More love affairs problem

Here are a bunch of Problems for Romeo and Juliet (kindly appropriated from Strogatz). Do the first one and try one more of the last four.

Consider the affair described by

 

1. Characterize the romantic styles of Romeo and Juliet
2. Sketch R(t) and J(t) starting at R(0)=1, J(0)=1.
3. Try to sketch a rough phase portrait for this problem over the range -2<R<2, -2<J<2. then use STELLA to check your guess (hint: use the ``sensi specs'' to do a series of runs across phase space).
4. Classify the fixed point at the origin.

For one of the following problems, predict the course of the love affair, depending on the signs and relative sizes of a and b If you get stuck, just pick some arbitrary values for a and b and use Stella to test your guesses. The question is how to understand how the parameters might change the overall behavior of the system.

1. (Out of touch with their own feelings) Suppose Romeo and Juliet react to each other, but not to themselves: , .
2. (Fire and water) Do opposites attract? Analyze ,
3. (Peas in a pod) If Romeo and Juliet are romantic clones ( , ), should they expect boredom or bliss?
4. (Romeo the robot) Nothing could ever change the way Romeo feels about Juliet: , . Does Juliet end up loving him or hating him?