兩個(gè)耦合Van der Pol振子系統(tǒng)的一階近似守恒量*

樓智美

(紹興文理學(xué)院物理系, 紹興　312000)

?

兩個(gè)耦合Van der Pol振子系統(tǒng)的一階近似守恒量*

樓智美?

(紹興文理學(xué)院物理系, 紹興312000)

用直接積分法計(jì)算兩個(gè)耦合Van der Pol振子系統(tǒng)的一階近似守恒量,將兩個(gè)耦合Van der Pol振子系統(tǒng)看成是未受微擾系統(tǒng)與微擾項(xiàng)的迭加,先通過坐標(biāo)變換將未受微擾系統(tǒng)解耦,并對(duì)解耦系統(tǒng)的3種可能狀態(tài)進(jìn)行討論,得到未受微擾系統(tǒng)的13個(gè)精確守恒量,再考慮微擾項(xiàng)對(duì)精確守恒量的影響,運(yùn)用一階近似守恒量的性質(zhì),得到1個(gè)穩(wěn)定的一階近似守恒量.另外,由13個(gè)精確守恒量直接得到13個(gè)平凡的一階近似守恒量.

Van der Pol振子系統(tǒng),精確守恒量,一階近似守恒量

引言

許多實(shí)際力學(xué)系統(tǒng)的運(yùn)動(dòng)微分方程中常常含非線性微擾項(xiàng),由于微擾項(xiàng)的存在,使力學(xué)系統(tǒng)的對(duì)稱性遭到破損,一些精確守恒量的形式發(fā)生變化或消失,精確解不再成立,穩(wěn)定性受到影響.因此,研究實(shí)際力學(xué)系統(tǒng)的近似守恒量對(duì)于研究其力學(xué)特性至關(guān)重要[1-13].目前關(guān)于微分方程近似守恒量的研究主要采用近似Lie對(duì)稱性理論[1]、近似Noether對(duì)稱性理論[2]和直接積分法[3].引進(jìn)近似的群無限小變換,微分方程在此變換下近似保持不變則為近似Lie對(duì)稱性；哈密頓作用量在此變換下近似保持不變則為近似Noether對(duì)稱性,所得的守恒量為近似守恒量；直接積分法是從近似守恒量的性質(zhì)出發(fā),把受微擾系統(tǒng)視為未受微擾系統(tǒng)與微擾項(xiàng)的迭加,先選擇合適的方法求得未受微擾系統(tǒng)的精確守恒量,再考慮微擾項(xiàng)對(duì)精確守恒量的影響,最后利用近似守恒量的性質(zhì)求得守恒量.用近似對(duì)稱性理論求近似守恒量要用到Lagrange函數(shù)和近似的群無限小變換,并需解出近似的無限小生成元、規(guī)范函數(shù),計(jì)算較繁復(fù),理論性強(qiáng)又比較抽象.用直接積分法求近似守恒量,思想方法簡(jiǎn)單,物理意義明確,計(jì)算方法靈活.

兩個(gè)耦合Van der Pol振子系統(tǒng)是一實(shí)際的力學(xué)系統(tǒng)[4],已廣泛應(yīng)用于非線性動(dòng)力學(xué)和數(shù)學(xué)物理的研究中.本文采用直接積分法計(jì)算兩個(gè)耦合Van der Pol振子系統(tǒng)的一階近似守恒量,先通過坐標(biāo)變換對(duì)未受微擾系統(tǒng)進(jìn)行解耦,并對(duì)解耦系統(tǒng)的3種可能狀態(tài)進(jìn)行討論,直接得到未受微擾系統(tǒng)在新坐標(biāo)系下的13個(gè)精確守恒量,利用坐標(biāo)反變換得到未受微擾系統(tǒng)在原坐標(biāo)系下的13個(gè)精確守恒量,再考慮微擾項(xiàng)對(duì)精確守恒量的影響,運(yùn)用一階近似守恒量的性質(zhì),得到1個(gè)穩(wěn)定的一階近似守恒量.另外,根據(jù)一階近似守恒量的性質(zhì),由13個(gè)精確守恒量直接得到13個(gè)平凡的一階近似守恒量.

1　未受微擾作用系統(tǒng)的精確守恒量

兩個(gè)耦合Van der Pol振子系統(tǒng)的運(yùn)動(dòng)微分方程可以表示成[4]

(1a)

(1b)

其中0<ε?1,為微擾系數(shù).A,B為線性耦合系數(shù),且為不為0的實(shí)常數(shù).

