Phase portraits and bifurcations of the non-linear oscillator:

The non-linear oscillator + αx + γx²x + βx + δx³ = 0 is studied using the methods of differentiable dynamics to obtain qualitative behaviour. The case x, β<0; γ, δ> 0 is considered in some detail; it has physical relevance as a simple model in certain now-induced structural vibration problems in which the structural non-linearities act to maintain overall stability. The presence of local and global bifurcations is detected and their physical significance discussed.