6.2.6: Love Affairs

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The next application is one that was introduced in 1988 by Strogatz as a cute system involving relationships. \({ }^{1}\) One considers what happens to the affections that two people have for each other over time. Let \(R\) denote the affection that Romeo has for Juliet and \(J\) be the affection that Juliet has for Romeo. Positive values indicate love and negative values indicate dislike.¹

One possible model is given by

\[ \begin{aligned} &\dfrac{d R}{d t}=b J \\ &\dfrac{d J}{d t}=c R \end{aligned} \label{6.56} \]

with \(b>0\) and \(c<0\). In this case Romeo loves Juliet the more she likes him. But Juliet backs away when she finds his love for her increasing.

A typical system relating the combined changes in affection can be modeled as

\[ \begin{aligned} &\dfrac{d R}{d t}=a R+b J \\ &\dfrac{d J}{d t}=c R+d J \end{aligned} \label{6.57} \]

Several scenarios are possible for various choices of the constants. For example, if \(a>0\) and \(b>0\), Romeo gets more and more excited by Juliet’s love for him. If \(c>0\) and \(d<0\), Juliet is being cautious about her relationship with Romeo. For specific values of the parameters and initial conditions, one can explore this match of an overly zealous lover with a cautious lover.

¹Steven H. Strogatz introduced this problem as an interesting example of systems of differential equations in Mathematics Magazine, Vol. 61, No. 1 (Feb. 1988) p 35. He also describes it in his book Nonlinear Dynamics and Chaos (1994).