= Solution
The <phase plane> system is $\dot x=y$, $\dot y=-cy-\sin x$. Its <equilibrium points> are \b[$(k\pi,0)$ for every integer $k$]. At such a point the <Jacobian matrix> and characteristic equation are
$$
J=\begin{pmatrix}0&1\\-(-1)^k&-c\end{pmatrix},\qquad r^2+cr+(-1)^k=0.
$$
For even $k$, the <equilibrium point> is a \b[<stable focus> for $0<c<2$], with <eigenvalues> $-c/2\pm i\sqrt{1-c^2/4}$; it is a \b[<stable node> for $c>2$], with two distinct negative real <eigenvalues>. The excluded value $c=2$ gives the repeated critical case.
For odd $k$, the <eigenvalues> are $(-c\pm\sqrt{c^2+4})/2$. One is positive and one negative, so each is a \b[<saddle equilibrium> for every $c>0$]. These classifications follow from the <linearization of a dynamical system>, since all the relevant <eigenvalues> have nonzero real part for $c>0$. The even equilibria remain hyperbolic at $c=2$ too, although their <linearization> then has a repeated eigenvalue; that critical case is excluded from the requested classification.
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