Fold-Hopf bifurcation (${\lambda }_0=0,{\lambda }_{\pm }=\pm i\omega ,\omega \ne 0$)

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

The amplitude equations $u^{\prime }=-2uv$, $v^{\prime }={\lambda }_2+v^2+u^2$ satisfy $u^{\prime }=F_v$, $v^{\prime }=-F_u$, hence preserve the displayed first integral. For ${\lambda }_2=-a^2<0$ and $u\ge 0$, the centre equilibrium is $(a,0)$ with $F=2a^3/3$, while the two boundary saddle equilibria are $(0,\pm a)$ with $F=0$. The level $F=0$ contains the boundary connection and the arc $u^2=3(a^2-v^2)$ joining the two saddle equilibria. Levels $0<F<2a^3/3$ form closed periodic orbits. Adding an independent rotating phase turns these closed amplitude trajectories into invariant tori.