Bifurcations in the vicinity of a “fused focus”

Abstract Conditions are presented for the existence of bifurcation of a singular point of the type of a “fused focus”. The fusing is accomplished with ordinary trajectories under the assumption that the general integrals are known for both systems forming the “fused system”. In the approximation of analytical characteristics in the equations of motion of dynamic systems by piecewise linear or relay functions on the lines of fusing, singular points can arise which are fused from ordinary or singular trajectories of systems to be fused. When the parameters of the system change, analogies arise here to bifurcations which are known for systems with analytical right-hand sides (birth of limit cycles from the fused focus, from the fused loop of the separatrix etc.).