Separatrix Diagrams of Flows

• Separatrix diagrams are combinatorial-topological objects that encode critical points and invariant manifolds, providing a complete invariant for the global structure of flows.
• They classify Morse–Smale and holomorphic flows by detailing connection data, cyclic ordering of edges, and invariant attachment information.
• They play a pivotal role in analyzing codimension-1 bifurcations and delineating basin boundaries in diverse systems from fluid dynamics to neural networks.

A separatrix diagram is a combinatorial-topological object encoding the global structure of invariant manifolds, their connections, and the types of critical points in a dynamical flow. It provides a finite, often graphical, invariant that completely characterizes the phase portrait up to topological equivalence for large classes of Morse and holomorphic flows on surfaces, complex manifolds, and high-dimensional systems. Separatrix diagrams serve as the primary tool for the classification of structurally stable flows, the analysis of codimension-1 bifurcations, and the delineation of basin boundaries—hence, they are fundamental in both mathematical theory and applied dynamical systems analysis.

1. Formal Definitions and Fundamental Structures

The separatrix diagram of a flow $X$ on a manifold $M$ is most generally a graph (or multi-graph) $\mathcal{D}$, embedded in $M$, with vertices corresponding to singular (critical) points (sources, sinks, saddles) and edges to 1-dimensional invariant manifolds (separatrices) connecting these singularities or approaching boundaries (Loseva et al., 2024, Ovtsynov et al., 11 Jan 2026). In Morse–Smale gradeint flows on compact surfaces, $\mathcal{D}$ is typically specified as a quadruple:

$\mathcal{D} = (V_0, V_1, E_s, E_u)$

where:

• $V_0$ is the set of non-saddle singularities (sources and sinks/nodes).
• $V_1$ is the set of saddle-type singularities.
• $E_s$ is the set of stable separatrices (invariant manifolds approaching a saddle).
• $E_u$ is the set of unstable separatrices (invariant manifolds leaving a saddle).

Each edge is an oriented arc (possibly on the boundary or connecting two saddles), colored according to its stability (commonly, red for stable, green for unstable, black for bifurcating/heteroclinic connections). For surfaces with boundary, vertices on $\partial M$ are distinguished, and the Poincaré–Hopf index formula imposes arithmetic constraints on allowed critical point combinations (Loseva et al., 2024, Loseva et al., 2024, Ovtsynov et al., 11 Jan 2026).

For rational, holomorphic, or meromorphic flows on Riemann surfaces or $\mathbb{CP}^1$, the separatrix diagram (or separatrix graph $\Gamma_R$) extends to encode not only Morse indices, but also the type and order of poles and multiple zeros, with faces in the embedding corresponding to topological zones of the flow (center, elliptic, annular, parallel) (Dias et al., 2020, Heitel et al., 2019, Kainz et al., 20 May 2025).

2. Encoding, Invariants, and Classification

A separatrix diagram constitutes a complete topological invariant for Morse–Smale flows: two flows are topologically equivalent if and only if their diagrams are isomorphic as embedded, directed, colored graphs (Ovtsynov et al., 11 Jan 2026). The invariants recorded include:

• List and type of critical points (with multiplicity, location: interior/boundary).
• Attachment data: which separatrix of each saddle connects to which node or saddle.
• The local valence (degree) and cyclic order of edges at each vertex.
• For flows on punctured or closed surfaces, the embedding data (distinguishing homology classes or sheets).
• For rational/holomorphic flows, the sector structure at singularities (number and type of hyperbolic, elliptic, parabolic sectors) and adjacency of basin boundaries (Kainz et al., 20 May 2025, Dias et al., 2020).

Classification proceeds by exhaustive enumeration (for small $N$), subject to index, stability, and transversality constraints. For example:

• On the Möbius strip, all possible separatrix diagrams with up to six critical points have been explicitly enumerated, yielding tables of possible Morse flows and codimension-1 bifurcations (Loseva et al., 2024).
• On the punctured torus, the possible diagrams with up to six singularities are organized by the combinatorics of their half-edges and cyclic boundary order (Loseva et al., 2024).
• On the Girl's surface (the projective plane immersed in $\mathbb{R}^3$), the full classification yields exactly 534 non-homeomorphic Morse–Smale flows with four fixed points, and 118 in the projective (descended) class (Loseva et al., 2022).

3. Bifurcation Theory and Dynamics via Separatrix Diagrams

Separatrix diagrams encode not only static flows but also dynamic transitions—specifically, codimension-1 bifurcations such as saddle–node (SN) and saddle–connection (SC). In these cases:

• An SN bifurcation is specified by choosing a separatrix connecting a saddle and a node; its collapse identifies these two vertices and their adjacent edges, encoding the coalescence and subsequent annihilation (Loseva et al., 2024, Ovtsynov et al., 11 Jan 2026, Bilun et al., 2023, Loseva et al., 2024).
• An SC (heteroclinic) bifurcation is encoded by marking the connecting separatrix (typically in black) at the moment two saddles share an invariant manifold. The diagram immediately before and after the bifurcation differ in which pairs of orbits are connected (Loseva et al., 2024, Ovtsynov et al., 11 Jan 2026, Bilun et al., 2023).

