Local geometry of Equilibria and a Poincaré-Bendixson-type Theorem for Holomorphic Flows
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914815213043712 |
|---|---|
| author | Kainz, Nicolas Lebiedz, Dirk |
| author_facet | Kainz, Nicolas Lebiedz, Dirk |
| contents | In this paper, we explore the local geometry of dynamical systems $\dot{x}=F(x)$ with real time parameterization, where $F$ is holomorphic on connected open subsets of $\mathbb{C}\stackrel{\sim}{=}\mathbb{R}^2$. We describe the geometry of first-order equilibria. For equilibria of higher orders, we establish an equivalent condition for "definite directions", allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in [Differential equations and dynamical systems, Lawrence Perko (1990)] under the additional condition of holomorphy. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincaré-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local geometry of Equilibria and a Poincaré-Bendixson-type Theorem for Holomorphic Flows Kainz, Nicolas Lebiedz, Dirk Dynamical Systems Complex Variables Geometric Topology 30A99, 30C10, 30C15, 30D30 (Primary), 32M25, 37F10, 37F75 (Secondary) In this paper, we explore the local geometry of dynamical systems $\dot{x}=F(x)$ with real time parameterization, where $F$ is holomorphic on connected open subsets of $\mathbb{C}\stackrel{\sim}{=}\mathbb{R}^2$. We describe the geometry of first-order equilibria. For equilibria of higher orders, we establish an equivalent condition for "definite directions", allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in [Differential equations and dynamical systems, Lawrence Perko (1990)] under the additional condition of holomorphy. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincaré-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic. |
| title | Local geometry of Equilibria and a Poincaré-Bendixson-type Theorem for Holomorphic Flows |
| topic | Dynamical Systems Complex Variables Geometric Topology 30A99, 30C10, 30C15, 30D30 (Primary), 32M25, 37F10, 37F75 (Secondary) |
| url | https://arxiv.org/abs/2402.07612 |