Another look at the saddle-centre bifurcation: Vanishing twist

In the saddle-centre bifurcation a pair of periodic orbits is created “out of nothing” in a Hamiltonian system with two degrees of freedom. It is the generic bifurcation with multiplier one. We show that “out of nothing” should be replaced by “out of a twistless torus”. More precisely, we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits. Vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes. We explicitly derive the position of the twistless torus in phase and in parameter space near the saddle-centre bifurcation. The theory is applied to the area preserving Hénon map.