Linear stability analysis on a simple pendulum

Question

So I have a simple pendulum (rod has no weight, point mass, no frictional forces) and I’m measuring the angle theta from the downward vertical, hence I have the governing equation $$\ddot{\theta}+sin(\theta)=0$$ I want to perform linear stability analysis to show that the point (0,0) (first component theta, second time derivative of theta) is stable. After using the small angle approximation for theta close to zero and then performing linear stability analysis my eigenvalues are +/- i giving me a center. Would it be right to say this shows that the point (0,0) is stable?

Answer 1

No, you can't draw any conclusions from a linear stability analysis if it's a center, since the higher-order terms that you have neglected could in fact very easily destroy the closed orbits of the center and make the actual orbits spiral inwards or outwards.

In this case, however, it's a conservative system, and the energy constant of motion $$E=\frac12 \dot\theta^2 - \cos\theta$$ shows that the origin is an fact surrounded by closed orbits (the level curves of $E$). So it's a stable equilibrium (but not asymptotically stable).