On infinitesimal stability and instability of pendulum type oscillations

For the pendulum type of oscillations governed by the equation ? + φ(x) = 0, with φ(x) an odd function, it is shown that according to the linearized disturbance equation, stability is predicted if and only if dTd_x = 0. where T is the period and α is the amplitude of the non-linear steady-state oscillations. From this it follows that for a given non-linear function φ(x). infinitesimal stability can at most be predicted only for certain discrete values of α. It is shown analytically that for a simple pendulum, a power-law spring and a cubic hard or soft spring, the oscillations are infinitesimally unstable for all α. It is further shown, however, that particular cases of non-linear restoring forces do exist for which infinitesimal stability is predicted for certain α's, in contrast to the actual Liapunov instability in these cases.