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Chua’s系统的Bogdanov-Takens分岔分析
Analysis of Bogdanov-Takens Bifurcation in Chua’s System
DOI: 10.12677/AAM.2018.712187, PDF, HTML, XML, 下载: 1,161  浏览: 3,747

Abstract: We present explicit formulae for normal form and universal unfolding of the Bogdanov-Takens bifurcation in Chua’s system by a homological method, and plot the corresponding bifurcation di-agram.

1. 引言

1983年，蔡少棠教授首次提出蔡氏电路 [1] ，这个系统拥有复杂的动力学行为，广泛应用于电子学方面 [2] 。本文主要基于参数依赖的中心流形，利用参数依赖的归一化方法 [3] [4] 来分析Chua’s系统的Bogdanov-Takens分岔。文章第二和第三部分分别介绍了对称系统Bogdanov-Takens (BT)分岔规范型和普适开折的计算公式，第四部分计算Chua’s系统相应分岔的规范型和普适开折，并画出它的分岔图。

2. 规范型计算公式

$\stackrel{˙}{x}=f\left(x,\alpha \right),$ (1)

$\stackrel{˙}{x}=Ax+\frac{1}{6}C\left(x,x,x\right)+\cdots ,$ (2)

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}={x}_{2},\\ {\stackrel{˙}{x}}_{2}={c}_{30}{x}_{1}^{3}+{c}_{21}{x}_{1}^{2}{x}_{2},\end{array}$ (3)

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}={x}_{2},\\ {\stackrel{˙}{x}}_{2}={\eta }_{1}{x}_{1}+{\eta }_{2}{x}_{2}+{c}_{30}{x}_{1}^{3}+{c}_{21}{x}_{1}^{2}{x}_{2}.\end{array}$ (4)

${c}_{30}=\frac{1}{6}{p}_{2}^{Τ}C\left({q}_{1},{q}_{1},{q}_{1}\right),$ (5)

${c}_{21}=\frac{1}{2}{p}_{2}^{Τ}C\left({q}_{1},{q}_{\text{1}},{q}_{\text{2}}\right)+\frac{1}{2}{p}_{1}^{Τ}C\left({q}_{1},{q}_{1},{q}_{1}\right),$ (6)

$A{q}_{1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}A{q}_{2}={q}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}^{Τ}{p}_{2}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}^{Τ}{p}_{1}={p}_{2},$ (7)

${p}_{1}^{Τ}{q}_{1}={p}_{2}^{Τ}{q}_{2}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{1}^{Τ}{q}_{2}={p}_{2}^{Τ}{q}_{1}=0.$ (8)

3. 普适开折的计算公式

$\stackrel{˙}{x}=Ax+\frac{1}{6}C\left(x,x,x\right)+{A}_{1}\left(x,\Delta \right)+\frac{1}{6}{C}_{1}\left(x,x,x,\Delta \right)+\frac{1}{2}{A}_{2}\left(x,\Delta ,\Delta \right)+\cdots ,$ (9)

${\Delta }_{i}={V}_{i}\left(\eta \right)={a}_{i}{\eta }_{1}+{b}_{i}{\eta }_{2}+{c}_{i}{\eta }_{1}^{2}+{d}_{i}{\eta }_{1}{\eta }_{2}+{e}_{i}{\eta }_{2}^{2}+Ο\left({‖\eta ‖}^{3}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2.$ (10)

${p}_{2}^{Τ}{A}_{1}\left({q}_{1},a\right)=1,$ (11)

${p}_{2}^{Τ}{A}_{1}\left({q}_{1},b\right)=0,$ (12)

${p}_{1}^{Τ}{A}_{1}\left({q}_{1},a\right)+{p}_{2}^{Τ}{A}_{1}\left({q}_{2},a\right)=0,$ (13)

${p}_{1}^{Τ}{A}_{1}\left({q}_{1},b\right)+{q}_{2}^{Τ}{A}_{1}\left({q}_{2},b\right)=1.$ (14)

