Conservative generalized bifurcation diagrams

Abstract

Bifurcation cascades in conservative systems are shown to exhibit a generalized diagram, which contains all relevant informations regarding the location of periodic orbits (resonances), their width (island size), irrational tori and the infinite higher-order resonances, showing the intricate way they are born. Contraction rates for islands sizes, along period-doubling bifurcations, are estimated to be $\alpha_I\sim 3.9$. Results are demonstrated for the standard map and for the continuous H\'enon-Heiles potential. The methods used here are very suitable to find periodic orbits in conservative systems, and to characterize the regular, mixed or chaotic dynamics as the nonlinear parameter is varied.