理论数学  >> Vol. 10 No. 5 (May 2020)

一类超混沌的Faraday圆盘发电机的Zero-Zero-Hopf分支
Zero-Zero-Hopf Bifurcation of a Hyperchaotic Faraday Disk Dynamo

DOI: 10.12677/PM.2020.105063, PDF, HTML, XML, 下载: 229  浏览: 376 

作者: 余环宇:华南理工大学数学学院,广东 广州

关键词: Faraday圆盘发电机超混沌Zero-Zero-Hopf分支周期解平均理论Faraday Disk Dynamo Hyperchaos Zero-Zero-Hopf Bifurcation Periodic SolutionAveraging Theory

摘要: 本文主要研究了一类四维的self-exciting Faraday圆盘发电机,它描述了azimuthal eddy流的作用。首先通过计算Lyapunov指数,发现该系统是一个超混沌的系统。然后研究了系统的zero-zero-Hopf分支。利用平均理论,获得了在zero-zero-Hopf分支点存在两个周期解的充分条件,并进一步讨论了周期解的稳定性。
Abstract: The paper investigates the bifurcation of periodic solutions at the zero-zero-Hopf equilibrium of a hyperchaotic Faraday disk dynamo. By means of the averaging theory, the paper obtains the suffi-cient conditions that two periodic solutions will appear at the bifurcation point and discusses the stability of the two orbits.

文章引用: 余环宇. 一类超混沌的Faraday圆盘发电机的Zero-Zero-Hopf分支[J]. 理论数学, 2020, 10(5): 518-523. https://doi.org/10.12677/PM.2020.105063

1. 引言

1996年,Hide等提出了一个self-exciting Faraday圆盘单极发电机,它是一个非线性的三维动力系统 [1]。在此基础上,逐渐发展了很多改进的模型 [2] - [10],其中Hide和Moroz提出了一类四维的发电机,它描述了方位涡流(azimuthal eddy currents)的作用。其模型如下:

(1.1)

其中x和y分别表示圆盘和线圈的磁通量,z表示电动机电枢的角转速,w表示圆盘的角转速。

Moroz在文献 [11] 中已经做了一些数值分析去研究系统(1.1)的zero-zero-Hopf分支,然而并没有严格的证明。本文将给出系统存在zero-zero-Hopf分支的证明,并把相应的周期解求出来。

一个zero-Hopf或zero-zero-Hopf奇点意味着可能出现混沌。近年来,虽有一些文献研究三维混沌系统的zero-Hopf分支 [12] [13] [14],但由于高维系统的复杂性,对四维及以上系统的zero-zero-Hopf分支的研究非常少 [15]。Cid-Montiel等在文献 [15] 中研究了一个四维超混沌的Lorenz系统的zero-Hopf分支。Chen 等在文献 [16] 中研究了一个广义Lorenz-Stenflo超混沌系统zero-zero-Hopf分支。

2. 超混沌和zero-zero-Hopf分支

系统(1.1) 总有一个奇点 E 0 = ( 0 , 0 , 0 , 0 ) 。当 k ( b d c d ) r d > 0 时,还有如下的两个奇点:

E ± = ( ± x 1 , ± x 1 , ± x 1 d , r x 1 2 k ) ,

其中 x 1 = k ( b d c d ) r d

对初始值 ( 0 , 0 , 0.2511 , 0 ) 和参数 ( a , b , c , d , k , r , μ ) = ( 16 , 60 , 15 , 0.1 , 1 , 6 , 0.5 ) ,我们计算出来Lyapunov指数是0.2927,0.1826,0,−19.0777。因此系统是超混沌的,图1显示了一个超混沌吸引因子。

图1是系统(1.1)的超混沌吸引因子,其中初始值 ( 0 , 0 , 0.2511 , 0 ) ,参数 ( a , b , c , d , k , r , μ ) = ( 16 , 60 , 15 , 0.1 , 1 , 6 , 0.5 )

Figure 1. Hyperchaotic attraction factor of system (1.1)

图1. 系统(1.1)的超混沌吸引因子

下面我们应用平均理论去研究系统(1.1)的zero-zero-Hopf分支。当 μ a ,一些表达式过于庞大。为了表达的方便,我们令 μ = a

理论1 对 c = 1 b , d = 1 , k = 0 , b < 0 ,原点 E 0 是系统(1.1)的一个zero-zero-Hopf奇点。在 E 0 的特征值是0,0, ± ω i ,其中 ω = b