${p}_{2}^{Τ}{A}_{1}\left({q}_{1},c\right)={p}_{2}^{Τ}\left({h}_{0110}-{A}_{1}\left({h}_{1010},a\right)-\frac{1}{2}{A}_{2}\left({q}_{1},a,a\right)\right),$ (15)

${p}_{2}^{Τ}{A}_{1}\left({q}_{1},d\right)={p}_{2}^{Τ}\left({h}_{0101}-{A}_{1}\left({h}_{1001},a\right)-{A}_{1}\left({h}_{1010},b\right)-{A}_{2}\left({q}_{1},a,b\right)\right),$ (16)

${p}_{2}^{Τ}{A}_{1}\left({q}_{1},e\right)=-{p}_{2}^{Τ}\left({A}_{1}\left({h}_{1001},b\right)+\frac{1}{2}{A}_{2}\left({q}_{1},b,b\right)\right),$ (17)

$\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({q}_{1},c\right)\\ {A}_{1}\left({q}_{2},c\right)\end{array}\right)={p}_{1}^{Τ}{h}_{0110}-\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({h}_{1010},a\right)\\ {A}_{1}\left({h}_{0110},a\right)\end{array}\right)-\frac{1}{2}\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{2}\left({q}_{1},a,a\right)\\ {A}_{2}\left({q}_{2},a,a\right)\end{array}\right),$ (18)

$\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({q}_{1},d\right)\\ {A}_{1}\left({q}_{2},d\right)\end{array}\right)=\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{h}_{0101}\\ {h}_{0110}\end{array}\right)-\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({h}_{1001},a\right)+{A}_{1}\left({h}_{1010},b\right)\\ {A}_{1}\left({h}_{0101},a\right)+{A}_{1}\left({h}_{0110},b\right)\end{array}\right)-\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{2}\left({q}_{1},a,b\right)\\ {A}_{2}\left({q}_{2},a,b\right)\end{array}\right),$ (19)

$\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({q}_{1},e\right)\\ {A}_{1}\left({q}_{2},e\right)\end{array}\right)={p}_{2}^{Τ}{h}_{0101}-\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{1}\left({h}_{1001},b\right)\\ {A}_{1}\left({h}_{0101},b\right)\end{array}\right)-\frac{1}{2}\left({p}_{1}^{Τ},{p}_{2}^{Τ}\right)\left(\begin{array}{l}{A}_{2}\left({q}_{1},b,b\right)\\ {A}_{2}\left({q}_{2},b,b\right)\end{array}\right),$ (20)

${p}_{2}^{Τ}\left({h}_{1010},{h}_{1001}\right)=-{p}_{1}^{Τ}\left({A}_{1}\left({q}_{1},a\right),{A}_{1}\left({q}_{1},b\right)\right),$ (21)

${p}_{2}^{Τ}\left({h}_{0110},{h}_{0101}\right)={p}_{1}^{Τ}\left({h}_{1010},{h}_{1001}\right)-{p}_{1}^{Τ}\left({A}_{1}\left({q}_{2},a\right),{A}_{1}\left({q}_{2},b\right)\right).$ (22)

${|\frac{\partial \left({\eta }_{1},{\eta }_{2}\right)}{\partial \left({\alpha }_{1},{\alpha }_{2}\right)}|}_{\alpha ={\alpha }^{\ast }}={|\frac{\partial \left({\eta }_{1},{\eta }_{2}\right)}{\partial \left({\Delta }_{1},{\Delta }_{2}\right)}|}_{\Delta }=\frac{1}{|\begin{array}{cc}{a}_{1}& {b}_{1}\\ {a}_{2}& {b}_{2}\end{array}|}\ne 0.$

4. Chua’s系统Bogdanov-Takens分岔分析

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}=\beta \left({x}_{2}-\gamma {x}_{1}-\delta {x}_{1}^{3}\right),\\ {\stackrel{˙}{x}}_{2}={x}_{1}-{x}_{2}+{x}_{3},\\ {\stackrel{˙}{x}}_{3}=-h{x}_{2}.\end{array}$ (23)

