p}_{±}}|={\lambda }^{3}+{\lambda }^{2}\left(a+am+1\right)+\lambda \left(am+b\right)+amb+ab=0$ (4)

$|\lambda E-{J}_{{p}_{0}}|={\lambda }^{3}+{\lambda }^{2}\left(a+ad+1+3ac{x}_{0}^{2}\right)+\lambda \left(ad+b+3ac{x}_{0}^{2}\right)+abd+ab+3abc{x}_{0}^{2}=0$ (5)

$a=2.75,\text{\hspace{0.17em}}b=3,\text{\hspace{0.17em}}c=0.5,\text{\hspace{0.17em}}d=-1.05,\text{\hspace{0.17em}}m=2,\text{\hspace{0.17em}}n=1.54,\text{\hspace{0.17em}}{x}_{0}=0.75$ (6)

4. 双频激励的相空间吸引子

Figure 1. Bifurcation diagram of x about α

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Figure 2. Is about the phase diagram of x-y, x-z, y-z, when $a=2.75,a=3.5,a=6.5$

5. 双频激励的簇发现象

5.1. 两个激励频率相同时的簇发振荡

(a) (b) (c) (d) (e) (f)

Figure 3. Is about the phase diagram of x-t, y-t, when $\mu =0.35,\text{\hspace{0.17em}}\mu =0.45,\text{\hspace{0.17em}}\mu =0.65$.

5.2. 两个激励频率不同时的簇发振荡

(a) (b) (c) (d) (e) (f) (g) (h)

Figure 4. (a), (b), (c) and (d) shows the system phase diagram and time series diagram when ${\omega }_{1}=0.008$, ${\omega }_{2}=0.004$, and (e), (f), (g) and (h) shows time series diagrams when ${\omega }_{1}=0.03$, ${\omega }_{2}=0.06$ and ${\omega }_{1}=0.1$, ${\omega }_{2}=0.2$

6. 结论

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