Global period-doubling bifurcation in the standard map

While there has been great effort to establish universal behavior of the se quence of period-doubling bifurcation in Hamiltonian systems with few degre es of freedom, the nature of the period-doubling bifurcation is far more co mplicated in two-dimensional maps. Though the onset of instability is deter mined by a local, linear property of the system, the area of a bifurcated r egion in the phase space increases gradually when the control parameter inc reases beyond the critical threshold. Scaling laws for the growth process o f the period-doubling bifurcation are elucidated for the period-2 step-1 ac celerator mode and for the fundamental fixed orbit in the standard map.