Normal forms and complex periodic orbits in semiclassical expansions of Hamiltonian systems

Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space dynamics in their neighborhood. We provide a pedestrian presentation of this classical theory and extend it by including systematically the periodic orbits lying in the complex plane on each side of the bifurcation. This allows for a more coherent and unified treatment of contributions of periodic orbits in semiclassical expansions. The contribution of complex fixed points is find to be exponentially small only for a particular type of bifurcation (the extremal one). In all other cases complex orbits give rise to corrections in powers of $\hbar$ and, unlike the former one, their contribution is hidden in the ``shadow'' of a real periodic orbit.