Separatrix configurations in holomorphic flows

TL;DR

This work analyzes boundary separatrices for holomorphic planar flows with real time by proving transit-time continuity along canonical-region boundaries and classifying boundary path components. It then provides gap-free proofs for separatrix configurations in three canonical settings—centers, nodes/foci, and global elliptic sectors—highlighting finite-time blow-up behavior and supplying a counterexample that blow-up need not occur in both time directions. The center-basin result bounds the total transit time of boundary double-sided separatrices by the center period $T(a)=\frac{2\pi i}{F'(a)}$, while the node/focus case shows blow-up in the same direction as convergence and demonstrates that bidirectional blow-up is not guaranteed. For elliptic sectors, the boundary structure comprises a central multiple equilibrium, one incoming and one outgoing separatrix, and at most countably many double-sided separatrices. Collectively, these results provide geometric tools for constructing blow-up scenarios, clarify the holomorphic separatrix notion, and deepen understanding of global phase portraits and basin geometry.

Abstract

We investigate properties of boundary orbits (separatrices) of canonical regions in holomorphic flows with real-valued time. We establish the continuity of transit times along these boundary orbits and classify possible path components of the boundary of flow-invariant domains. Thus, we provide central tools for geometric constructions aimed at examining the role of blow-up scenarios in separatrix configurations of basins of simple equilibria and global elliptic sectors: First, we prove that the separatrices of basins of centers is entirely composed of double-sided separatrices with a blow-up in finite positive and finite negative time. Second, we show that the separatrices of node and focus basins (sinks and sources) exhibit a finite-time blow-up in the same time direction in which the orbits within the basin tend towards the equilibrium. Additionally, we propose a counterexample to the claim in Theorem 4.3 (3) in [The structure of sectors of zeros of entire flows, K. Broughan (2003)], demonstrating that a blow-up does not necessarily have to occur in both time directions. Third, we describe the boundary structure of global elliptic sectors. It consists of the multiple equilibrium, one incoming and one outgoing separatrix attached to it, and at most countably many double-sided separatrices.

Figures (7)

• Figure 1: Geometrical visualization of the construction in the proof of Theorem \ref{['thm:separatrices_node']}, for the case where $a$ is attracting. The gray paths are transversals through $x$ and $y$ (both black), respectively. $\Gamma_n$ (red) is the separatrix. $C$ (purple) is the circle without contact with the equilibrium $a$ (black) in its interior. The interior of the closed curve $J$ (blue) is simply connected.
• Figure 2: Local phase portrait of system \ref{['eq:planarODE']} with $F(x)=xe^x$, plotted with Matlab. The equilibrium is yellow. Separatrices are red and green. The separatrices on the boundary of $\mathcal{N}$ are green. The orbits within $\mathcal{N}$ are blue. Due to the exponential term, all orbits in $\mathbb{C} \setminus[0,\infty)$ tend towards the left half-plane for positive time. The exponential term draws the blue and green orbits towards the negative real axis.
• Figure 3: Visualization of the construction of the closed Jordan curve $J_y$ (blue). $\Gamma_y$ (blue) connects the points $P$ and $P_y$ (purple) via the negative separatrix $\Gamma$ (green). The straight line segments $\eta$ and $\eta_y$ connect $\Gamma$ to the real axis orthogonally. The point $a$ (red) is the equilibrium.
• Figure 4: kainz2024basins Geometrical objects of a local elliptic sector $S$ (light blue) with counterclockwise direction in a multiple equilibrium $a$ (black). $\Gamma_1$ and $\Gamma_2$ (red) are the characteristic orbits forming the boundary of $S$. $\Lambda_1$ and $\Lambda_2$ (purple) are the transversals. $\Xi=\Gamma(E_1)$ (green) is the homoclinic orbit. The black arrows indicate the direction of the vector field.
• Figure 5: Geometrical visualization of the construction in Step 3 of the proof of Theorem \ref{['thm:separatrices_sectors']}, for the case where the local elliptic sector (light blue) in the multiple equilibrium $a$ (black) has counterclockwise direction. The gray paths are transversals through $x$ and $y$ (both black), respectively. $\Gamma_1$ and $\Gamma_2$ (red) are the separatrices. $\Lambda_1$ and $\Lambda_2$ (purple) are the transversals of the local elliptic sector. $\Xi=\Gamma(E_1)$ (green) is the homoclinic sector-forming orbit. The interior of the closed curve $J$ (yellow) is simply connected.

Theorems & Definitions (38)

• Definition 3.1: transit time broughan2003structure
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Definition 4.1: Separatrix broughan2003structure
• Remark 4.2