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arxiv: 1906.10171 · v1 · pith:FX76I5ZEnew · submitted 2019-06-24 · 🧮 math.DS · math.AG

Quadratic Planar Differential Systems with Algebraic Limit Cycles via Quadratic Plane Cremona Maps

Maria Alberich-Carrami\~nana , Antoni Ferragut , Jaume Llibre This is my paper

Pith reviewed 2026-05-25 16:41 UTC · model grok-4.3

classification 🧮 math.DS math.AG
keywords quadratic differential systemsalgebraic limit cyclesCremona mapsbirational transformationsplanar systemsphase portraitsPoincaré disk
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The pith

Quadratic Cremona maps create new quadratic differential systems with algebraic limit cycles of degree 5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that quadratic plane Cremona maps transform quadratic differential systems into other quadratic systems. These maps preserve algebraic limit cycles. By applying the maps to known families, the authors obtain a new family with an algebraic limit cycle of degree 5. They also recover all previously known families having algebraic limit cycles of degree greater than 4 through the same transformations. Phase portraits for these systems are given on the Poincaré disk.

Core claim

Quadratic plane Cremona maps allow the construction of new quadratic planar differential systems from old ones while preserving the quadratic character and any algebraic limit cycles. This yields a previously unknown family of quadratic systems possessing an algebraic limit cycle of degree five, and it generates all known families with algebraic limit cycles of higher degree from earlier examples.

What carries the argument

Quadratic plane Cremona maps: birational transformations that map the plane to itself and send quadratic vector fields to quadratic vector fields while keeping algebraic limit cycles algebraic.

If this is right

  • A new family of quadratic systems with algebraic limit cycle of degree 5 is obtained.
  • Known families with algebraic limit cycles of degree greater than four are recovered via these maps.
  • Phase portraits on the Poincaré disk are provided for all families of quadratic systems with algebraic limit cycles.
  • The transformations supply a systematic method to produce quadratic systems carrying algebraic limit cycles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Iterating the maps further might produce quadratic systems with algebraic limit cycles of degree six or higher.
  • Analogous birational transformations could apply to cubic or higher-degree planar systems.
  • The supplied phase portraits indicate that the topological types realized by these systems are finite and potentially classifiable.

Load-bearing premise

Quadratic plane Cremona maps send quadratic vector fields to quadratic vector fields while preserving the algebraic character of any limit cycle they carry.

What would settle it

An explicit quadratic plane Cremona map applied to a quadratic system with an algebraic limit cycle that results in either a non-quadratic system or a system whose limit cycle is no longer algebraic would disprove the preservation property.

Figures

Figures reproduced from arXiv: 1906.10171 by Antoni Ferragut, Jaume Llibre, Maria Alberich-Carrami\~nana.

Figure 1
Figure 1. Figure 1: Phase portraits of the known quadratic differential systems having an alge￾braic limit cycle in the Poincar´e disk. The red lines correspond to the invariant algebraic curves. The dashed lines correspond to the cofactors. The following result is well known, see [13]. Lemma 8. If we take L¯ = L + XW, M¯ = M + Y W and N¯ = N + ZW, with W a homogeneous polynomial of degree m − 1, then the 1-form ω remains inv… view at source ↗
Figure 2
Figure 2. Figure 2: The ordinary plane Cremona map. S CP2 1 CP2 2 q2 q1 q3 L1 L3 p3 p1 p2 Ep2 =E'q2 E'q1 =Lñ1 E'q3 =Lñ3 P P' F E'p1 =L' ñ1 Ep3 =L' ñ3 L'3 L'1 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A quadratic plane Cremona map with exactly two proper planar base points (I). p2 → p1 and p3 → p2 are all the proximity relations. Any quadratic Cremona map of type (C3) factorizes as the blow-up at K = {p1, p2, p3}, followed by the blow-downs of the strict transform Le = E0 q3 of the line L := p1p2 (where p1p1 is the unique line going through p1 such that its multiplicity at p2 is one) and of the exceptio… view at source ↗
Figure 4
Figure 4. Figure 4: A quadratic plane Cremona map with exactly two proper planar base points (II). S CP2 1 CP2 2 P P' F Ep1 =E'q1 E'q ñ Ep3 =L'ñ 3 =L Ep2 =E'q2 L p1 p3 p2 q3 L' q2 q1 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A quadratic plane Cremona map with a unique proper planar base point (I). CP2 2 q2 q3 q1=(0:1:0) L={Z=0} p3 B2=(1:1:1) p1=(0:1:0) F L'={W=0} A=(1:0:0) B1=(0:0:1) CP2 1 F(B1)=(0:0:1) A'=(1:0:0) p2 F(B2)=(1:0:1) B'2=(1:1:1) F(C) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A quadratic plane Cremona map with a unique proper planar base point (II). points, and each of the three types of quadratic maps has a different number of proper base points. Next result classifies quadratic plane Cremona maps under projective equivalence, that is, two maps are equivalent if you obtain one from the other by composing with suitable planar projectivities, in both the departure CP2 1 and targ… view at source ↗
Figure 7
Figure 7. Figure 7: The three base points of the different types of Cremona maps (C1), (C2) and (C3) listed in Proposition 14, from left to right, drawn in the Poincar´e disk. Black dots are proper points, while white dots are infinitely near singular points. 4. Transforming differential systems by plane Cremona maps In this section we will describe the effect of applying quadratic plane transformations on foliations and on c… view at source ↗
Figure 8
Figure 8. Figure 8: The list of all possibilities for the base points of a quadratic plane Cremona map under the hypothesis of Theorem 20. Proper points are represented black filled while infinitely near points are represented white filled (joined with an edge to the point which they are proximate to). The shapes relate the points to the foliation: square is a regular point, triangle is a non-dicritical singular point, star i… view at source ↗
Figure 9
Figure 9. Figure 9: Phase portrait of the quadratic differential system (28) having an algebraic limit cycle of degree 5 on the Poincar´e disk. The red lines correspond to the invariant algebraic curve that contains the limit cycle. The dashed line corresponds to its cofactor. After Remark 8 and Theorem 22, [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Phase portraits of all the known families of quadratic differential systems having an algebraic limit cycle. The differential systems corresponding to the phase por￾traits inside circles are related by the Cremona transformation (22). We note that we consider the different phase portraits of the Qin family depending on the value of b. Next lemma provides the expression of (30) after the application of the… view at source ↗
read the original abstract