证明. 在 E 0 的特征方程是

λ 4 + ( k + 1 + d ) λ 3 + ( k + a + c + d + k d a b a c ) λ 2 + ( a c + a d + a k + c k + k d a b d a b k a c k ) λ k a ( k d c d ) = 0 (2.1)

c = 1 b , d = 1 , k = 0 , b < 0 时,(2.1)有根0,0, ± ω i ,其中 ω = b

利用平均理论,我们能证明下面的定理。

理论2 令

( c , d , k ) = ( 1 b + ε c 1 , 1 + ε d 1 , ε k 1 ) , ρ = c 1 + d 1 b d 1 , N = ρ 2 + 2 d 1 ( a + b a b ) ρ + d 1 2 ( b 1 ) ( b ( a 1 ) 2 a 2 ) ,

这里 c 1 d 1 k 1 0

假设 a N 0 , k 1 ρ r > 0 , b < 0 ,那么系统(1.1)有一个zero-zero-Hopf分支。当 ε > 0 ε 足够小时,在 E 0 产生两个周期解。如果 d 1 > 0 , k 1 > 0 , a ρ < 0 , N > 0 ,那么这两个周期解是稳定的。

证明. 作变换

c = 1 b + ε c 1 , d = 1 + ε d 1 , k = ε k 1 ,

系统(1.1)变成

{ x ˙ = a ( y x ) , y ˙ = ( b a ) x ( 1 a ) y + ( b 1 ) z x w c 1 z ε , z ˙ = a x + ( 1 a ) y + z d 1 z ε , w ˙ = r a x 2 + r ( 1 a ) x y k 1 w ε . (2.2)

( x , y , z , w ) T = ( ε u , ε v , ε p , ε q ) T 。我们把 ( u , v , p , q ) T 又表示为 X = ( x , y , z , w ) T ,那么系统(2.2)变为

X ˙ = F 0 ( t , X ) + ε F 1 ( t , X ) = ( a ( y x ) ( b a ) x + ( a 1 ) y + ( b 1 ) z a x + ( 1 a ) y + z 0 ) ε ( 0 c 1 z + x w d 1 z k 1 w r a x 2 + r ( a 1 ) x y ) . (2.3)

下面利用平均理论来研究系统(2.2)的动力学行为。首先我们考虑下面这个未扰动系统的初值问题

X ˙ = F 0 ( t , X ) , X 0 = ( x 0 , y 0 , z 0 , w 0 ) . (2.4)

系统(2.4)的解是 X ( t , X 0 ) = ( x ( t ) , y ( t ) , z ( t ) , w ( t ) ) ,其中

m = b ( 1 a ) x 0 + a y 0 + a ( 1 b ) z 0 b , x ( t ) = ( a b x 0 a y 0 + a ( b 1 ) z 0 ) cos ( ω t ) b + + ω a ( x 0 y 0 ) sin ( ω t ) b + m ,

y ( t ) = ( b ( a 1 ) x 0 + ( b a ) y 0 + ( b 1 ) a z 0 ) cos ( ω t ) b + ( ( b a ) x 0 + ( a 1 ) y 0 + ( b 1 ) z 0 ) sin ( ω t ) ω + m , z ( t ) = ( b ( 1 a ) x 0 + a y 0 + ( a + b a b ) z 0 ) cos ( ω t ) b + ( a x 0 + ( 1 a ) y 0 + z 0 ) sin ( ω t ) ω m , w ( t ) = w 0 .

X 0 0 时,系统(2.4)的所有解 X ( t , X 0 ) 是周期的,其中周期为 T = 2 π ω 。沿着一个周期解 X ( t , X 0 )

其线性化系统 Y ˙ = D X F 0 ( t , X ( t , X 0 ) ) Y 的基解矩阵 M X 0 ( t )

M X 0 ( t ) = 1 b ( a 11 a 12 a 13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 0 0 0 0 b ) ,

其中

a 11 = a b cos ( ω t ) + ω a sin ( ω t ) + b ( 1 a ) , a 12 = a cos ( ω t ) ω a sin ( ω t ) + a , a 13 = a ( b 1 ) ( cos ( ω t ) 1 ) , a 21 = ( a 1 ) b cos ( ω t ) + ω ( a b ) sin ( ω t ) + b ( 1 a ) , a 22 = ( b a ) cos ( ω t ) + ω ( 1 a ) sin ( ω t ) + a ,

a 23 = ( b 1 ) ( a cos ( ω t ) ω sin ( ω t ) a ) , a 31 = b ( 1 a ) cos ( ω t ) ω a sin ( ω t ) b ( 1 a ) , a 32 = a cos ( ω t ) ω ( 1 a ) sin ( ω t ) a , a 33 = ( b + a a b ) cos ( ω t ) ω sin ( ω t ) + a ( b 1 ) .

