Romeo and Juliet's relationship is modeled mathematically. Their love and hate for each other oscillates over time based on a set of differential equations. Mercutio tries to interfere by negatively influencing Romeo's feelings for Juliet. This changes the model and results in a different outcome for Romeo and Juliet's relationship. The model is further complicated by Mercutio developing feelings for Juliet and Juliet having mixed feelings for both Romeo and Mercutio, creating a love triangle. The dynamics of this new system are analyzed using eigenvalues and phase planes. Additional models examine planetary orbits and competition between rabbits and sheep.

1. HW 5: Math 405. Due: beginning of class – Monday, 10 Oct
2016
1. Strogatz (1988). Consider lovers, Romeo and Juliet. Let:
R(t) = Romeo’s love/hate for Juliet at time t;
J(t) = Juliet’s love/hate for Romeo at time T;
where positive values of R(t), J(t), signify love, and negative
values signify hate, at
a time t. Consider this model for a “fickle” lover, in which “the
more Romeo loves
Juliet, the more she 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 Juliet: he warms up when she loves him and
grows cold when
she hates him.”
R′(t) = aJ,
J′(t) = −bR, (1)
where a,b > 0 .
(a) Rewrite (1) as a system.
(b) Find the fixed point(s).
(c) Find the eigenvalues.

2. (d) Find the eigenvectors.
(e) Write the general solution. Show that it can be written as a
real-valued solution
like: [
R(t)
J(t)
]
=
{
cos(
√
ab t)
[
k1
√
a/b
k2
]
+ sin(
√
ab t)
[
k2
√
a/b

3. −k1
]}
(f) Show that the trajectories in phase space are ellipses,
governed by the equation
R2
aC2
+
J2
bC2
= 1,
where C2 > 0 is an arbitrary constant.
(g) Classify the fixed point and its stability.
(h) In what direction do Romeo’s and Juliet’s feelings go
around the ellipse?
(i) Discuss the possible outcome for different initial conditions
of love/hate.
2. (Doug Wright, Drexel U.) Romeo’s best friend, Mercutio,
doesn’t like Juliet’s fick-
leness and thinks that Romeo is too good for her. He has
decided to try to break
them up for good. So, he has started telling Romeo how awful
Juliet is. Romeo
trusts Mercutio, and so, his ardor for Juliet wanes a bit when
Mercutio tells him
such things, though he still really likes Juliet. On the other
hand, Juliet dislikes

4. 1
Mercutio and the more he disapproves of her relationship with
Romeo, the more
she likes Romeo. Let R and J be as before, and let M(t) be
Mercutio’s disapproval
of Romeo and Juliet’s relationship at time t, with positive
values of M signifying
disapproval. Then a model for this complicated saga is:
R′(t) = J − 2M,
J′(t) = −R + 4M,
M ′(t) = R + 4J (2)
(a) Rewrite (2) as a system.
(b) Find the fixed point(s).
(c) Find the eigenvalues.
(d) Without further calculation, describe what happens to
Romeo and Juliet’s
relationship now. Does Mercutio’s tampering have the effect he
wants? Do
Romeo and Juliet continue to oscillate between love and hate as
before?
3. (Doug Wright, Drexel U.) So, now, it turns out that Mercutio
has feelings for
Juliet. Let R(t) be Romeo’s love/hate for Juliet at time t, as
before; JR(t) be
Juliet’s love/hate for Romeo at time t; M(t) be Mercutio’s
love/hate for Juliet at

5. time t; and JM (t) be Juliet’s love/hate for Mercutio at time t.
The situation is that:
• Romeo still likes/dislikes Juliet more the more she
likes/dislikes him;
• Romeo doesn’t know about the Mercutio/Juliet leg of the
triangle, so his
feelings are unaffected by Mercutio’s feelings for Juliet and
Juliet’s feelings for
Mercutio.
• Juliet sill gets more interested in Romeo when he is not
interested in her and
vice versa. But, she also likes the intrigue of being in a love
triangle and so
feels more for Romeo when Mercutio likes her and more for
Mercutio when
Romeo likes her. (Weird.)
• Mercutio thinks Juliet’s distaste for him is “a major turn-on”,
so, the more
she hates him, the more he likes her. But, he is feeling guilty,
so he tries to
slow his feelings for Juliet when they are strong. Also, he’s the
jealous type,
so the more Juliet likes Romeo, the more he likes Juliet.
• Juliet pretty much just doesn’t like Mercutio, but she responds
to him the
same way she does to Romeo.
A mathematical model for this saga is:
R′(t) = JR,

