Complex potentials and holomorphic differential equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914237830397952 |
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| author | Rondón, Gabriel da Silva, Paulo R. |
| author_facet | Rondón, Gabriel da Silva, Paulo R. |
| contents | A complex potential is a holomorphic function $Ω:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} = \overline{Ω'(z)}$. In this work, we generalize the concept of potential to the broader class of dynamical systems of the form $\dot{z} = f(z)$, with $f:\mathbb{C} \to \mathbb{C}$ holomorphic. The resulting potential induces a rectification mapping providing a natural framework for the topological classification of phase portraits of planar polynomial vector fields. The existence of complex potentials serves as a powerful tool in addressing fundamental problems, such as the establishment of bounds for the number of limit cycles in piecewise-smooth systems, and the local configuration of curvature lines around umbilic points, among others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03404 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex potentials and holomorphic differential equations Rondón, Gabriel da Silva, Paulo R. Dynamical Systems Complex Variables 32A10, 34C20, 34A34, 34A36, 34C05 A complex potential is a holomorphic function $Ω:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} = \overline{Ω'(z)}$. In this work, we generalize the concept of potential to the broader class of dynamical systems of the form $\dot{z} = f(z)$, with $f:\mathbb{C} \to \mathbb{C}$ holomorphic. The resulting potential induces a rectification mapping providing a natural framework for the topological classification of phase portraits of planar polynomial vector fields. The existence of complex potentials serves as a powerful tool in addressing fundamental problems, such as the establishment of bounds for the number of limit cycles in piecewise-smooth systems, and the local configuration of curvature lines around umbilic points, among others. |
| title | Complex potentials and holomorphic differential equations |
| topic | Dynamical Systems Complex Variables 32A10, 34C20, 34A34, 34A36, 34C05 |
| url | https://arxiv.org/abs/2601.03404 |