As our next example we take a fourth order well, with . We get which leads to The first integral is which again is related to the mechanical energy. The extremal points in x-space are with The period is now which is more complicated. It can be done, but here we only concentrate on the energy dependence. This is characteristic for all non-linear systems and has important consequences when we add perturbations.
Harmonic systems are exceptional and well behaved.
It is only fortunate that many systems are harmonic in first approximation. Our lives would have been much more chaotic otherwise.