Abstract 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.
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