In 1986, Dupret and Marchal (J. Mel. Theor. Appl., 5 (1986) 403) determined two tensorial conditions in order to analyze the loss of evolution and the change of type for Maxwell fluid flows. We perform here a quasi-linear analysis of the characteristics in flows of FENE-P fluids. Mathematical and easily computable conditions have been derived and can be used as criteria in complex flows. A similar result is also available for the Chilcott-Rallison model. The change of type is analyzed for flows through an abrupt contraction. Numerical results show that the vortex enhancement mechanism may be coupled with the change of type. A numerical simulation of the tree-like jet flow has also been performed.