计算积分

G ( X 0 ) = 1 T 0 T M X 0 1 ( t ) F 1 ( t , X ( t , x 0 ) ) d t = 1 2 b 2 ( G 1 ( X 0 ) , G 2 ( X 0 ) , G 3 ( X 0 ) , G 4 ( X 0 ) ) ,

其中

G 1 ( X 0 ) = 2 a b ( 2 a 1 ) x 0 w 0 a 2 ( b + 3 ) y 0 w 0 3 a 2 ( 1 b ) z 0 w 0 ( ( 4 a 3 ) ρ + a d 1 ( b 1 ) ) a b x 0 + ( 3 a ρ + b c 1 ( a 1 ) ) a y 0 a ( ( 3 a b 3 a b ) ρ + c 1 b ) z 0 , G 2 ( X 0 ) = a b ( 4 a b 3 ) x 0 w 0 a ( a b + 3 a 2 b ) y 0 w 0 + a ( b 1 ) ( 3 a b ) z 0 w 0 b ( a ( 3 a 4 ) ρ + c 1 ( a 2 a b + b ) ) x 0 + ( ( 3 a 2 a b + b ) ρ + a b c 1 ( a 2 ) ) y 0 + ( b 1 ) ( ( a b 3 a 2 b ) ρ + b d 1 ( a b 2 a b ) ) z 0 ,

G 3 ( X 0 ) = a b ( 3 4 a ) x 0 w 0 + a ( a b + 3 a b ) y 0 w 0 3 a 2 ( b 1 ) z 0 w 0 + ( 4 a b ρ ( a 1 ) + b d 1 ( a 2 b a 2 b ) ) x 0 ( ( a 1 ) 2 b c 1 + 3 a 2 ρ + b d 1 ) y 0 ( ( b 3 a 2 b + 3 a 2 ) ρ 2 b d 1 ( a b a b ) ) z 0 , G 4 ( X 0 ) = b r ( 3 a 2 b a 2 4 a b + 2 b ) x 0 2 a 2 r ( b 3 ) y 0 2 + 3 a 2 r ( b 1 ) 2 z 0 2 4 a b r ( a 1 ) x 0 y 0 + 2 a b r ( b 1 ) ( 3 a 2 ) x 0 z 0 6 a 2 r ( b 1 ) y 0 z 0 2 k 1 b 2 w 0

解方程 G ( X 0 ) = 0 ,我们得到下面的解

S 0 = ( 0 , 0 , 0 , 0 ) , S 1 = ( k 1 ρ r , k 1 ρ r , k 1 ρ r , ρ ) , S 2 = ( k 1 ρ r , k 1 ρ r , k 1 ρ r , ρ ) .

S 0 对应奇点 E 0 = ( 0 , 0 , 0 , 0 ) 。对其他两个解 S 1 S 2 ,我们有

G ( S 1 ) = G ( S 2 ) = 0 ,

det ( ( G / X 0 ) ( S 1 ) ) = det ( ( G / X 0 ) ( S 2 ) ) = a k 1 ρ N 2 b 2 .

因此根据平均理论,系统(2.3)存在的两个周期为T的解 X 1 ( t , ε ) X 2 ( t , ε ) ,当时,有 X 1 ( t , ε ) S 1 X 2 ( t , ε ) S 2 。考虑到系统(2.2)和(2.3)的关系,系统(2.3)的两个周期解分别对应系统(2.2)的两个周期解 ε X 1 ( t , ε ) ε X 2 ( t , ε )

最后,我们计算Jacobian矩阵 ( G / X 0 ) ( S 1 ) ( G / X 0 ) ( S 2 ) 的特征值来判断两个周期解的稳定性。对这两个矩阵而言,特征多项式是相同的,如下

( b λ 2 + k 1 b λ + 2 a k 1 ρ ) ( 4 b λ 2 + 4 b d 1 λ N ) = 0. (2.5)

d 1 > 0 , k 1 > 0 , a ρ < 0 , N > 0 时,方程(2.5)的根都有负实部,这意味着在 E 0 产生的两个周期解 X 1 ( t , ε ) X 2 ( t , ε ) 是稳定的。

3. 结束语

通过选取适当的参数,我们计算发现系统有两个正的Lyapunov指数,是一个超混沌的系统。不同于前人的数值工作,本文对系统存在zero-zero-Hopf分支给出了严格的证明,并给出了周期解的表达式,以及周期解稳定的条件。

NOTES

*通讯作者。

参考文献

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