 Pendulum with quadratic damping

In nondimensional time the quadratic damping removes energy from the pendulum at rate $-c|\omega {|}^3$. Downward resting positions are nonhyperbolic asymptotically stable spirals, even though the linearized equation has a centre. Upward resting positions remain saddle equilibria. Each fixed-sign branch of $\omega$ has a linear equation for ${\omega }^2$ as a function of angle.