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

Bifurcations of periodic orbits as an external parameter is varied are a ch aracteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of ph ase space dynamics in their neighborhood. We provide a pedestrian presentat ion 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 fi xed points is found to be exponentially small only for a particular type of bifurcation (the extremal one). In all other cases complex orbits give ris e to corrections in powers of A and, unlike the former one, their contribut ion is hidden in the "shadow" of a real periodic orbit. (C) 1999 Academic P ress.