STEP-BY-STEP RESOLUTION OF SINGULARITIES AND STUDYING INTERACTIONS BETWEEN THEM

An explicit method, based on subsequent small perturbations, allowing one to study the algebraic and geometric nature of multiple isolated singularities of a polynomial vector field, is discussed. The main ingredients of the method are (i) establishing a canonical form of a singularity, (ii) explicit decomposition of a compound singularity into simpler ones, and (iii) deriving asymptotic laws of decomposition/collision of singularities. In particular, the saddle-node, pitchfork, and quadruple bifurcations of zeros of a polynomial vector field are considered from the various novel and perhaps unexpected angles. Several examples of subsequent phase portraits illustrating possible interactions between equilibrium of ODEs are also discussed.