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\lhead{\sf MATH 677/CSI 749}
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\rhead{\sf FALL 1997}
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\begin{center}   \Large \bf
Homework 12 \\
\Large \bf Hopf bifurcation in the FitzHugh-Nagumo equation
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\bigskip

The FitzHugh-Nagumo equation
\begin{eqnarray*}
\dot{x}&=&10(c-x(x-1/2)(x-1)-y)\\
\dot{y}&=&x-y-0.2
\end{eqnarray*}
is often used in biophysical modeling, where quiescent motion
coexists or alternates with rhythmical motion, as in heartbeats,
spiking neurons, biorhythms, etc. In this assignment you are asked to
study the Hopf bifurcation which forms the boundary between the two
types of dynamics.

\begin{enumerate}

\item There is a stable equilibrium in the unit square when the
parameter $c$ is set to $0.12$. Use matlab's {\bf ode45} routine to 
find the equilibrium to 4 correct digits. Find the Jacobian matrix at
the equilibrium, and use its eigenvalues to classify the type of
equilibrium.

\item There is a stable limit cycle when $c=0.13$. Plot the
limit cycle. Graph only the cycle, not the transient curves
leading to it. As in previous assignments, it is a good idea to
traverse the cycle several times before plotting.

\item This family of equations moves through a Hopf bifurcation as $c$
goes from $0.12$ to $0.13$. Find the Hopf bifurcation value $c_H$ {\it
analytically}, and report it to 6 correct digits. [Hint: First, find
the $x$ coordinate of the equilibrium $(x_H, y_H)$ at the Hopf
bifurcation point, using the fact that the Jacobian matrix has purely
imaginary eigenvalues there. Then solve for $y_H$ and $c_H$. List
$(x_H, y_H)$ also in your report.]

\item It is known that the amplitude of the limit cycle created at a
Hopf bifurcation grows approximately as $A(h) = Kh^p$, where $h=c-c_H$ is the
distance the parameter is beyond the bifurcation point $c_H$ (at least
when $h$ is small -- keep $c$ below $0.13$. ). Verify
the amplitude result for this example, and find $p$. (It's a number you have
heard of.) Plot  $A$ versus $h$ for as many different $h$ as you
can. Estimate $K$ and $p$ by drawing a log-log version of the same
plot. As above, when calculating the amplitude, be sure the system has
completed its transient and is on the new cycle before measuring the
amplitude. Include an explanation of your strategy with your results.

\end{enumerate}


Due: Mon., December 1
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