${q}_{1}=\left(\begin{array}{c}-v\\ 0\\ v\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}=\left(\begin{array}{c}-u-\frac{v}{\beta }\\ -\frac{v}{\beta }\\ u\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{1}=\left(\begin{array}{c}\frac{u\beta }{{v}^{2}}+\frac{\beta }{v}\\ -\frac{\beta }{v}\\ \frac{u\beta }{{v}^{2}}+\frac{\beta }{v}+\frac{1}{v}\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{2}=\left(\begin{array}{c}-\frac{\beta }{v}\\ 0\\ -\frac{\beta }{v}\end{array}\right),$

$\begin{array}{l}C\left(y,z,\omega \right)=\left(\begin{array}{c}-6\beta \delta {y}_{1}{z}_{1}{\omega }_{1}\\ 0\\ 0\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{1}\left(y,u\right)=\left(\begin{array}{c}-\beta {y}_{1}{u}_{2}+{y}_{2}{u}_{1}\\ 0\\ 0\end{array}\right),\\ {C}_{1}\left(y,z,w,u\right)=\left(\begin{array}{c}-6\delta {y}_{1}{z}_{1}{w}_{1}{u}_{1}\\ 0\\ 0\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{A}_{2}\left(y,u,v\right)=\left(\begin{array}{c}-{y}_{1}{u}_{1}{v}_{2}-{y}_{1}{u}_{2}{v}_{1}\\ 0\\ 0\end{array}\right).\end{array}$

$\left\{\begin{array}{l}{c}_{30}=\frac{1}{6}{p}_{2}^{Τ}\left({q}_{1},{q}_{1},{q}_{1}\right)=-{v}^{2}{\beta }^{2}\delta ,\\ {c}_{21}=\frac{1}{2}{p}_{2}^{Τ}\left({q}_{1},{q}_{1},{q}_{2}\right)=3{v}^{2}\beta \delta \left(\beta -1\right).\end{array}$ (24)

${a}_{1}=1-\frac{1}{\beta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{1}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{2}=-\frac{1}{{\beta }^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}=0,$

${h}_{1010}=\left(\begin{array}{c}\frac{-u-v+\left({C}_{4}-{C}_{5}+{C}_{6}-{C}_{8}\right){\beta }^{2}}{\beta }\\ -\frac{u}{\beta }\\ -\left({C}_{4}-{C}_{3}+{C}_{6}-{C}_{8}\right)\beta \end{array}\right),$

${h}_{1001}=\left(\begin{array}{c}u+\left({C}_{7}-{C}_{8}\right)\beta +\left(\frac{1}{2}{C}_{1}-\frac{1}{2}{C}_{2}+\frac{1}{2}{C}_{3}-{C}_{5}\right){\beta }^{2}\\ -u-\left({C}_{7}-{C}_{8}\right)\beta +\left(\frac{1}{2}{C}_{1}-\frac{1}{2}{C}_{2}+\frac{1}{2}{C}_{3}-{C}_{5}\right){\beta }^{2}\end{array}\right),$

${h}_{0110}=\left(\begin{array}{c}\frac{-2u+2\left({C}_{4}-{C}_{5}+{C}_{6}-{C}_{8}\right)\beta +\left({C}_{1}-{C}_{2}+{C}_{3}\right){\beta }^{2}}{2\beta }\\ {C}_{4}-{C}_{5}+{C}_{6}-{C}_{8}\\ -\left({C}_{4}-{C}_{5}+{C}_{6}-{C}_{8}\right)\beta \end{array}\right),$

