Abstract In this paper, we study the phase portraits of the current fields J J defined by J (z) = i (z) ' (z) J (z) = i ψ (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 J, addressing holomorphic, meromorphic, and essential singularity cases. Regarding the global dynamics, we provide a necessary and sufficient condition for all orbits of J J to be bounded when ψ is meromorphic. Furthermore, we establish that J J admits a global center if and only if ψ is either of the form (z) =c (z-z₀) ᵏ ψ (z) = c (z - z 0) k or (z) =c/ (z-z₀) ᵏ ψ (z) = c / (z - z 0) 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 J associated with complex polynomial functions ψ of degrees 2 and 3.
Braga et al. (Mon,) studied this question.