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 
作者: 苏彩娴:广东技术师范学院计算机科学学院,广东 广州
关键词: Bogdanov-Takens分岔规范型普适开折Chua’s系统Bogdanov-Takens Bifurcation Normal Form Universal Unfolding Chua’s System
摘要: 应用同调方法显式计算Chua’s系统Bogdanov-Takens分岔的规范型和普适开折,并画出对应的分岔图。
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.
文章引用:苏彩娴. Chua’s系统的Bogdanov-Takens分岔分析[J]. 应用数学进展, 2018, 7(12): 1600-1606. https://doi.org/10.12677/AAM.2018.712187

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

考虑以下 Z 2 对称系统

x ˙ = f ( x , α ) , (1)

其中 x R n ( n 2 ) α R 2 f : R n × R 2 R n 充分光滑。向量场(1)在变换 x x 下保持不变,当 α = α 时,有平衡点 x = x = 0 ,系统的雅可比矩阵 J ( x , α ) ( x , α ) 处非零并且有二重零特征值,由向量场(1)的对称性,其可展开为如下形式:

x ˙ = A x + 1 6 C ( x , x , x ) + , (2)

其中 A = J ( 0 , α ) C ( x , z , w ) = i , j , k = 1 n 3 f ( x , α ) x i x j x k y i z j w k ,其余类似。设(1)的规范型和普适开折分别 [5] [6] [7] [8]:

{ x ˙ 1 = x 2 , x ˙ 2 = c 30 x 1 3 + c 21 x 1 2 x 2 , (3)

{ x ˙ 1 = x 2 , x ˙ 2 = η 1 x 1 + η 2 x 2 + c 30 x 1 3 + c 21 x 1 2 x 2 . (4)

定理1:根据临界中心流形不变性和Fredholm择一性定理,向量场(1)相应于BT分岔规范型(3),其系数计算公式如下 [9]:

c 30 = 1 6 p 2 Τ C ( q 1 , q 1 , q 1 ) , (5)

c 21 = 1 2 p 2 Τ C ( q 1 , q 1 , q 2 ) + 1 2 p 1 Τ C ( q 1 , q 1 , q 1 ) , (6)

其中 q 1 , q 2 p 1 , p 2 分别为A和AT的广义特征向量,且满足:

A q 1 = 0 , A q 2 = q 1 , A Τ p 2 = 0 , A Τ p 1 = p 2 , (7)

p 1 Τ q 1 = p 2 Τ q 2 = 1 , p 1 Τ q 2 = p 2 Τ q 1 = 0. (8)

3. 普适开折的计算公式

考虑参数 α α 附近扰动,扰动量为 Δ = α α ,此时(1)式可展开为如下形式:

x ˙ = A x + 1 6 C ( x , x , x ) + A 1 ( x , Δ ) + 1 6 C 1 ( x , x , x , Δ ) + 1 2 A 2 ( x , Δ , Δ ) + , (9)

其中 A , C 如前所述, A 1 ( y , Δ ) = i = 1 n j = 1 2 2 f ( 0 , α ) x i α j y i Δ j C 1 , A 2 , 类似。

将原始参数和开折参数的关系表示为 Δ = V ( η ) ,进一步采用如下平方逼近:

Δ i = V i ( η ) = a i η 1 + b i η 2 + c i η 1 2 + d i η 1 η 2 + e i η 2 2 + Ο ( η 3 ) , i = 1 , 2. (10)

定理2:设(1)的BT分岔是非退化的,相应于普适开折(4),根据Fredholm择一定理,比较同调代数方程中 ω 1 i ω 2 j η 3 k η 4 l ( i + j = 1 , k + l = 1 , 2 ) 项的系数得到如下线性代数方程 [10]:

p 2 Τ A 1 ( q 1 , a ) = 1 , (11)

p 2 Τ A 1 ( q 1 , b ) = 0 , (12)

p 1 Τ A 1 ( q 1 , a ) + p 2 Τ A 1 ( q 2 , a ) = 0 , (13)

p 1 Τ A 1 ( q 1 , b ) + q 2 Τ A 1 ( q 2 , b ) = 1. (14)

由(11)~(14)可解得a,b,以下方程可以求得c,d和e:

p 2 Τ A 1 ( q 1 , c ) = p 2 Τ ( h 0110 A 1 ( h 1010 , a ) 1 2 A 2 ( q 1 , a , a ) ) , (15)

p 2 Τ A 1 ( q 1 , d ) = p 2 Τ ( h 0101 A 1 ( h 1001 , a ) A 1 ( h 1010 , b ) A 2 ( q 1 , a , b ) ) , (16)

p 2 Τ A 1 ( q 1 , e ) = p 2 Τ ( A 1 ( h 1001 , b ) + 1 2 A 2 ( q 1 , b , b ) ) , (17)