${h}_{0101}=\left(\begin{array}{c}\frac{-2v+2\left({C}_{7}-{C}_{8}\right)\beta +\left({C}_{1}-{C}_{2}+{C}_{3}-2{C}_{5}+2{C}_{8}\right){\beta }^{2}}{2\beta }\\ \left(\frac{1}{2}{C}_{1}-\frac{1}{2}{C}_{2}+\frac{1}{2}{C}_{3}-{C}_{5}\right)\beta \\ -u-\left({C}_{7}-{C}_{8}\right)\beta -\left(\frac{1}{2}{C}_{1}-\frac{1}{2}{C}_{2}+\frac{1}{2}{C}_{3}-{C}_{5}\right){\beta }^{2}\end{array}\right).$

${c}_{1}=\frac{1}{{\beta }^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{1}=-\frac{2}{\beta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}_{1}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{2}=\frac{2}{{\beta }^{3}}-\frac{1}{{\beta }^{4}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{2}=\frac{1}{{\beta }^{2}}+\frac{1}{{\beta }^{3}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}_{2}=0.$

${\eta }_{i}={f}_{i}{\Delta }_{1}+{g}_{i}{\Delta }_{2}+{h}_{i}{\Delta }_{1}^{2}+{j}_{1}{\Delta }_{1}{\Delta }_{2}+{k}_{i}{\Delta }_{2}^{2}+Ο\left({‖\Delta ‖}^{3}\right),i=1,2$，并代入 $\Delta =V\left(\eta \right)$，将以上两式代入(10)式，比较两端系数可以得到开折参数如下：

$\left\{\begin{array}{l}{\eta }_{1}=-{\beta }^{2}{\Delta }_{2}+{\beta }^{4}{\Delta }_{2}^{2}-\left(\beta -{\beta }^{2}\right){\Delta }_{1}{\Delta }_{2}+Ο\left({‖\Delta ‖}^{3}\right),\\ {\eta }_{2}={\Delta }_{1}-\left(\beta -{\beta }^{2}\right){\Delta }_{2}-{\Delta }_{1}^{2}-\left(1-2\beta +3{\beta }^{2}\right){\Delta }_{1}{\Delta }_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({\beta }^{3}-2{\beta }^{4}\right){\Delta }_{2}^{2}+Ο\left({‖\Delta ‖}^{3}\right).\end{array}$ (25)

$R=\left\{\left({\eta }_{1}\left(\Delta \right),{\eta }_{2}\left(\Delta \right)\right)|{\eta }_{1}=0,{\eta }_{2}\ne 0\right\}=\left\{\left({\Delta }_{1},{\Delta }_{2}\right)|{\Delta }_{2}=0,{\Delta }_{1}\ne 0\right\},$

$H=\left\{\left({\eta }_{1}\left(\Delta \right),{\eta }_{2}\left(\Delta \right)\right)|{\eta }_{1}=0,{\eta }_{2}<0\right\}=\left\{\left({\Delta }_{1},{\Delta }_{2}\right)|{\Delta }_{2}=-\frac{1}{2}{\Delta }_{1}+\frac{5{\Delta }_{1}^{2}}{4}+Ο\left({\Delta }_{1}^{3}\right),{\Delta }_{1}<0\right\},$

$HL=\left\{\left({\eta }_{1}\left(\Delta \right),{\eta }_{2}\left(\Delta \right)\right)|{\eta }_{2}=\frac{{c}_{21}}{5|{c}_{30}|}{\eta }_{1}+Ο\left({\eta }_{1}^{3/2}\right),{\eta }_{1}<0\right\}=\left\{\left({\Delta }_{1},{\Delta }_{2}\right)|{\Delta }_{2}=-\frac{5}{4}{\Delta }_{1}+\frac{205{\Delta }_{1}^{2}}{8}+Ο\left({\Delta }_{1}^{3}\right),{\Delta }_{1}<0\right\}.$

Figure 1. Bifurcation curves of system (23) at ${\alpha }^{\ast }=\left(\beta ,\gamma ,\delta ,h\right)=\left(\beta ,0,\delta ,\beta \right)=\left(2,0,1,2\right)$ with bifurcation parameter $\left(\delta ,\beta \right)$

5. 结束语

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