Bifurcation of the period-4 orbits in the standard map

The period doubling bifurcation process in the two-dimensional area preserving mapping is investigated on the basis of symmetry structure analysis. In particular a case of the peirod-4 orbits in the standard map has been studied thoroughly to analyze boundary islands formation around the principal period-4 island, and the onset of the hyperbolic bifurcation without reflection. It is illustrated explicity that the hyperbolic bifurcation without reflection gives rise to the birth of twin orbits with the periodicity of the mother orbit.