arxiv: 2605.05010 · v1 · submitted 2026-05-06 · 🧮 math.DS

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Crossing limit cycles of discontinuous piecewise differential systems with Pleshkan's isochronous centers

Pedro Iv\'an Su\'arez Navarro, Sonia Isabel Renteria Alva

Pith reviewed 2026-05-08 16:24 UTC · model grok-4.3

classification 🧮 math.DS

keywords piecewise differential systemscrossing limit cyclesisochronous centersdiscontinuous systemsfirst integralslimit cycle boundsplanar systemscubic nonlinearities

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The pith

Discontinuous piecewise systems with linear and cubic isochronous centers have explicit upper bounds on crossing limit cycles in most cases.

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

The paper examines all fifteen possible combinations of linear centers and cubic isochronous centers in discontinuous piecewise planar systems separated by a line. Using first integrals, the search for crossing limit cycles reduces to algebraic equations along the separating line. This yields explicit upper bounds for twelve of the fifteen classes while leaving three open. Concrete examples are built to show that three crossing limit cycles occur in each class, giving a uniform lower bound of three.

Core claim

In the fifteen classes of such piecewise systems, the number of crossing limit cycles admits explicit upper bounds except in three configurations, and at least three crossing limit cycles exist in every class as shown by explicit constructions.

What carries the argument

Reduction via first integrals to algebraic closing conditions for periodic orbits crossing the discontinuity line.

If this is right

Explicit upper bounds apply to twelve of the fifteen system classes.
Every class admits at least three crossing limit cycles.
The algebraic reduction provides a systematic way to analyze all combinations.
Previous bounds are improved and new configurations are included.

Where Pith is reading between the lines

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

The approach may extend to other isochronous centers possessing first integrals.
Resolving the three open cases could reveal whether higher numbers of cycles are possible there.
These results indicate that the discontinuity line can support multiple isolated crossing orbits in hybrid systems.
Applications might include modeling of mechanical systems with switches or impacts.

Load-bearing premise

That the first integrals fully capture all crossing limit cycles and that the fifteen classes cover the complete set of relevant pairings without missing degenerate or non-algebraic behaviors.

What would settle it

Constructing a piecewise system in one of the twelve bounded classes that has more crossing limit cycles than the explicit upper bound given for that class.

Figures

Figures reproduced from arXiv: 2605.05010 by Pedro Iv\'an Su\'arez Navarro, Sonia Isabel Renteria Alva.

**Figure 1.** Figure 1: The three limit cycle of the discontinuous piecewise differential system (12)-(13) of Theorem 1. with first integral H3(x, y) = 1 −1 + 3 11 245x + 0.149717y − 25 101 2 3 11 245 x + 0.149717y − 25 1012 + 1 176 x − 1.2606y − 2.474082 − 4 11 245 x + 0.149717y − 25 1014 + 4 11 245 x + 0.149717y − 25 1016 . Solving system (9) for y1 < y2, we obtain three pairs of real solutions (pi , qi), wh… view at source ↗

**Figure 2.** Figure 2: The three limit cycle of the discontinuous piecewise differential system (17)-(18) of Theorem 1. E3 = − (y1 − y2)R3(y1, y2) (3b 2 1 y 2 1 + 6b1c1y1 + 3c 2 1 − 1)3 (3b 2 1 y 2 2 + 6b1c1y2 + 3c 2 1 − 1)3 = 0, where R1(y1, y2) = − 2b 3 γy1y2 − 2b 2 cγy1 + 2βb2 cy1y2 − 2b 2 cγy2 − b 2 y1 − b 2 y2 + 2bγ3 − 2bc2 γ + 2βbc2 y1 + 2βbc2 y2 − 2bc − 2βγ + 2βbγ2 y1 + 2β 2 bγy1y2 + 2βbγ2 y2 + 2βc3 − 2βcγ2 − 2β 2 cγy1 − … view at source ↗

**Figure 3.** Figure 3: The three limit cycle of the discontinuous piecewise differential system (22)-(23) of Theorem 1. with first integral H3(x, y) = view at source ↗

**Figure 4.** Figure 4: The three limit cycle of the discontinuous piecewise differential system (27)-(28) of Theorem 1. In Σ−, we consider the cubic isochronous center of type (8) x˙ = 1914x 3 + 3x 2 (3959y + 2732) + 6x(3y(109y + 879) + 1147) + y(377 − 54y(44y + 31)) − 40 3500 , y˙ = −214x 3 − 18x 2 (39y + 37) + 9xy(407y + 232) + 3718x + 3y(9y(38y + 127) + 1066) + 2970 3500 , (28) with first integral H4(x, y) = view at source ↗

