Contours of the normalized cross-correlation of the perturbed surface density in the radius-angular frequency plane for models C05 (left) and W05 (right) at different epochs. The solid and dashed lines in each panel plot instantaneous ﹩{\rm{\Omega }}(R)﹩ and ﹩{\rm{\Omega }}\pm \kappa /2﹩. While the pattern speed ﹩{{\rm{\Omega }}}_{b}﹩ of the bars is almost independent of R, corresponding to rigid-body rotation, the pattern speed of the spirals arms is close to Ω at early time and gradually becomes similar to the bar pattern speed at late time. The color bar labels ﹩C(R,\theta ,t)/C(R,{\theta }_{\max },t)﹩.