In high-dimensional or complex systems, the boundary between basins—the separatrix manifold—can be located via numerical computation of invariant manifolds (e.g., stable manifolds of edge states in fluid turbulence (Davis et al., 2020), or zero sets of Koopman eigenfunctions in high-dimensional neural or ecological networks (Dabholkar et al., 21 May 2025)).

4. Canonical Examples and Explicit Diagram Types

Surfaces with Boundary

A systematic enumeration exists for small $N$ for various surfaces. As an example, for the Möbius strip (Loseva et al., 2024):

n (critical points)	Morse flows	SN bifurcations	SC bifurcations	BSN	BDS	HN	HS	HSC	BSC
3	1	0	0	0	0	0	0	0	0
4	4	2	0	2	1	2	0	2	1
5	15	10	14	6	2	4	10	4	2
6	42	36	15	48	21	30	30	14	2

Key features:

• Each interior saddle is 4-valent (two stable, two unstable separatrices).
• Multiple separatrices between two critical points are constrained—collapsing all simultaneously would violate gradient structure.
• Doubling the Möbius strip embeds the diagrams into the Klein bottle, distinguishing boundary from interior points via their behavior under reflection (Loseva et al., 2024).

Holomorphic and Rational Vector Fields

Separatrix graphs for rational flows $R(z) d/dz$ on $\mathbb{CP}^1$ are characterized via the concept of admissible planar graphs. Such a graph must satisfy local valence and orientation-reversal parity at each vertex, while globally obeying the Poincaré index formula and face decomposition into recognized phase-portrait types (center, elliptic, annular, parallel) (Dias et al., 2020).

Explicit construction of the separating curves (via blow-up and compactification) yields a combinatorial model that is unique up to Möbius equivalence. In polynomial flows, the only pole is at infinity, so the graphs are always connected. In Newton flows, an extra bipartite structure appears (Heitel et al., 2019, Dias et al., 2020).

5. Application Domains and Computational Methodologies

Separatrix diagrams are used to:

• Enumerate and classify all Morse–Smale flows with a fixed number of singularities on a given surface. This underpins the global topological classification theory for dynamical systems on surfaces (Loseva et al., 2022, Loseva et al., 2024, Ovtsynov et al., 11 Jan 2026).
• Analyze bifurcation structure in low-dimensional systems, including systematic accounting for changes in phase portrait across parameter space (Bilun et al., 2023).
• Identify basin boundaries and design minimal perturbations to cross the separatrix in high-dimensional systems (e.g., RNNs, ecological networks), using data-driven approximation of Koopman eigenfunctions whose zero-level set coincides with the separatrix (Dabholkar et al., 21 May 2025).
• Formulate transition and boundary structures in fluid turbulence, with deterministically computed edge states anchoring the separatrix between laminar and turbulent regimes (Davis et al., 2020).

Typical computational methodologies include symbolic manipulation of graph data (for enumeration), Newton–Krylov solvers for coherent structures, PDE-constrained neural network training for high-dimensional Koopman eigenfunctions, and numerical integration/MMC sampling to visualize separatrix location in phase space (Dabholkar et al., 21 May 2025, Davis et al., 2020).

6. Generalizations, Constraints, and Enumeration Rules

Separatrix diagram enumeration is governed by:

• Index theorems (Poincaré–Hopf for Morse, and extended index sums accounting for boundary components).
• Degree (valence) rules: interior saddles must have four separatrices; boundary saddles three.
• Cyclic ordering constraints for multi-boundary components: every boundary point's incident half-edges must be recorded in the correct order (Loseva et al., 2024, Loseva et al., 2024).
• No forbidden cycles or heteroclinic connections in the gradient case (Morse–Smale restriction).
• Admissible coloring and vertex-valence configuration for rational flows on $\mathbb{CP}^1$ (Dias et al., 2020).

Enumeration becomes combinatorially intensive at higher $N$, but is tractable through code-based (rotation systems, flow codes) or three-color graph encodings.

7. Limitations and Special Cases

While the separatrix diagram is a complete invariant for Morse–Smale and many holomorphic flows, limitations arise in non-generic, non-gradient, or structurally unstable situations. In holomorphic systems, the relationship to complex time and singular Riemann surfaces governs more nuanced bifurcations and branching phenomena (Heitel et al., 2019). The classification also does not directly apply to systems with chaotic attractors or structurally unstable flows, where the set of invariant manifolds is not finite or their non-transversality precludes diagrammatic encoding.

• (Loseva et al., 2024) Structure of the codimension one gradient flows with at most six singular points on the Möbius strip
• (Ovtsynov et al., 11 Jan 2026) The structure of Morse flows and co-dimension one gradient flows on the sphere with holes
• (Dias et al., 2020) On the separatrix graph of a rational vector field on the Riemann sphere
• (Heitel et al., 2019) On Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamical Systems and Complex-Time Newton Flows
• (Dabholkar et al., 21 May 2025) Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions
• (Kainz et al., 20 May 2025) Separatrix configurations in holomorphic flows
• (Loseva et al., 2022) Topological structure of optimal flows on the Girl's surface
• (Loseva et al., 2024) Structure of Morse flows with at most six singular points on the torus with a hole
• (Bilun et al., 2023) Typical one-parameter bifurcations of gradient flows with at most six singular points on the 2-sphere with holes
• (Davis et al., 2020) Dynamics of laminar and transitional flows over slip surfaces: effects on the laminar-turbulent separatrix