In this paper, we study the phase portraits of the current fields <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {J}(z) = i \psi (z) \overline{\psi '(z)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">J</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mover> <mrow> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> is a complex-valued function called wave function. We provide a comprehensive characterization of the local phase portraits near equilibrium and singular points of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation>, addressing holomorphic, meromorphic, and essential singularity cases. Regarding the global dynamics, we provide a necessary and sufficient condition for all orbits of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> to be bounded when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> is meromorphic. Furthermore, we establish that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> admits a global center if and only if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> is either of the form <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\psi (z)=c(z-z_0)^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>c</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\psi (z)=c/(z-z_0)^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where c is a complex number and k is a positive integer. Finally, we present the complete global classification and bifurcation diagrams for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> associated with complex polynomial functions <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> of degrees 2 and 3.

Phase Portraits of Current Fields

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2026年5月4日

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

摘要

In this paper, we study the phase portraits of the current fields $\mathcal {J}$ $J$ defined by $\mathcal {J}(z) = i \psi (z) \overline{\psi '(z)}$ $J (z) = i ψ (z) \bar{ψ^{'} (z)}$ , where $\psi $ $ψ$ is a complex-valued function called wave function. We provide a comprehensive characterization of the local phase portraits near equilibrium and singular points of $\mathcal {J}$ $J$ , addressing holomorphic, meromorphic, and essential singularity cases. Regarding the global dynamics, we provide a necessary and sufficient condition for all orbits of $\mathcal {J}$ $J$ to be bounded when $\psi $ $ψ$ is meromorphic. Furthermore, we establish that $\mathcal {J}$ $J$ admits a global center if and only if $\psi $ $ψ$ is either of the form $\psi (z)=c(z-z_0)^k$ $ψ (z) = c {(z - z_{0})}^{k}$ or $\psi (z)=c/(z-z_0)^k$ $ψ (z) = c / {(z - z_{0})}^{k}$ , where c is a complex number and k is a positive integer. Finally, we present the complete global classification and bifurcation diagrams for $\mathcal {J}$ $J$ associated with complex polynomial functions $\psi $ $ψ$ of degrees 2 and 3.