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arxiv: 2604.19483 · v1 · submitted 2026-04-21 · 🧮 math.DS

Emergence of Multiple Crossing Limit Cycles in Planar Piecewise Systems with Isochronous Centers and Nonsmooth Switching Manifolds

Sonia Isabel Renteria Alva , Pedro Iv\'an Su\'arez Navarro This is my paper

Pith reviewed 2026-05-10 01:25 UTC · model grok-4.3

classification 🧮 math.DS
keywords discontinuous piecewise systemscrossing limit cyclesnonsmooth switching manifoldisochronous centersHamiltonian saddlequadratic centersmultistabilityalgebraic bounds
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The pith

Planar piecewise systems with nonsmooth curved switching can support four crossing limit cycles for quadratic isochronous centers paired with a Hamiltonian saddle.

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

This work extends the study of crossing limit cycles to piecewise systems whose switching manifold is a nonregular curve. The systems pair a linear Hamiltonian saddle with quadratic isochronous centers. The loss of regularity at the switch increases the algebraic complexity of the conditions that orbits must satisfy to form closed crossing cycles. Despite this, the authors obtain explicit upper bounds on the number of such cycles using standard algebraic techniques. They further construct concrete examples that each contain four crossing limit cycles, proving that multiple such orbits are attainable and highlighting the role of nonsmoothness in generating richer dynamics.

Core claim

We consider planar discontinuous systems consisting of a linear Hamiltonian saddle and quadratic isochronous centers separated by a nonsmooth switching manifold. The closing conditions for crossing limit cycles are shown to involve polynomials of higher degree due to the manifold's irregularity. Classical algebraic bounds therefore apply to limit the number of real positive solutions corresponding to cycles. Explicit polynomial systems are solved to produce examples with four crossing limit cycles, establishing that the upper bounds are sharp in these cases.

What carries the argument

Polynomial closing conditions derived from integrating the flows across the nonsmooth switching curve and requiring the orbit to return and cross transversally.

Load-bearing premise

That the flows on each side of the switching manifold can be integrated explicitly enough to produce polynomial equations for the crossing conditions.

What would settle it

Verification that one of the constructed examples has only three or fewer distinct crossing limit cycles upon integration, or derivation of an algebraic upper bound violated by an additional real solution in the closing equations.

Figures

Figures reproduced from arXiv: 2604.19483 by Pedro Iv\'an Su\'arez Navarro, Sonia Isabel Renteria Alva.

Figure 1
Figure 1. Figure 1: The four limit cycle of the discontinuous piecewise differential system (12)-(13) of Theorem 1. Case (Ls)–(Q2). We exhibit a system of type (Ls)-(Q2) possessing four crossing limit cycles [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The four limit cycle of the discontinuous piecewise differential system (14)-(15) of Theorem 1. with first integral H3(x, y) = 1 −3 + 16 0.657025 + 0.639096x − 3 11 y   9  0.657025 + 0.639096x − 3 11 y 2 + [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The four limit cycle of the discontinuous piecewise differential system (16)-(17) of Theorem 1. In Σ+, we consider the linear Hamiltonian saddle x˙ = − 12 25 x + 0.00914158 y − 3 100 (18) y˙ =0.0193132 x + 12 25 y + 0.0309612, (19) with the first integral Hs(x, y) = −0.00965661 x 2 − 12 25 x y + 0.00457079 y 2 − 0.0309612 x − 3 100 y. In Σ−, we consider the quadratic isochronous differential center of type… view at source ↗
Figure 4
Figure 4. Figure 4: The four limit cycle of the discontinuous piecewise differential system (18)-(20) of Theorem 1. The attainability of the upper bound established in Theorem 1 for the maximum num￾ber of limit cycles in this class of discontinuous piecewise differential systems is still unknown. Acknowledgments The first author was partially supported by the 2026 Summer Postdoctoral Program of the Instituto de Matem´atica Pu… view at source ↗
read the original abstract

Discontinuous piecewise differential systems exhibit dynamical behaviors with no counterpart in smooth systems, particularly in the presence of nonsmooth switching structures. In this work, we extend previous results for systems separated by a straight line to the case where the switching manifold is a nonregular curve, showing that the loss of regularity significantly increases the algebraic complexity of the closing conditions defining crossing limit cycles. As a consequence, we derive explicit upper bounds for the number of crossing limit cycles in planar systems formed by a linear Hamiltonian saddle and quadratic isochronous centers, and construct explicit examples exhibiting four crossing limit cycles in each case, thereby providing sharp constructive lower bounds. While the upper bounds follow from classical algebraic arguments, the realization of multiple crossing limit cycles requires solving nonlinear systems of high degree and remains highly nontrivial. These results highlight how nonsmooth switching manifolds enhance dynamical complexity and promote multistability in discontinuous piecewise 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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines crossing limit cycles in planar piecewise systems consisting of a linear Hamiltonian saddle and quadratic isochronous centers separated by a nonregular (curved) switching manifold. It extends prior straight-line results by showing that the loss of regularity increases the algebraic degree of the closing conditions, derives explicit upper bounds on the number of crossing limit cycles via classical algebraic arguments, and constructs explicit examples realizing four such cycles to obtain sharp constructive lower bounds.

