Computer Algebra Recipes

18 CHAPTER 1. PHASE-PLANE PORTRAITS
1.1.1 Romeo and Juliet
I am convinced we do not only love ourselves in others
but hate ourselves in others too.
G. C. Lichtenberg, German physicist, philosopher (1742{1799)
The mathematician Steven Strogatz [Str88] [Str94] has suggested a simple dynamic
model to create di®erent scenarios for the love a®air between Romeo and
Juliet. In his model, R(t) andJ(t) represents Romeo's love/hate for Juliet and
Juliet's love/hate for Romeo, respectively, at time t. Positive values of R and
J indicate love, while negative values indicate hate. The love a®air equations
take the form
_R(t) =aR+ bJ; J(t) _ =cR+ dJ;
where a, b, c, anddarereal coe±cients that may have either sign. For the sake
of de¯niteness, let's take a =2,b =1,c = ¡1, and d = ¡2 in the following
problem, leaving other coe±cient values for you to explore.
(a) Is the ODE system linear or nonlinear? Locate the ¯xed point(s).
(b) Create a tangent ¯eld plot and identify the nature of the ¯xed point(s).
(c) Create a phase-plane portrait that contains the four trajectories corresponding
to the following initial conditions: (i) R(0) = ¡0:25, J(0) = 1,
(ii) R(0) = ¡0:27, J(0) = 1, (iii) R(0) = 0:27, J(0) = ¡1, and ¯nally
(iv) R(0) = 0:25, J(0) = ¡1. Consider t =0to4timeunits.
(d) Plot R versus t over the interval t = 0 to 2 for initial condition (i).
(e) Derive analytic solutions for R(t) andJ(t) for initial condition (i).
To plot the tangent ¯eld and create the phase-plane portrait, the dfieldplot
and phaseportrait commands, respectively, will be used. These specialized
di®erential equation plotting tools are contained in the DEtools library package,
which is now loaded. The colon may be replaced with a semicolon to
display the complete list of available commands in this package.
> restart: with(DEtools):
The general love a®air di®erential equations are entered in de1 and de2, each
¯rst-order derivative with respect to t being entered with the diff command.
> de1:=diff(R(t),t)=a*R(t)+b*J(t);
de1 := d
R(t) =a R(t)+b J (t)
dt
> de2:=diff(J(t),t)=c*R(t)+d*J(t);
de2 := d
J (t) =c R(t)+d J (t)
dt
Before proceeding further with solving our problem, a few words should
be said about Maple's derivative command. Using exp, ln, andsin to enter
the exponential, natural logarithm, and sine functions, suppose that Romeo's
love/hate for Juliet depended on time in the following way.