The birth process of periodic orbits in non-twist maps

We study the birth process of periodic orbits in non-twist systems, by means of a model map which contains all the typical features of such a system. The most common form of the birth process, or standard scenario, is described in detail. This scenario involves several steps: first one “dimerized” chain of saddle-center pairs is born, then a second, and eventually these two chains are reconnected into two Poincaré-Birkhoff chains. We also discuss several variations on this standard scenario. These variations can give rise to arbitrarily many chains, intertwined in a complex fashion, and the reconnection of these chains can be highly non-trivial. Finally we study the effect of dissipation on the birth process. For sufficiently small dissipation one can still recognize the birth and reconnection processes, but with several new features. In the first place, the chains do not consist anymore of conservative saddles and centers, but rather of dissipative saddles and nodes. Furthermore, the dissipation destrtoys the symmetry between the inner and outer chains, and as a result the reconnection does not take place in one single step anymore, but in three.