( p 1 Τ , p 2 Τ ) ( A 1 ( q 1 , c ) A 1 ( q 2 , c ) ) = p 1 Τ h 0110 ( p 1 Τ , p 2 Τ ) ( A 1 ( h 1010 , a ) A 1 ( h 0110 , a ) ) 1 2 ( p 1 Τ , p 2 Τ ) ( A 2 ( q 1 , a , a ) A 2 ( q 2 , a , a ) ) , (18)

( p 1 Τ , p 2 Τ ) ( A 1 ( q 1 , d ) A 1 ( q 2 , d ) ) = ( p 1 Τ , p 2 Τ ) ( h 0101 h 0110 ) ( p 1 Τ , p 2 Τ ) ( A 1 ( h 1001 , a ) + A 1 ( h 1010 , b ) A 1 ( h 0101 , a ) + A 1 ( h 0110 , b ) ) ( p 1 Τ , p 2 Τ ) ( A 2 ( q 1 , a , b ) A 2 ( q 2 , a , b ) ) , (19)

( p 1 Τ , p 2 Τ ) ( A 1 ( q 1 , e ) A 1 ( q 2 , e ) ) = p 2 Τ h 0101 ( p 1 Τ , p 2 Τ ) ( A 1 ( h 1001 , b ) A 1 ( h 0101 , b ) ) 1 2 ( p 1 Τ , p 2 Τ ) ( A 2 ( q 1 , b , b ) A 2 ( q 2 , b , b ) ) , (20)

其中 h i j k l ( i + j = 1 , k + l = 1 ) 是如下奇异线性代数方程的任意解:

p 2 Τ ( h 1010 , h 1001 ) = p 1 Τ ( A 1 ( q 1 , a ) , A 1 ( q 1 , b ) ) , (21)

p 2 Τ ( h 0110 , h 0101 ) = p 1 Τ ( h 1010 , h 1001 ) p 1 Τ ( A 1 ( q 2 , a ) , A 1 ( q 2 , b ) ) . (22)

通过以上方程计算,并消去 h 10 k l ( k + l = 2 ) ,可以确定参数变换 Δ = V ( η ) ,从而求得开折参数 η = V 1 ( Δ ) ,且分岔的横截性条件为:

| ( η 1 , η 2 ) ( α 1 , α 2 ) | α = α = | ( η 1 , η 2 ) ( Δ 1 , Δ 2 ) | Δ = 1 | a 1 b 1 a 2 b 2 | 0.

如果(1)的双零分岔满足非退化和横截性条件,则当 α α 附近扰动,普适开折(4)对不同的 c 30 ( Δ ) c 21 ( Δ ) ,其分岔图和相图的拓扑结构与 c 30 = c 30 ( 0 ) c 21 = c 21 ( 0 ) 时相同 [13] 。

4. Chua’s系统Bogdanov-Takens分岔分析

以下为本文研究的立方非线性Chua’s系统 [11] 。

{ x ˙ 1 = β ( x 2 γ x 1 δ x 1 3 ) , x ˙ 2 = x 1 x 2 + x 3 , x ˙ 3 = h x 2 . (23)

容易验证:当 α = ( β , 0 , δ , β ) 时,系统(23)有平衡点 x = 0 ,它的Jacobi矩阵在平衡点处为 A = ( 0 β 0 1 1 1 0 β 0 ) 。令 | A λ I | = 0 ,解得特征值 λ 1 , 2 = 0 λ 3 = 1 ,根据(7)和(8)得到广义特征向量为:

q 1 = ( v 0 v ) , q 2 = ( u v β v β u ) , p 1 = ( u β v 2 + β v β v u β v 2 + β v + 1 v ) , p 2 = ( β v 0 β v ) ,

其中 u , v 0 为任意非零实数,计算 C , A 1 , C 1 , A 2 为:

C ( y , z , ω ) = ( 6 β δ y 1 z 1 ω 1 0 0 ) , A 1 ( y , u ) = ( β y 1 u 2 + y 2 u 1 0 0 ) , C 1 ( y , z , w , u ) = ( 6 δ y 1 z 1 w 1 u 1 0 0 ) , A 2 ( y , u , v ) = ( y 1 u 1 v 2 y 1 u 2 v 1 0 0 ) .

由(5)和(6)得到(23)的系数规范型

{ c 30 = 1 6 p 2 Τ ( q 1 , q 1 , q 1 ) = v 2 β 2 δ , c 21 = 1 2 p 2 Τ ( q 1 , q 1 , q 2 ) = 3 v 2 β δ ( β 1 ) . (24)

