Romeo&Juliet愛情的動力系統(tǒng)模型 Lover affaires and differential equations

相似文件 換一批

是 否 +2 論壇幣

k人 參與回答

經(jīng)管之家送您一份

應(yīng)屆畢業(yè)生專屬福利!

求職就業(yè)群

趙安豆老師微信：zhaoandou666

經(jīng)管之家聯(lián)合CDA

送您一個(gè)全額獎學(xué)金名額~ !

立即領(lǐng)取

感謝您參與論壇問題回答

經(jīng)管之家送您兩個(gè)論壇幣！

+2 論壇幣

原文最早是由Steven H. Strogatz發(fā)表在1988年第61期的Mathematics Magazine上，名為"Lover affaires and differential equations"。據(jù)說做數(shù)學(xué)的人碰到一起，一般只談兩個(gè)話題：數(shù)學(xué)和女人。所以這種既有學(xué)術(shù)又有女人的題目，很是能夠吸引這批人的眼球。
      故事是這樣的。Romeo與Juliet相愛，但是Juliet在感情方面是一個(gè)變幻無常的人：Romeo越是喜歡她，對她越好，她就會覺得這個(gè)男人很賤，便愈加地討厭他，躲避他；Romeo便開始心灰意冷，漸漸地遠(yuǎn)離她，冷落她，這時(shí)Juliet卻發(fā)現(xiàn)Romeo有著特殊的吸引力，一心要去親近他；當(dāng)Romeo想去回應(yīng)Juliet的主動時(shí)，又一輪循環(huán)開始了。
     如果將這場風(fēng)花雪月看作是一個(gè)線性動力系統(tǒng)的話，可以用一個(gè)簡單的數(shù)學(xué)模型加以表示，首先定義一下變量和參數(shù)：R(t)代表Romeo對Juliet的愛/恨；J(t)代表Juliet對Romeo的愛/恨；正值代表愛，負(fù)值代表恨；a和b為參數(shù)，皆取正值。那么該模型可以寫為：
dR/dt=aJ
dJ/dt=-bR
      該系統(tǒng)的解取決于初始值。在如果初值處于原點(diǎn)以外，那么這場愛情是一個(gè)閉合的循環(huán)軌道（closed orbit），在這條軌道上，不是R追J就是J追R，總之兩人是有緣無份。如果初始點(diǎn)是在原點(diǎn)，原點(diǎn)是一個(gè)不動點(diǎn)（fixed point），那么他們處于相對穩(wěn)定的狀態(tài)，有四分之一的機(jī)會出現(xiàn)彼此相愛的情形。由此看來，如果兩人的初始狀態(tài)不在不動點(diǎn)上，需要有外來的擾動才能成就姻緣。
     其實(shí)這個(gè)模型還可以做得再復(fù)雜一點(diǎn)，比如：
dR/dt=aR+bJ
dJ/dt=cR+dJ
      如果a>0和b>0，那么Romeo是一個(gè)對感情特別賣力的人（eager beaver），Juliet對他一分好，他便對人家雙倍地好。如果a<0和b>0，那么Romeo是一個(gè)謹(jǐn)慎的愛人（cautious lover）：自己好象不是帥哥，人家為什么要對自己好？由此可以討論出很多有趣的例子，并且更具有理論挑戰(zhàn)性的是：eager beaver和cautious lover之間是否能夠產(chǎn)生真愛？


英文論文如下

本帖隱藏的內(nèi)容

Love Affairs and Differential Equations
STEVEN H. STROGATZ, Harvard University, Cambridge, MA 02138
The purpose of this note is to suggest an unusual approach to the teaching of some standard material about systems of coupled ordinary differential equations. The approach relates the mathematics to a topic that is already on the minds of many college students: the time-evolution of a love affair between two people. Students seem to enjoy the material, taking an active role in the construction, solution, and interpretation of the equations.
The purpose of this note is to suggest an unusual approach to the teaching of some standard material about systems of coupled ordinary differential equations. The approach relates the mathematics to a topic that is already on the minds of many college students: the time-evolution of a love affair between two people. Students seem to enjoy the material, taking an active role in the construction, solution, and interpretation of the equations.
The essence of the idea is contained in the following example.

Juliet is in love with Romeo, but in our version of this story, Romeo is a fickle lover. The more Juliet loves him, the more he begins to dislike her. But when she loses interest, his feelings for her warm up. She, on the other hand, tends to echo him: her love grows when he loves her, and turns to hate when he hates her.
A simple model for their ill-fated romance is

dr/dt =-aj,
dj/dt = br,

where
r(t) = Romeo's love/hate for Juliet at time t j(t) = Juliet's love/hate for Romeo at time t.

Positive values of r, j signify love, negative values signify hate. The parameters a, b are positive, to be consistent with the story.
The sad outcome of their affair is, of course, a neverending cycle of love and hate; their governing equations are those of a simple harmonic oscillator. At least they manage to achieve simultaneous love one-quarter of the time.
As one possible variation, the instructor may wish to discuss the more general second-order linear system
dr/dt =a11r+a12j
Positive values of r, j signify love, negative values signify hate. The parameters a, b are positive, to be consistent with the story.
The sad outcome of their affair is, of course, a neverending cycle of love and hate; their governing equations are those of a simple harmonic oscillator. At least they manage to achieve simultaneous love one-quarter of the time.
As one possible variation, the instructor may wish to discuss the more general second-order linear system
dr/dt =a11r+a12j
dj/dt=a21r +a22j,

where the parameters aik (i, k = 1,2) may be either positive or negative. A choice of sign specifies the romantic style. As named by one of my students, the choice a11, a12 > 0 characterizes an "eager beaver"   someone both excited by his partner's love for him and further spurred on by his own affectionate feelings for her. It is entertaining to name the other three possible styles, and also to contemplate the romantic forecast for the various pairings. For instance, can a cautious lover (a11 < 0, a12 > 0) find true love with an eager-beaver?

Additional complications may be introduced in the name of realism or mathemati-cal interest. Nonlinear terms could be included to prevent the possibilities of unbounded passion or disdain. Poets have long suggested that the equations should be nonautonomous ("In the spring, a young man's fancy lightly turns to thoughts of love"---Tennyson). Finally, the term "many-body problem" takes on new meaning in this context.

關(guān)于愛情的動力系統(tǒng) Lover affaires and differential equations
Steven H. Strogatz發(fā)表在1988年第61期的Mathematics Magazine上。

      

掃碼加我 拉你入群

請注明：姓名-公司-職位

以便審核進(jìn)群資格，未注明則拒絕

分享0 收藏0 回帖

關(guān)鍵詞：Differential Different Equations equation DIFFER 感情方面 吸引力 愛情 模型 故事

up....

好家伙！

樓主掌握的知識可真是淵博�。�

贊賞！

what is it?

還好還好

look see

看看這個(gè)東西

Love Affairs and Differential Equations

看看