Bifurcation of the period-4 orbits in the standard map

The period doubling bifurcation process in the two-dimensional area preserv ing 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 stud ied thoroughly to analyze boundary islands formation around the principal p eriod-4 island, and the onset of the hyperbolic bifurcation without reflect ion. It is illustrated explicity that the hyperbolic bifurcation without re flection gives rise to the birth of twin orbits with the periodicity of the mother orbit. (C) 2001 Elsevier Science Ltd. All rights reserved.