= Fold-Hopf bifurcation
{title2=$\lambda_0=0,\quad\lambda_\pm=\pm i\omega,\quad\omega\ne0$}
= Zero-Hopf bifurcation
{synonym}
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>.
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