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Local bifurcations of a quasiperiodic Orbit

We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincare section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.