pend.m

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

$$\begin{eqnarray*} \begin{array}{rcrcr} x' &=& y &=& F(x,y) \\ y' &=& -\omega^2\... ... \right]\, \left[\begin{array}{c}\sin x \\ y\end{array}\right] \end{eqnarray*}$$

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

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

$$\begin{eqnarray*} F_x(x,y) \;=\; 0, && F_y(x,y) \;=\; 1 \\ G_x(x,y) \;=\; -\omega^2\,\cos x, && G_y(x,y) \;=\; -\gamma \end{eqnarray*}$$

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

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

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

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

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

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