[en] The stochasticity and symmetry of the standard map have been discussed to explore the generic aspect of the statistical properties inherent to the low dimensional classical Hamiltonian systems. The accelerator modes of the standard map give rise to anomalous enhancement of the stochastic diffusion. Systematic analysis of symmetry of the standard map is undertaken to investigate structure of the periodic orbits around the stable fixed point, and the method is applied to the Poincare-Birkhoff chains around the accelerator mode. The squeeze effect of the period-3 accelerator orbits is examined explicitly, in terms of the reduced accelerator mode map. Furthermore, it is shown that the period-3 accerator orbits are born as intrinsic nonlinear multifurcation, differing from birth of the Poincare-Birhoff chains with the higher periodicity q≥4. (author)