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\centerline{Workshop \#2}
\centerline{APM 115: mathematical modeling}

\bigskip
Equilibrium Stability for Two Dimensional Dynamic Systems 

\begin{enumerate}
\item In this workshop, we are going to analyze the stability of a
  system dynamical model of love affairs. After we fully characterize
  and solve the linear model, you are going to develop a new
  non-linear model of love affairs by improving on the existing model.

  Modeling Romeo's love for Juliet

  $R=$ Romeo's love for Juliet  $J=$ Juliet's love for Romeo (NOTE:
  positive means love, negative means hate)
  \begin{eqnarray}
    dR(t)/dt &=& a R + b J \\
    dJ(t)/dt &=& c R + d J 
  \end{eqnarray}
  a, b, c, and d are proportionality constants that can be either
  positive or negative (or zero).
  \begin{enumerate}
  \item Find the fixed points and analyze the stability of those fixed points.
  \item Write a matlab code that uses ode45 to solve the system with
    for a variety of values of a, b, c, \& d. The relative signs of a,
    b, c, \& d dictate what kind of lovers you are modeling.
  \begin{enumerate}
  \item As an initial example, consider $b>0$, $c<0$, and $a=d=0$.
    What kind of lovers are we modeling?
  \item Plot the phase plane of Romio's love and Juliet's love
  \item Change the parameter values to model a pair of ``Cautious
    Lovers.''
  \end{enumerate}
\item Now, improve the model of some particular pair of lovers by
  making it non­linear.
  \begin{enumerate}
  \item Find the fixed points and stability of the new non-linear model
  \item Solve the new model with ode45.
  \item FOR TF: provide the following verbal description and let the
    students find the equation: Romeo is excited by Juliet's love (or
    hate) and reciprocates.  Juliet is more complicated.  On the one
    hand she is turned off by Romeo's love.  However, she is excited
    by her own love to Romeo, as long as his feelings (love or hate)
    for her are not too strong.  If his feelings are getting stronger
    (again, whether love or hate) she cannot handle the emotional
    stress and feels scared of her own feelings.  ah...
    \begin{eqnarray}
      dR/dt &=& J \\
      dJ/dt &=& -a (R^2-1) J - R 
    \end{eqnarray}
  \end{enumerate}
\item Write a script to use the matlab function quiver to plot the
  stability of the fixed points of the new model.  Integrate the
  equations using ode45.
\end{enumerate}
\end{enumerate}
\end{document}