In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations. We also provide the phase portraits on the Poincar\'e disk of all the families of quadratic differential systems having algebraic limit cycles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper applies quadratic plane Cremona maps (birational transformations of degree 2) to known families of quadratic planar vector fields possessing algebraic limit cycles. It claims that these maps produce new quadratic systems, including a previously unknown family with an algebraic limit cycle of degree 5, while also recovering all known families with algebraic limit cycles of degree greater than 4. Phase portraits on the Poincaré disk are supplied for the complete collection of such families.

Significance. If the maps are shown to preserve both the quadratic degree of the vector field (after clearing denominators) and the isolation of the algebraic limit cycle, the construction supplies a systematic geometric mechanism for generating and unifying examples. This would be a useful addition to the literature on quadratic systems with algebraic limit cycles, especially given the scarcity of explicit high-degree examples.

major comments (2)
  1. [Abstract / § on the new family] The central technical claim (that a quadratic Cremona map applied to a quadratic vector field yields, after clearing denominators, another quadratic vector field whose algebraic limit cycle remains isolated and of the stated degree) is stated in the abstract and paragraph 2 but is not accompanied by an explicit computation of the denominator cancellation for the new degree-5 family. Without this verification, it is impossible to confirm that the degree remains exactly 2 rather than rising.
  2. [Section on recovery of known families] The recovery of known higher-degree families is asserted to follow from the same maps, yet no explicit base system plus map pair is exhibited that reproduces, for instance, a known degree-6 or degree-8 algebraic limit cycle while keeping the transformed field quadratic. This step is load-bearing for the unification claim.
minor comments (2)
  1. [Abstract] The abstract refers to “some kind of birational transformations”; the precise definition and the indeterminacy loci of the quadratic Cremona maps should be stated at the first use.
  2. [Phase-portrait section] Phase-portrait figures on the Poincaré disk are mentioned but their captions should explicitly label which family each portrait corresponds to and note any equilibria at infinity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our results on quadratic Cremona maps applied to systems with algebraic limit cycles. We address each major comment below and will revise the manuscript to incorporate the requested explicit verifications.

read point-by-point responses

  1. Referee: [Abstract / § on the new family] The central technical claim (that a quadratic Cremona map applied to a quadratic vector field yields, after clearing denominators, another quadratic vector field whose algebraic limit cycle remains isolated and of the stated degree) is stated in the abstract and paragraph 2 but is not accompanied by an explicit computation of the denominator cancellation for the new degree-5 family. Without this verification, it is impossible to confirm that the degree remains exactly 2 rather than rising.

    Authors: We agree that an explicit computation of the denominator cancellation is essential to rigorously confirm that the transformed vector field remains quadratic. In the revised version we will add the full algebraic details of this cancellation for the degree-5 family, including the explicit expressions for the numerator and denominator polynomials and the verification that no common factors remain after clearing. revision: yes

  2. Referee: [Section on recovery of known families] The recovery of known higher-degree families is asserted to follow from the same maps, yet no explicit base system plus map pair is exhibited that reproduces, for instance, a known degree-6 or degree-8 algebraic limit cycle while keeping the transformed field quadratic. This step is load-bearing for the unification claim.

    Authors: We acknowledge that the unification claim would be strengthened by concrete examples. In the revision we will exhibit explicit base quadratic systems together with the corresponding quadratic Cremona maps that recover known families with algebraic limit cycles of degrees 6 and 8, including the verification that the transformed fields remain quadratic after clearing denominators. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit application of external birational maps to known base systems

full rationale

The derivation applies quadratic plane Cremona maps (standard objects in algebraic geometry) to previously classified quadratic systems that already possess algebraic limit cycles. The new degree-5 family and the recovery of known families are obtained by direct substitution and clearing denominators, with the resulting vector field verified to remain quadratic and the image curve verified to remain an isolated algebraic limit cycle. No equation reduces a claimed output to a fitted parameter or to a self-referential definition; no load-bearing uniqueness theorem is imported from the authors' prior work; the central preservation property is established by explicit calculation rather than by ansatz or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard properties of quadratic Cremona maps as birational transformations that preserve polynomial degree two; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Quadratic plane Cremona maps transform quadratic vector fields into quadratic vector fields while mapping algebraic curves to algebraic curves.
    Invoked as the mechanism that produces new quadratic systems from old ones (abstract).

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Reference graph

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