Phase Portraits of Current Fields

In this paper, we study the phase portraits of the current fields J J defined by J(z) = i ψ (z) ψ '(z)̄ J ( z ) = i ψ ( z ) ψ ′ ( z ) ¯ , where ψ 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 ψ is meromorphic. Furthermore, we establish that J J admits a global center if and only if ψ ψ is either of the form ψ (z)=c(z-z₀)ᵏ ψ ( z ) = c ( z - z 0 ) k or ψ (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 ψ of degrees 2 and 3.