Global Period-Doubling Bifurcation in the Standard Map

While there has been great effort to establish universal behavior of the sequence of period-doubling bifurcation in Hamiltonian systems with few degrees of freedom, the nature of the period-doubling bifurcation is far more complicated in two-dimensional maps. Though the onset of instability is determined by a local, linear property of the system, the area of a bifurcated region in the phase space increases gradually when the control parameter increases beyond the critical threshold. Scaling laws for the growth process of the period-doubling bifurcation are elucidated for the period-2 step-1 accelerator mode and for the fundamental fixed orbit in the standard map.