(10 points)
Nonlinear Pendulum: Small Friction Case
The equation of a pendulum having an attached mass $m > 0$, a massless rod of length $l > 0$, and swinging in a medium with damping constant
$d \ge 0$ is
$ml\theta'' = -mg\sin(\theta) - d\theta'$,
where $\theta(t)$ is the angular position of the pendulum as function of time, measured from the vertical downwards position, positive counter-
clockwise, and $g$ is the acceleration of gravity. We consider the following particular case:
$\frac{g}{l} = 1$ and $m = 1$.
\begin{itemize}
\item Small friction, $0 < d < 2$.
\end{itemize}
Part 1: First Order Reduction
Part 2: Critical Points
Part 3: The Derivative Matrix
Part 4: The Derivative Matrix at Even Critical Points
Part 5: The Derivative Matrix at Odd Critical Points
(c1) Find the Jacobian matrix at odd-valued critical points $O_k = ((2k + 1)\pi, 0)$.
$\begin{bmatrix}
0 & 1\\1 & -d
\end{bmatrix}$
(d1) Find the eigenvalues of matrix $DF(O_k)$ in the case $0 < d < 2$ (small friction).
$\lambda_1, \lambda_2 = (\frac{1}{2})(-d - i\sqrt{4 + d^2}), (\frac{1}{2})(-d + i\sqrt{4 + d^2})$
(e1) Each odd critical point $O_k$ is a Saddle Node
