To arouse your interest in the classification of linear systems, the following model for the dynamics of love affairs was introduced by Strogatz in 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 lovers 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. Then a model for their star-crossed romance is R' = aJ (0.1) J' = -bR, where positive values of R and J signify love from one person to the other (negative values signify the opposite of love). Suppose that the parameters a and b are positive, to be consistent with the story. The sad outcome of their affair is, of course, a never-ending cycle of love and hate; the governing system has a center at the origin. They seem to manage to achieve simultaneous love one-quarter of the time. A more general forecast of love is governed by the system R' = aR +b.J (0.2) J' = CR. + d.J. where the parameters a, b, c, d may have either sign. A choice of signs specifies the romantic styles. It's entertaining to name all of the romantic styles and to predict the outcomes for the various pairings. For example, can a "cautious lover" find true love with an eager beaver