6. J′R(t) = −R + JM,
M ′(t) = −R + JR −M −JM
J′M (t) = JR −M (3)
2
(a) Rewrite (3) as a system and state the one fixed point.
(b) Show the work to find the eigenvalues, which are {−2, i,−i,
1}.
(c) Use computer software to find the corresponding
eigenvectors and write the
solution in general form without simplifying. (So, you can leave
things with
imaginary eigenvalues.) Identify the four fundamental solutions.
(d) Discuss the role of the four fundamental solutions (two of
them can be dis-
cussed together). How do you predict the story will end as time
gets large.
4. Determine type and stability of the critical point. Find the
general solution. Sketch
the eigenvectors and some trajectories in the phase plane.
dy1
dt
= 4y1 + y2
dy2
dt

7. = 4y1 + 4y2
Solve the initial value problem given that y(0) =
[
1
−1
]
.
5. General relativity and planetary orbits: The relativistic
equation for the orbit
of a planet around the sun is
d2u
dθ2
+ u = α + �u2
where u = 1/r and r,θ are the polar coordinates of the planet in
its plane of motion.
The parameter α is positive and can be found explicitly from
classical Newtonian
mechanics; the term �u2 is Einstein’s correction. Here � is a
very small positive
parameter.
(a) Rewrite the equation as a system in (u,v) phase plane, where
v = du
dθ
.
(b) Find all the equilibrium points of the system.

8. (c) Show that one of the equilibria is a center in the (u,v) phase
plane, according
to the linearization. Do you think that allowing for nonlinearity
would cause
the center to be a spiral or a nonlinear center (all trajectories
sufficently close
to it remain closed). What does a spiral correspond to?
(d) Show that the equilibrium point found in (c) corresponds to
a elliptical plan-
etary orbit that are close to circular.
6. Rabbit versus Sheep: Consider the following “rabbits vs.
sheep” problem, with
y1, y2 ≥ 0 representing the number of rabbits and sheep at time
t. This model
allows for competition within species as well as competition
between the species:
y1
′ = r1y1(1 −
y1
K1
) − b1y1y2;
3
y2
′ = r2y2(1 −
y2

9. K2
) − b2y1y2.
Here the rj (j=1,2) are rates of reproduction of the rabbits and
sheep; Kj (j=1,2)
represent competition within species, and bj (j=1,2) represent
competition between
species.
(a) Nondimensionalize the model: Let
Y1 = y1/K1 ≥ 0
Y2 = y2/K2 ≥ 0
T = r1t ≥ 0
R = r2/r1 > 0
α1 = b1K2/r1 > 0
α2 = b2K1/r1 > 0,
where Y1 is a nondimensional measure of the number of rabbits,
Y2 is a nondi-
mensional measure of the number of sheep, and T is a
nondimensional measure
of time. Show that the resulting nondimensional model is:
Y1
′ = Y1(1 −Y1) −α1Y1Y2;
Y2
′ = RY2(1 −Y2) −α2Y1Y2.
(b) Find the fixed points.
(c) Show that there are four qualitatively different phase

10. portraits possible for
when t →∞. That is, show that, depending on the relative sizes
of α1,α2,R,
the end state can be any of: (i) either there will be all rabbits or
there will
be all sheep; (ii) only sheep will survive; (iii) only rabbits will
survive; (iv)
rabbits and sheep coexist stably together.
(d) Discuss the biological meaning of these states.
4
Change Initiative Proposal Grading Guide
MGT/330 Version 9
3
Change Initiative Proposal Grading Guide
MGT/330 Version 9
Management Theory, Practice, and Application
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11. endorsement, sponsorship, or affiliation.
Edited in accordance with University of Phoenix® editorial
standards and practices.
Learning Team: Change Initiative ProposalPurpose of
Assignment
This assignment provides students with an opportunity to show
their understanding of course material related to driving change
and innovation from earlier in the course (Week 2) and
application of important management tools for driving that
change from the current week’s readings on power, politics and
motivation, and effective conflict resolution.
Resources Required
Chapter 6, 10 and 11, Week 4 selected readings, University
LibraryGrading Guide
Content
Met
Partially Met
Not Met
Comments:
Explain the importance of change and innovation to the
workforce.
Explain the type of approaches that you think are best to
optimize the use of power and politics in your organization to
support the change initiative.

12. Analyze what sorts of approaches to conflict management will
be used as the change initiative will likely cause confusion in
roles and responsibilities during the implementation process.
Explain what sorts of motivation your team recommends for
management to help the workforce become adaptive and open to
not only this initiative but also future innovations in the
organization.
The memo is 1,575 words in length.
Total Available
Total Earned
10.5
#/10.5
Writing Guidelines
Met
Partially Met

13. Not Met
Comments:
The paper—including tables and graphs, headings, title page,
and reference page—is consistent with APA formatting
guidelines and meets course-level requirements.
Intellectual property is recognized with in-text citations and a
reference page.
Paragraph and sentence transitions are present, logical, and
maintain the flow throughout the paper.
Sentences are complete, clear, and concise.
Rules of grammar and usage are followed including spelling and
punctuation.
Total Available
Total Earned

14. 4.5
#/4.5
Assignment Total
#
15
#/15
Additional comments: