Nonlinear Sciences – Chaotic Dynamics
Scientific paper
1998-09-18
Nonlinear Sciences
Chaotic Dynamics
better ps figures available at http://www.phys.univ-tours.fr/~mouchet or on request to mouchet@celfi.phys.univ-tours.fr
Scientific paper
10.1006/aphy.1999.5916
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.
Leboeuf Patricio
Mouchet Amaury
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