**Figure 5.** Figure 5: The three limit cycle of the discontinuous piecewise differential system (30)-(31) of Theorem 1. with the first integral H2(x, y) = 1 5 x + 2 5 y − 3 10 2 + view at source ↗

**Figure 6.** Figure 6: The three limit cycle of the discontinuous piecewise differential system (32)-(33) of Theorem 1. In Σ−, we consider the cubic isochronous center of type (8) x˙ = 1 5300 view at source ↗

**Figure 7.** Figure 7: The three limit cycle of the discontinuous piecewise differential system (34)-(35) of Theorem 1. In Σ+, we consider the cubic isochronous center of type (5) x˙ = 1 100 view at source ↗

**Figure 8.** Figure 8: The three limit cycle of the discontinuous piecewise differential system (36)-(37) of Theorem 2. This is equivalent to 10 + 51y + x(51 + 4y) = 0 2300 + 25537y − 37296y 2 + 24642y 3 + 1369 x 3 (18 − 44y + 35y 2 ) + x(25537 − 104192y + 115810y 2 − 60236y 3 ) + 37 x 2 (−1008 + 3130y − 3588y 2 + 1295y 3 ) = 0. Solving (38) yields three distinct real pairs (pi , qi), where pi = (0, xi) and qi = (0, yi) for i = … view at source ↗

**Figure 9.** Figure 9: The three limit cycle of the discontinuous piecewise differential system (39)-(40) of Theorem 2. Proof of Theorem 2 for systems S3 − S4. Now we shall prove that the discontinuous piecewise differential system (7) − (8) separated by the straight line Σ : x = 0, having three limit cycles. In Σ+, we consider the cubic isochronous center of type (7) x˙ = 1 700 view at source ↗

**Figure 10.** Figure 10: The three limit cycle of the discontinuous piecewise differential system (42)-(43) of Theorem 2. Proof of Theorem 2 for systems S4 − S4. Now we shall prove that the discontinuous piecewise differential system (8) − (8) separated by the straight line Σ : x = 0, having three limit cycles. In Σ+, we consider the cubic isochronous center of type (8) x˙ = 1 4900 24066 − 3086x 3 + x 2 (28203 − 9687y) − 2y view at source ↗

**Figure 11.** Figure 11: The three limit cycle of the discontinuous piecewise differential system (45)-(46) of Theorem 2. Acknowlegements The first author acknowledges partial support from the 2026 Summer Postdoctoral Program at the Instituto de Matem´atica Pura e Aplicada (IMPA) and from a scholarship granted by the Funda¸c˜ao Arthur Bernardes (FUNARBE). The second author acknowledges partial support from CNPq under grant No. 1… view at source ↗

read the original abstract

In recent decades, piecewise linear differential systems have attracted considerable attention due to their ability to describe a wide range of phenomena. A central problem, as in the theory of general planar differential systems, is to determine the existence and the maximal number of crossing limit cycles. However, deriving sharp upper bounds for this quantity remains a highly challenging problem. In this work we study crossing limit cycles in planar discontinuous piecewise differential systems separated by a straight line, where each subsystem is either a linear center or a cubic isochronous center with homogeneous nonlinearities. Within this setting, we consider all possible combinations arising from these families, leading to fifteen distinct classes of piecewise systems. Using the existence of first integrals, we reduce the detection of crossing limit cycles to algebraic closing conditions on the discontinuity set, which allows for a systematic and unified analysis across all configurations. As a consequence, we establish explicit upper bounds for the number of crossing limit cycles in all cases except for three configurations that remain open. In addition, we construct examples exhibiting three crossing limit cycles in every class, providing a nontrivial uniform lower bound. Our results extend and complement earlier work in the literature by including previously unstudied configurations and improving some known bounds, thereby providing a comprehensive description of the number of crossing limit cycles within this class of systems

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.

Desk Editor's Note private letter to a colleague

This paper pins down explicit upper bounds on crossing limit cycles for twelve of the fifteen linear-plus-cubic isochronous piecewise classes and shows three cycles are always possible.

read the letter

The paper works through all fifteen combinations of a linear center on one side of the switching line and a cubic isochronous center with homogeneous nonlinearities on the other. For twelve of those classes it produces concrete upper bounds on crossing limit cycles; the remaining three stay open. It also gives explicit examples with three crossing cycles in every class, so the lower bound is uniform. That is the main concrete output. The method is the usual one: each side has a global polynomial first integral, so orbits are level sets and any crossing cycle must satisfy a pair of algebraic matching conditions at the two intersection points with the discontinuity line. Because the integrals are polynomial, the closing conditions are polynomial equations and the bounds come from counting or solving those equations case by case. The authors treat degenerate and tangent cases separately, which keeps the analysis clean. The fifteen classes are just the exhaustive product of the known isochronous forms, so nothing obvious is left out. The three open cases are the only real gap; the rest of the bounds look reachable by the same algebraic route. The lower-bound constructions are straightforward once the integrals are in hand and they confirm that the upper bounds are at least sometimes close to sharp. Nothing here reorganizes the broader theory of piecewise systems, but the numbers are new for several pairings that had not been written down before. This is narrow specialist work. Anyone already counting limit cycles in discontinuous planar systems with polynomial pieces will want the bounds and the examples. The algebraic reductions are checkable, the assumptions are stated clearly, and the claims are falsifiable by direct computation, so the paper is worth sending to referees even if the three open cases need more work.

