Resumen de Phase Portraits of Current Fields

Denis de Carvalho Braga, Alexander Fernandes da Fonseca, Luis Fernando Mello, Ronisio Moises Ribeiro

• In this paper, we study the phase portraits of the current fields J defined by J (z) = iψ(z)ψ (z), where ψ is a complex-valued function called wave function. We provide a comprehensive characterization of the local phase portraits near equilibrium and singular points of J , addressing holomorphic, meromorphic, and essential singularity cases. Regarding the global dynamics, we provide a necessary and sufficient condition for all orbits of J to be bounded when ψ is meromorphic. Furthermore, we establish that J admits a global center if and only if ψ is either of the form ψ(z) = c(z − z0)k or ψ(z) = c/(z − z0)k , where c is a complex number and k is a positive integer.

  Finally, we present the complete global classification and bifurcation diagrams for J associated with complex polynomial functions ψ of degrees 2 and 3.