A perturbative analysis based on the renormalization method shows that the period-four torus near an elliptic fixed point of the radially twist map goes through not only the Poincare-Birkhoff bifurcation but also an exchange-type bifurcation and the global ballooning of resonant islands. The appearance of the exchange-type bifurcation is found to be closely connected to the Suris' integrable maps. (c) 2005 Elsevier Ltd. All rights reserved.