Normal forms for nonlinear vector fields. II. Applications

For pt.I see ERL memo UCB/ERL, M87/81 (1987). The normal form theory for nonlinear vector fields presented in pt.I, is applied to several examples of vector fields whose Jacobian matrix is a typical Jordan form, which gives rise to interesting bifurcation behavior. The normal forms derived from these examples are based on Ushiki´s method, which is a refinement of Takens´ method. A comparison of the normal forms derived by Poincare´s method, F. Takens´ method (1974), and S. Ushiki´s method (1984) is also given. For vector fields imbued with some form of symmetry, additional constraints are imposed in the normal-form algorithm from pt.I so that the resulting normal form will inherit the same form of symmetry. The normal forms of a given vector field are then used to derive its versal unfoldings in the form of an n-parameter family of vector fields. Such unfoldings are powerful tools for analyzing the bifurcation phenomena of vector fields when the parameter changes. Moreover, since the local bifurcation structure around a highly degenerate singularity can include some bifurcation phenomena of a global nature which have been observed from a less degenerate family of vector fields, it follows that the concepts of normal form and versal unfolding are useful tools for analyzing such degenerate singularities.<>