This problem is shamelessly stolen from Strogatz's excellent book on non-linear dynamics , and originally appeared in his article in Math Magazine . This is an example of a linear system because the rate equations depend only on linear combinations of the variables. Nevertheless it forms an important basis for understanding non-linear systems. I quote from Strogatz directly
To arouse your interest in the classification of linear systems, we now discuss a simple model for the dynamics of love affairs (Strogatz 1988). The following story illustrates the idea.
Romeo is in love with Juliet, but in our version of this story, Juliet is a fickle lover, The more Romeo loves her, the more Juliet wants to run away and hide. But when Romeo gets discouraged and backs off, Juliet begins to find him strangely attractive. Romeo, on the other hand tends to echo her: he warms up when she loves him and grows cold when she hates him.
Positive values of R,J signify love, negative values signify hate. Then a model for their star-crossed romance is
Let's analyze this problem now and then use Stella to test our intuition. I will do this in class because it's easier to show than to talk about.