Phase portraits of (2;1) reversible vector fields of low codimension

In this paper we study the phase portraits and bifurcation diagram of the symmetric singularities of codimensions zero, one and two of planar reversible vector fields having a line of reversibility.

💡 Research Summary

The paper investigates planar reversible vector fields of type (2;1), i.e., systems that are reversible with respect to the involution ϕ(x, y) = (x, −y). After fixing a coordinate system that makes the line y = 0 the reversibility line, the authors consider the space X of C^k‑germs of such vector fields and introduce a notion of topological equivalence that respects the reversible symmetry. The main goal is to classify all low‑codimension symmetric singularities (codimension 0, 1 and 2) and to describe their global phase portraits on the Poincaré compactification (the unit disk with four distinguished “poles”).

The classification relies on the normal‑form theory developed by Teixeira (2006). For codimension‑0 the only topological types are X₀₁ = (0, ½) (a trivial flow) and X₀₂ = (y, δx) with δ = ±1 (a linear saddle on the symmetry line). Codimension‑1 yields five normal families X₁₁–X₁₄, each depending on a single parameter λ. The paper analyses each family in detail: the eigenvalues of the linear part determine whether the origin is a center, a focus, a saddle or a node; the sign of λ governs the bifurcations (center–focus, saddle–node, Hopf‑like). In particular, X₁₂ = (δxy, ½(2δy² + x + λ)) is studied extensively. When λ = 0 a quasihomogeneous blow‑up with weights (2, 1) reveals four finite singularities (a hyperbolic saddle and two nodes) and a precise separatrix configuration dictated by the reversible symmetry. The authors prove that X₁₂ has no limit cycles, using the invariance of the y‑axis and a standard argument that a limit cycle cannot surround a single saddle.

For codimension‑2 the authors list twelve normal families X₂₁–X₂₅ (some with sub‑cases). These families involve cubic or higher nonlinearities and depend on two parameters (α, β) or (b, β, α). The paper determines the number and ordering of real roots of the cubic polynomial p(x) = x³ + βx + α that appears in X₂₁, using the discriminant D = ¼α² + 1/27 β³. The sign of D decides whether the system has one or three finite equilibria, and the relative positions of these equilibria are described explicitly. Similar analyses are carried out for the other families, identifying cusp, nodal, saddle and focus symmetric singularities.

A crucial part of the work is the global picture. The authors compactify the plane by the Poincaré sphere, introduce four charts (U₁, U₂, V₁, V₂) corresponding to the north, south, east and west poles, and study the vector fields in each chart. They show that the “infinity” circle is filled with equilibria for many families, and they compute the flow on the regularized infinity using desingularization (division by the radial coordinate). The separatrix structure is then traced: because of the symmetry, separatrices appear in symmetric pairs, and their connections are determined by the direction of the flow across the symmetry line and by the sign of the parameters.

All the results are summarized in a series of figures (19–36). Each figure displays the phase portrait on the Poincaré disk for a particular normal family, both for the generic parameter values and for the bifurcation values (λ = 0 or (α, β) ≈ (0, 0)). Thick lines denote separatrices, thin lines generic orbits, large dots isolated equilibria, and dotted lines indicate a line of equilibria when it occurs (e.g., in X₁₂ and X₂₄). The figures make clear that the bifurcation diagrams are global: they are valid for all real values of the unfolding parameters.

The paper concludes that every symmetric singularity of codimension ≤ 2 in a (2;1) reversible vector field is topologically equivalent to one of the portraits shown in the figures, when the parameters are close to the critical values. Moreover, the authors remark that without a bound on the degree of the vector field, arbitrarily many singularities can appear, so the presented portraits are not exhaustive for all possible reversible systems, but they do give a complete description for the low‑codimension cases.

Finally, the authors suggest that the methodology can be extended to higher‑degree reversible systems or to systems with multiple symmetry lines, opening a path for future research on the global dynamics of reversible differential equations.