Stagnation graphs and separatrices of local bifurcations in velocity and current density planar vector fields

Abstract

An elementary introduction to the notion of separatrix as a topologically closed surface bounding disjointed domains of a dynamic system is given within the theory of local bifurcations. Accordingly, local bifurcations of a two-dimensional dynamic system, occurring as a consequence of a small smooth change made to the value of a parameter, and causing a sudden qualitative change in its behavior, have been analyzed by visualizing velocity vector fields, related stagnation graphs, which collect stagnation lines, i.e., sets of equilibrium points corresponding to vanishing flow, and separatrices. According to a theorem by Gomes, the Poincare index is conserved in the saddle-center and saddle-node bifurcations, which constitutes an important piece of information. The results obtained have mainly a didactic value and are preparatory to more advanced investigations of harder problems, encountered in the analysis of the quantum mechanical current densities $$ {\varvec{J}}^\mathbf{B} $$ and $$ {\varvec{J}}^{{ \mathbf m}_{I}} $$ induced in the electrons of a molecule by an applied magnetic field $$ {\varvec{B}}$$ and by the magnetic dipole $$ {\bf{m}}_{I} $$ at a nucleus I, and in the rationalization of magnetic response properties such as magnetizabilities, nuclear magnetic shielding, and nuclear spin–spin coupling. Applications to the study of bifurcation phenomena in chemical reactions are also possible.