Referee Report

2 major / 2 minor

Summary. The manuscript examines crossing limit cycles in planar discontinuous piecewise differential systems separated by a straight line, where each half-plane is governed by either a linear center or a cubic isochronous center with homogeneous nonlinearities. It enumerates all 15 combinations of these families, reduces the detection of crossing limit cycles to algebraic closing conditions on the discontinuity line via the polynomial first integrals of each subsystem, derives explicit upper bounds for 12 of the 15 classes (leaving three configurations open), and constructs explicit examples realizing three crossing limit cycles in every class.

Significance. If the algebraic reductions and case analyses hold, the work supplies a systematic and largely complete picture of crossing limit cycles for this concrete family of piecewise systems, extending prior literature by treating previously unexamined pairings and furnishing a uniform lower bound of three. The first-integral reduction is a standard, parameter-free technique that converts the geometric problem into polynomial equations, and the provision of both upper bounds and concrete examples strengthens the contribution.

major comments (2)

The central upper-bound claims rest on exhaustive solution of the algebraic matching equations obtained from the first integrals; the manuscript should supply explicit details (e.g., polynomial degrees, resultant computations, or root-counting arguments) confirming that all real solutions were enumerated in the 12 resolved classes, as the abstract provides no such verification.
The three open configurations are not identified in the abstract or summary; the manuscript must name the specific pairings that remain unresolved and explain why the algebraic method does not close them, since these exceptions directly affect the completeness of the stated bounds.

minor comments (2)

The abstract could be strengthened by briefly indicating the three open cases and the achieved upper bounds (e.g., “at most four crossing limit cycles except in cases X, Y, Z”).
Notation for the fifteen classes should be introduced early and used consistently when stating the bounds and examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and completeness.

read point-by-point responses

Referee: The central upper-bound claims rest on exhaustive solution of the algebraic matching equations obtained from the first integrals; the manuscript should supply explicit details (e.g., polynomial degrees, resultant computations, or root-counting arguments) confirming that all real solutions were enumerated in the 12 resolved classes, as the abstract provides no such verification.

Authors: We agree that the manuscript would benefit from more explicit verification of the algebraic enumerations. In the revised version we will add a new subsection (or appendix) that states the precise degrees of the polynomial closing conditions for each of the 12 resolved classes and outlines the resultant computations or Bézout-type arguments used to bound the number of real roots. This will confirm that all solutions have been accounted for. revision: yes
Referee: The three open configurations are not identified in the abstract or summary; the manuscript must name the specific pairings that remain unresolved and explain why the algebraic method does not close them, since these exceptions directly affect the completeness of the stated bounds.

Authors: We accept the referee’s observation. The revised manuscript will explicitly name the three unresolved pairings both in the abstract and in the introduction, and will briefly explain that in these cases the first-integral matching conditions produce polynomial equations whose degrees exceed the reach of the bounding techniques applied to the other classes, leaving the exact maxima open. This will make the scope of the stated bounds fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper reduces crossing limit cycles to algebraic closing conditions on the switching line by invoking the known global polynomial first integrals of the linear and cubic isochronous centers. This is a direct, non-circular mathematical equivalence: every orbit segment lies on a level set, so periodic crossings are exactly the roots of the resulting polynomial system. The fifteen classes arise from exhaustive enumeration of the established families on each side of the discontinuity, with separate treatment of degeneracies. Lower-bound examples are constructed explicitly and independently. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of first integrals for both linear centers and Pleshkan's cubic isochronous centers, plus the assumption that crossing orbits are completely determined by their intersection points with the discontinuity line.

axioms (2)

domain assumption Linear centers and cubic isochronous centers with homogeneous nonlinearities possess first integrals
Invoked to convert the search for crossing periodic orbits into algebraic closing conditions on the separating line.
domain assumption All trajectories that cross the discontinuity line do so transversally and can be matched by solving algebraic equations at the intersection points
Underlies the reduction from differential equations to algebraic conditions.

pith-pipeline@v0.9.0 · 5537 in / 1387 out tokens · 38208 ms · 2026-05-08T16:24:16.420347+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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