The Lissajous transformation III. Parametric bifurcations

We examine a parametric family of cubic perturbed 1-1 resonant harmonic oscillators with an aim to understanding the phase flows of the reduced system. Variation of the parameters leads the system through five bifurcations of different types. Some bifurcations are due to passage through cases of discrete symmetry or integrability. A conjecture correlating degenerate equilibria in reduced systems with integrability is modified and reinforced.