Local dynamics of a family of quasilinear ODEs with folded singular equilibria

The local behavior of Newton’s method near singular equilibria can be addressed via normal form theory in a continuous-time framework. In this paper we analyze the dynamics of a recently obtained normal form for the continuous Newton method near so-called folded equilibrium points; these can be classified in terms of a Morse index which characterizes the nearby structure of pseudoequilibria and impasse points. Extremal values of this index yield the simplest cases, in which all surrounding singularities are either forward or backward impasse points. In more involved isotropic cases, the behavior of trajectories will be distinguished by two types of separatrices, one of them undergoing singularity crossing phenomena. This approach leads to a complete local description of the different attraction domains which can be displayed at singular solutions.