The birth of twin Poincaré-Birkhoff chains near 1:3 resonance

For a typical area-preserving map we describe the birth process of two twin Poincaré-Birkhoff chains, i.e. two rings consisting of center points alternated by saddles, wound around an elliptic fixed point. These twin chains are not born out of the elliptic fixed point, but in the plane, from an annular region where the rotation number has a rational extremum. This situation generically occurs near a 1:3 resonance. We find that the birth of two twin PB chains in such an annular region requires first the birth of two “dimerized” chains of saddle-center pairs, by a tangent bifurcation. The transition from two dimerized chains to two PB chains involves the breakup of homoclinic saddle connections and the formation of heteroclinic connections; it amounts to the reconnection phenomenon of Howard and Hohs. Our results can be regarded as a supplement to the Poincaré-Birkhoff theorem, for the case that the twist condition is not satisfied.