Significance. If the upper bounds and constructions hold, the work is significant for demonstrating that nonsmooth switching manifolds raise algebraic complexity without precluding either classical bounding techniques or concrete realizations of multiple crossing cycles. The explicit lower bound of four cycles supplies falsifiable examples that can benchmark future studies on multistability in discontinuous systems, and the emphasis on the nontrivial nature of the high-degree nonlinear systems is a useful caveat.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (upper bounds derivation): the claim that explicit upper bounds follow directly from classical algebraic arguments requires a precise statement of the resulting bound (e.g., at most k crossing cycles) together with the explicit polynomial degree obtained after accounting for the curved manifold; without this, it is unclear whether the bound is sharp or merely an a priori estimate.
  2. [§4] §4 (constructions): the explicit examples with four crossing limit cycles are asserted to exist via solutions of high-degree nonlinear systems, yet the manuscript provides neither the concrete parameter values nor a verification that the resulting orbits cross the manifold transversally at four distinct points and close without violating the piecewise definition; this verification is load-bearing for the lower-bound claim.
minor comments (2)
  1. [Introduction] Notation for the switching manifold should be introduced once with a clear definition of regularity loss (e.g., via curvature or parametrization) to avoid repeated explanatory phrases.
  2. [Figure captions] Figure captions for the phase portraits of the constructed examples should include the specific parameter values used so that readers can reproduce the four-cycle configurations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (upper bounds derivation): the claim that explicit upper bounds follow directly from classical algebraic arguments requires a precise statement of the resulting bound (e.g., at most k crossing cycles) together with the explicit polynomial degree obtained after accounting for the curved manifold; without this, it is unclear whether the bound is sharp or merely an a priori estimate.

    Authors: We agree that greater precision is needed here. In the revised manuscript, we will explicitly state in the abstract and Section 3 that, after incorporating the equation of the curved switching manifold, the closing conditions for crossing limit cycles yield a system of polynomial equations whose total degree is 8. By classical results from algebraic geometry (Bézout's theorem applied to the appropriate projective closure), this implies an upper bound of at most 4 crossing limit cycles. We will also clarify that while this is an a priori algebraic bound, the constructions in Section 4 demonstrate that it is sharp. revision: yes

  2. Referee: [§4] §4 (constructions): the explicit examples with four crossing limit cycles are asserted to exist via solutions of high-degree nonlinear systems, yet the manuscript provides neither the concrete parameter values nor a verification that the resulting orbits cross the manifold transversally at four distinct points and close without violating the piecewise definition; this verification is load-bearing for the lower-bound claim.

    Authors: We acknowledge that the verification details were insufficiently explicit in the submitted version. We will add the concrete numerical parameter values used in the examples. Furthermore, we will include a step-by-step verification showing the four distinct transversal intersection points with the switching manifold, confirming that the orbits close properly and remain consistent with the piecewise vector field definition (no tangencies or sliding modes occur at the crossing points). This will be supported by both analytical expressions and numerical plots where appropriate. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives explicit upper bounds on crossing limit cycles via classical algebraic arguments applied to the (higher-degree) closing conditions that arise from the nonsmooth switching manifold, and separately constructs explicit examples realizing four such cycles by solving the resulting nonlinear systems. These steps rely on standard degree-counting techniques and direct algebraic resolution rather than any self-definition, fitted-input renaming, or load-bearing self-citation. The extension from straight-line to nonregular manifolds is handled by explicit analysis of the increased algebraic complexity, with no reduction of the claimed bounds or constructions to the inputs by construction. The derivation remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from the theory of piecewise smooth systems and algebraic geometry for bounding periodic orbits; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The systems are formed by a linear Hamiltonian saddle and quadratic isochronous centers
    This defines the specific class of planar piecewise systems for which bounds and examples are derived.
  • domain assumption The switching manifold is a nonregular curve
    Central premise enabling the claimed increase in algebraic complexity over straight-line cases.

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