Romeo and Juliet, part 2

In spite of my pledge to never ever write about the womens, I will continue to allow some of my misguided editorials to sneak in to the blog under the guise of mathematical cuteness. In this case, you may even find it useful, as I'm going to turn this Theoretical treatment into an Applied Mathematics problem for your entertainment.

So I've previously described a simple model in which the romantic behavior of Romeo and Juliet over time is based on two parameters: their sensitivity to the other one's interest level. In this case, they oscillated endlessly, occasionally even simultaneously being interested in one another.

This begs the question, do they not have any self-awareness? Are their emotions completely dependent on external influences? If they are teenagers, then the answer is probably yes. Still, most normal human beings aren't so susceptible. Knowing that you love someone will prevent you from falling out of love too quickly, or so the logic goes.

Let's expand on the previous model to allow for self-awareness. The equation then becomes

where a is the influence of Romeo's current state on his future feelings and b is Juliet's influence on him, with d and c corresponding to the respective influences for Juliet. It makes sense, trust me.

Nearly all of the solutions are boring. Both of our lovebirds tend inevitably towards apathy (which means an R or J value of 0); in the best of these cases, the solution revolves around (0,0) at lower and lower orbits until it finally reaches mutual apathy. Boooring.

In some rare cases, though, both Romeo and Juliet might fall madly in love with each other. They also might revert to homicidal rage (not unlike what I feel about the races being cancelled this weekend). Their behavior would look something like the following plot, which I've labeled in order to clarify what's what.

The slopes of the straight lines will depend on the values of the parameters (a,b,c, and d). Furthermore, the corner to which R and J will tend is dependent on the initial conditions of R and J. Imagine setting a marble on a plate, which is shaped so that its highest points are at top-left and bottom right, with its lowest points at "love" (+,+) and "hate" (-,-). Depending on where you set the marble, it will end up either at "love" or at "hate". How fast it gets there depends on the parameters.

Now let's apply this knowledge to something useful. Let's say you go to a bar (I suggest Harvest Moon). There is an attractive young lady, standing with a group of friends. Already, the initial conditions are in the R>0 quadrants. Now the question is whether the parameters (your a and b, her c and d) will allow for love, or if they will condemn you to apathy.

So here's what you do: Walk up to the girl and ask her to rate, on a scale from -10 to 10, how much a guy's desire for her affects her interest in him. Then ask her to rate how much her feelings for that guy will affect her interest in him (this may be more difficult to explan). In fact, it may be easier to go with the classic "1 to 10" scale, and then normalize. It would be best to ask for -100 to 100 to get the highest resolution, but that may be asking too much.

So know you know all of the parameters of this model. Simply solve the following equation for lamdas 1 and 2, the eigenvalues of the system:

If, and only if, either lamda 1 or lamda 2 is positive, you may proceed to the next step. Now plug that positive lamda into the following system of equations and solve for v1 and v2, the system's eigenvectors

If, and only if, the sign of v1 is equal to the sign of v2, then there is a chance for that passionate love you're hoping for. Now the only unknown is the initial condition of J. If she is very disinterested, then the system may trend towards hatred... the bifurcation between love and hate is dependent on a,b,c, and d. Frankly, finding the bifurcation and your location relative to it will take too much time; besides, it is probably rude to ask a woman "How much do you like or dislike me?" I recommend throwing caution to the wind and skipping this step of the system characterization.

So there you have it. NinjaDon's foolproof guide to finding true love. Remember to always bring a scientific calculator wherever you go; you never know when Cupid will appear.