Noose bifurcation of periodic orbits

The authors investigate the consequences of putting together a period-doubling and saddlenode bifurcation to form a closed loop, the noose bifurcation, where the two branches of the period-doubling evolve so as to come together again and annihilate in a saddle-node. The noose is non-trivial because of the topological properties of periodic orbits in phase space: interlinking between orbits and 'self-linking' (twisting) of the manifolds of orbits. They ask whether the noose is 'generic', typical of all ordinary differential equations (ODES), or does it require special properties. In explaining the noose they introduce some useful ideas from the analytical literature on ODES: generic properties and bifurcations, and some simple applications of knot theory. The noose is a good place from which to hang these ideas.

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