系統(tǒng)(1)可以改寫成

=g1(ε0)+εg1(ε1)

(2a)

=g2(ε0)+εg2(ε1)

(2b)

與系統(tǒng)(2)相應(yīng)的未受微擾作用系統(tǒng)的運(yùn)動(dòng)微分方程可表示成

(3a)

(3b)

系統(tǒng)(2)可以看成是系統(tǒng)(3)與微擾項(xiàng)εg1(ε1)、εg2(ε1)的迭加, 系統(tǒng)(3)是兩個(gè)線性耦合的振子系統(tǒng).引進(jìn)坐標(biāo)變換

u1=x1+x2,u2=x1-x2

(4)

對(duì)系統(tǒng)(3)解耦后得

(5a)

(5b)

在新坐標(biāo)系下,系統(tǒng)(5)不再相互耦合,可以方便地求得系統(tǒng)(5)在新坐標(biāo)系下的精確守恒量, 再利用坐標(biāo)反變換可求得系統(tǒng)(5)在原坐標(biāo)系下的精確守恒量.

(5a)式表示諧振子的運(yùn)動(dòng)微分方程,在A、B任意取值下,(5a)式存在兩個(gè)基本的精確守恒量[14]

(6a)

(6b)

(7)

(5b)式表示線性阻尼振子的運(yùn)動(dòng)微分方程,其精確守恒量與A、B的取值有關(guān).(5b)式的解存在如下三種可能狀態(tài),在不同的狀態(tài)下,其中一些精確守恒量的表達(dá)式也不同.

1)B2>1+2A,為過阻尼狀態(tài),(5b)式存在兩個(gè)基本的精確守恒量

(8a)

(8b)

(9a)

(9b)

2)B2<1+2A, 為欠阻尼狀態(tài), (5b)式存在兩個(gè)基本的精確守恒量[15]

=[ω(x1-x2)sin(ωt)+

(10a)

=[ω(x1-x2)cos(ωt)-

(10b)

(11a)

(11b)

3)B2=1+2A, 為臨界阻尼狀態(tài), (5b)式存在一個(gè)基本的精確守恒量

(12)

在上述三種可能狀態(tài)下,可以統(tǒng)一構(gòu)建一個(gè)具有能量量綱的精確守恒量

(1+2A)(x1-x2)2]e2Bt

(13)

2　耦合Van der Pol 振子系統(tǒng)的一階近似守恒量

設(shè)耦合Van der Pol振子系統(tǒng)(2)的一階近似守恒量可表示成[5]

(14)

一階近似守恒量的性質(zhì)為[5]

(15)

將(14)式代入(15)式并展開,令ε0,ε1的系數(shù)分別等于0,忽略ε2以上項(xiàng),可得

(16a)

(16b)

(17a)

(17b)

(17c)

(17d)

(17e)

(17f)

(17g)

(17h)

(17i)

(17j)

(17k)

(17l)

(17m)

(18)

則系統(tǒng)存在1個(gè)穩(wěn)定的一階近似守恒量

(19)

(20)

3　結(jié)論

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*The project supported by the National Natural Science Foundation of China(11472177)

? Corresponding author E-mail: louzhimei@usx.edu.cn

16 April 2015,revised 02 July 2015.

FIRST ORDER APPROXIMATE CONSERVED QUANTITIES OF TWO COUPLED VAN DER POL OSCILLATORS SYSTEM*

Lou Zhimei?

(DepartmentofPhysics,ShaoxingUniversity,Shaoxing312000,China)

The first order approximate conserved quantities of two coupled Van der Pol oscillators system is investigated by using direct integral method. Two coupled Van der Pol oscillators system is taken as the combination of unperturbed system and perturbed terms. Firstly, the unperturbed system is decoupled by transforming coordinates, and thirteen exact conserved quantities of unperturbed system are obtain by discussing the three possible condition of uncoupled system. Second, the influence of perturbed terms on exact conserved quantities is examined. Finally, a stable first order approximate conserved quantity of the system is developed according to the characteristic of the first order approximate conserved quantities. In additional, thirteen trivial conserved quantities from thirteen exact conserved quantities are also received.

Van der Pol oscillators systems,exact conserved quantities,first order approximate conserved quantities

E-mail: louzhimei@usx.edu.cn

10.6052/1672-6553-2015-58

2015-04-16收到第1稿,2015-07-02收到修改稿.

* 國家自然科學(xué)基金資助項(xiàng)目(11472177)