关于爱情的动力系统 Lover affaires and differential equations

Steven H. Strogatz发表在1988年第61期的Mathematics Magazine上。

     原文最早是由Steven H. Strogatz发表在1988年第61期的Mathematics Magazine上，名为"Lover affaires and differential equations"。据说做数学的人碰到一起，一般只谈两个话题：数学和女人。所以这种既有学术又有女人的题目，很是能够吸引这批人的眼球。
     故事是这样的。Romeo与Juliet相爱，但是Juliet在感情方面是一个变幻无常的人：Romeo越是喜欢她，对她越好，她就会觉得这个男人很贱，便愈加地讨厌他，躲避他；Romeo便开始心灰意冷，渐渐地远离她，冷落她，这时Juliet却发现Romeo有着特殊的吸引力，一心要去亲近他；当Romeo想去回应Juliet的主动时，又一轮循环开始了。
    如果将这场风花雪月看作是一个线性动力系统的话，可以用一个简单的数学模型加以表示，首先定义一下变量和参数：R(t)代表Romeo对Juliet的爱/恨；J(t)代表Juliet对Romeo的爱/恨；正值代表爱，负值代表恨；a和b为参数，皆取正值。那么该模型可以写为：
dR/dt=aJ
dJ/dt=-bR
     该系统的解取决于初始值。在如果初值处于原点以外，那么这场爱情是一个闭合的循环轨道（closed orbit），在这条轨道上，不是R追J就是J追R，总之两人是有缘无份。如果初始点是在原点，原点是一个不动点（fixed point），那么他们处于相对稳定的状态，有四分之一的机会出现彼此相爱的情形。由此看来，如果两人的初始状态不在不动点上，需要有外来的扰动才能成就姻缘。
    其实这个模型还可以做得再复杂一点，比如：
dR/dt=aR+bJ
dJ/dt=cR+dJ
     如果a>0和b>0，那么Romeo是一个对感情特别卖力的人（eager beaver），Juliet对他一分好，他便对人家双倍地好。如果a<0和b>0，那么Romeo是一个谨慎的爱人（cautious lover）：自己好象不是帅哥，人家为什么要对自己好？由此可以讨论出很多有趣的例子，并且更具有理论挑战性的是：eager beaver和cautious lover之间是否能够产生真爱？

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.