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This represents the system associated with the 2nd order, nonlinear, homogeneous ODE with constant coefficients that describes the motion of a damped pendulum.

\begin{array}{rcrcr} x' &=& y &=& F(x,y) \\ y' &=&
... \right]\,
\left[\begin{array}{c}\sin x \\ y\end{array}\right]

\begin{displaymath}\; \Longrightarrow \;\; \mbox{\tt A = [$\;$];} \quad \mbox{\tt B = [$0$, $1$;
$-\omega^2$, $-\gamma$];}\end{displaymath}

The linearization of this system is accomplished via the four partial derivatives

F_x(x,y) \;=\; 0, && F_y(x,y) \;=\; 1 \\ G_x(x,y) \;=\;
-\omega^2\,\cos x, && G_y(x,y) \;=\; -\gamma

evaluated at any critical point $(x_0, y_0) \; \Longrightarrow \;$

\left[\begin{array}{c}x' \\ y'\end{array}\right] &\approx& \le...
...{array}{c}x \\ y\end{array}\right] \quad\mbox{near $(x_0,y_0)$}

Once the linearization has been formed, we use cclin.m with

\begin{displaymath}\mbox{\tt A = [0,1; $-\omega^2\,\cos x_0$,$-\gamma$];} \quad
\mbox{\tt B = [$\;$];}\end{displaymath}

to investigate stability about each $(x_0, y_0)$.

The symbolicplot option `eigenvals' is useful only for the linearization.

Michael Renardy