Normal forms for nonlinear vector fields. II. Applications

Abstract

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. >