How do I love thee? Let me model the ways!
- My apologies to Elizabeth Barrett Browning for nerdifying her lovely poem! |
(Adapted from various sources. But if you're seriously interested, check out this article!)
The scene
Juliet loves Romeo, and Romeo loves Juliet, at least at time t=0. The question is, how will their feelings evolve with time, and as they meet other, potentially fascinating, new people! Here is our modeling strategy:
To help you get comfortable with these love functions and their
phase plots, let us begin with a warmup exercise.
Each graph below shows the time history of various hypothetical love affairs between Juliet and Romeo. Interpret what is happening in each graph.
Fig. 1 | Fig. 2 |
Fig. 3 | Fig. 4 |
As usual, we want to think in terms of the derivatives $\displaystyle \frac{dJ}{dt}$ and $\displaystyle \frac{dR}{dt}$, in order to model $J(t)$ and $R(t)$. Here are some question to explore:
Here the negative signs on the 1st term suggest both lovers respond negatively when Juliet's love for Romeo increases! The positive signs on the 2nd term suggests both lovers fall deeper in love as Romeo's love for Juliet increases. True, it's weird, but there is plenty of weirdness in the real world!
Construct a few other simple models of this type. That is:
$J^\prime(t) =$ term(s) that^{ }
depend on $J$ and $R$
Explain what sign you want for the term(s) and why?
$R^\prime(t) =$ term(s)^{ }that
depend on $J$ and $R$
Explain what sign you want for the term(s) and why?
where all the coefficients denote constant parameter values that model different personality types and/or behavior traits. For example, here is one classification scheme based on Romeo's personality that has been proposed in the literature:
$J^\prime(t) = -0.2(J-3) - (R-4)$
$R^\prime(t) = (J-3) - 0.2(R-4)$
Pick any choice of initial values for $J$ and $R$ -- in fact,
try various combinations such as hate-hate, love-hate, etc.
Use the Sage script below to run these simulations and discuss
your results: what are the values of $J(t)$ and $R(t)$
in the long-term? arguing from a calculus perspective,
can you conjecture why? what are the personality types of
Romeo and Juliet in this model (e.g., Eager beaver,
Narcissistic nerd, etc.)? Can you think of a simple
change to the model that would make them both hate each
other in the long-term?
$J^\prime(t) = -0.2(J-3) - (R-4)$
$R^\prime(t) = (J-3) - 0.2(R-4)$
Next, let's look at the Jacobian matrix and is eigenvalues. Since the system is linear, the terms in the Jacobian (say, $\bf A$) are just the coefficients: ${\bf A} = \left[\begin{array}{cc} -0.2 & -1 \\ 1 & -0.2 \end{array}\right]$
Its eigenvalues are found using the Sage segment below.
An easy way to change the problem and make the solution
diverge would be to change the matrix so
its eigenvalues are positive. For example,
${\bf A} = \left[\begin{array}{cc} 0.2 & -1 \\
1 & 0.2 \end{array}\right]$ has complex conjugate
eigenvalues with positive real part. Thus, the following system
will have only diverging spiral solutions
$J^\prime(t) = 0.2(J-3) - (R-4)$
$R^\prime(t) = (J-3) + 0.2(R-4)$
Can you conjecture what type of real-life behavior the nonlinear terms might capture (assume b and c are positive)? Depending on the choice of parameters, different solution behaviors can be seen. For example, try to run some simulations with a=b=-2 and c=d=1. To really understand what is going on, you will need to try a few different initial values for J(0) and R(0).
J_{R}'(t) =
a J_{R} + b (R-S)
J_{S}'(t) =
a J_{S} + b (S-R)
R'(t) = c J_{R} + d R
S'(t) = e J_{S} + f S
where J_{R}(t) and J_{S}(t) are functions that represent Juliet's love for Romeo and Sky, respectively. As before, R(t) and S(t) represent the respective partners' love for Juliet. If you assume positive parameter values for a-f, can you conjecture meaningful interpretations for all the terms in this model?
Despite the seeming complexity of this last model, it is still linear, and has a fairly limited range of solution behaviors. To see chaotic solution behaviors it is necessary to include nonlinear terms in the model.