系统(23)满足BT分岔非退化条件 c 30 c 21 0 ,从而 β δ 0 β 1 时,然后扰动参数向量 ( β , γ , δ , h ) ,由临界值 ( β , 0 , δ , β ) 变成 ( β + Δ 1 , Δ 2 , δ , β ) 由(11)~(14)得到线性项的系数如下:

a 1 = 1 1 β , b 1 = 1 , a 2 = 1 β 2 , b 2 = 0 ,

将以上系数代入(21)和(22)得

h 1010 = ( u v + ( C 4 C 5 + C 6 C 8 ) β 2 β u β ( C 4 C 3 + C 6 C 8 ) β ) ,

h 1001 = ( u + ( C 7 C 8 ) β + ( 1 2 C 1 1 2 C 2 + 1 2 C 3 C 5 ) β 2 u ( C 7 C 8 ) β + ( 1 2 C 1 1 2 C 2 + 1 2 C 3 C 5 ) β 2 ) ,

h 0110 = ( 2 u + 2 ( C 4 C 5 + C 6 C 8 ) β + ( C 1 C 2 + C 3 ) β 2 2 β C 4 C 5 + C 6 C 8 ( C 4 C 5 + C 6 C 8 ) β ) ,

h 0101 = ( 2 v + 2 ( C 7 C 8 ) β + ( C 1 C 2 + C 3 2 C 5 + 2 C 8 ) β 2 2 β ( 1 2 C 1 1 2 C 2 + 1 2 C 3 C 5 ) β u ( C 7 C 8 ) β ( 1 2 C 1 1 2 C 2 + 1 2 C 3 C 5 ) β 2 ) .

这里 u , v C i ( i = 1 , 2 , , 8 ) 为任意实数,由上面(15)~(20),得到二次项系数为:

c 1 = 1 β 2 , d 1 = 2 β , e 1 = 1 , c 2 = 2 β 3 1 β 4 , d 2 = 1 β 2 + 1 β 3 , e 2 = 0.

η i = f i Δ 1 + g i Δ 2 + h i Δ 1 2 + j 1 Δ 1 Δ 2 + k i Δ 2 2 + Ο ( Δ 3 ) , i = 1 , 2 ,并代入 Δ = V ( η ) ,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:

{ η 1 = β 2 Δ 2 + β 4 Δ 2 2 ( β β 2 ) Δ 1 Δ 2 + Ο ( Δ 3 ) , η 2 = Δ 1 ( β β 2 ) Δ 2 Δ 1 2 ( 1 2 β + 3 β 2 ) Δ 1 Δ 2 + ( β 3 2 β 4 ) Δ 2 2 + Ο ( Δ 3 ) . (25)

由于 1 | a 1 b 1 a 2 b 2 | = 1 β 2 0 ,故横截性满足,从而当 β δ 0 β 1 时, β γ 可以作为系统(23)的分岔参数使系统发生完整双零分岔,本文可以计算2次精确度的开折参数,而文献 [12] 中计算的分岔参数仅是我们计算的线性部分。

最后,取 β = 2 δ = 1 ,则系统参数变成 ( β , γ , δ , h ) = ( 2 + Δ 1 , Δ 2 , 1 , 2 ) ,由上面(24)和(25)得相应系数 c 30 = 4 v 2 c 21 = 6 v 2 η 1 = 4 Δ 2 + 2 Δ 1 Δ 2 + 16 Δ 2 2 + Ο ( Δ 3 ) η 2 = Δ 1 + 2 Δ 2 Δ 1 2 9 Δ 1 Δ 2 24 Δ 2 2 + Ο ( Δ 3 ) .根据文献 [13] ,可得分岔曲线如下:

R = { ( η 1 ( Δ ) , η 2 ( Δ ) ) | η 1 = 0 , η 2 0 } = { ( Δ 1 , Δ 2 ) | Δ 2 = 0 , Δ 1 0 } ,

H = { ( η 1 ( Δ ) , η 2 ( Δ ) ) | η 1 = 0 , η 2 < 0 } = { ( Δ 1 , Δ 2 ) | Δ 2 = 1 2 Δ 1 + 5 Δ 1 2 4 + Ο ( Δ 1 3 ) , Δ 1 < 0 } ,

H L = { ( η 1 ( Δ ) , η 2 ( Δ ) ) | η 2 = c 21 5 | c 30 | η 1 + Ο ( η 1 3 / 2 ) , η 1 < 0 } = { ( Δ 1 , Δ 2 ) | Δ 2 = 5 4 Δ 1 + 205 Δ 1 2 8 + Ο ( Δ 1 3 ) , Δ 1 < 0 } .

其分岔图见图1

Figure 1. Bifurcation curves of system (23) at α = ( β , γ , δ , h ) = ( β , 0 , δ , β ) = ( 2 , 0 , 1 , 2 ) with bifurcation parameter ( δ , β )

图1. 系统(23)在临界参数 α = ( β , γ , δ , h ) = ( β , 0 , δ , β ) = ( 2 , 0 , 1 , 2 ) 附近以 ( δ , β ) 为分岔参数的分岔图

5. 结束语

本文利用同调代数方法计算Chua’s系统的规范型和普适开折,计算出来的开折参数精确到2次项,相对于其他研究的计算精确度更高,计算方法也更加简便。最后利用开折参数来分析Chua’s系统的分岔,并画出它的分岔图。

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