arxiv: 2604.26061 · v1 · submitted 2026-04-28 · 🧮 math.DS · math.CA· math.CV

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Limit cycles in piecewise smooth systems with circular switching manifold

Gabriel Rond\'on , Paulo R. da Silva , Jaume Llibre

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Pith reviewed 2026-05-07 14:08 UTC · model grok-4.3

classification 🧮 math.DS math.CAmath.CV

keywords limit cyclespiecewise smooth systemscircular switching manifoldantiholomorphic systemsMöbius transformationsalgebraic limit cyclesholomorphic systemsrigidity theorem

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The pith

Möbius transformations equate circular and straight switching manifolds in piecewise complex systems, yielding at most three limit cycles for linear antiholomorphic cases and ten for quadratic.

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

The paper examines limit cycles in piecewise smooth complex systems whose discontinuity occurs along the unit circle. Möbius transformations are used to map the circle to a straight line while preserving periods, stability, and algebraic character of orbits, allowing results from linear switching to transfer directly to the circular case. Upper bounds are proved for antiholomorphic piecewise systems, explicit algebraic limit cycles are constructed, and a rigidity theorem rules out crossing limit cycles when both pieces admit holomorphic normal forms. Lower bounds for holomorphic piecewise systems follow from averaging and Lyapunov quantities. These limits constrain the possible periodic behavior in models with circular discontinuities.

Core claim

Möbius transformations establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise antiholomorphic systems this yields at most three limit cycles in the linear case and at most ten in the quadratic case. Explicit algebraic limit cycles are constructed in the circular setting. When both component systems admit classical holomorphic normal forms at the origin, no crossing limit cycles exist. Lower bounds on the number of limit cycles are obtained for piecewise polynomial holomorphic systems via second-order averaging and Lyapunov quantities.

What carries the argument

Möbius transformations that map the circle to a straight line while preserving the periods, stability, and algebraic structure of limit cycles in the piecewise system.

If this is right

Algebraic limit cycles exist explicitly in the circular switching context.
Piecewise polynomial holomorphic systems admit lower bounds on limit cycles obtained from averaging methods.
No crossing limit cycles occur when both pieces have holomorphic normal forms at the origin.
Upper bounds of three and ten limit cycles hold respectively for linear and quadratic piecewise antiholomorphic systems.

Where Pith is reading between the lines

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

The preservation of algebraic structure under the equivalence implies that polynomial limit cycles found in the linear case correspond to algebraic periodic orbits after the Möbius map.
The rigidity result suggests that antiholomorphic terms are required to produce crossing periodic orbits in these piecewise settings.
The explicit algebraic examples supply concrete test cases for numerical continuation methods applied to discontinuous complex systems.

Load-bearing premise

The Möbius transformations must preserve periods, stability, and algebraic structure when mapping circular discontinuities to straight-line ones.

What would settle it

A concrete quadratic piecewise antiholomorphic system with a circular switching manifold possessing eleven or more limit cycles would disprove the stated upper bound.

Figures

Figures reproduced from arXiv: 2604.26061 by Gabriel Rond\'on, Jaume Llibre, Paulo R. da Silva.

**Figure 1.** Figure 1: ). q p r qq ′ ϕ θ view at source ↗

**Figure 2.** Figure 2: Three possible types of equilibrium points lying on the switching manifold S 1 : a center (left), a focus (center), and a saddle (right). The center gives rise to a continuum of periodic orbits that can bifurcate into limit cycles under perturbation (Theorem A). The focus allows for degenerate Hopf bifurcations (Theorem B). The saddle is the typical configuration for piecewise antiholomorphic systems (Theo… view at source ↗

**Figure 3.** Figure 3: The conformal map ϕ(z) = −z+i iz−1 . Consequently instead of analyzing the system with the unit circle as the discontinuity, we may equivalently study a system with the real axis as the discontinuity. In particular, the dynamics of PWHS (3) are equivalent to those of (14) ( w˙ = G+(w), when Im(w) ≥ 0, w˙ = G−(w), when Im(w) ≤ 0, view at source ↗

**Figure 4.** Figure 4: Figure illustrating the effect of a M¨obius transformation on a limit cycle of a piecewise holomorphic system. On the left the limit cycle (red) of the original system passes through a point where the M¨obius transformation ϕ is not defined (e.g., −i). On the right the image of this limit cycle (red) under the transformation ϕ is shown. The transformation ϕ maps the original limit cycle to an orbit that do… view at source ↗

**Figure 5.** Figure 5: Phase portrait of a piecewise holomorphic system with a discontinuity on the unit circle S 1 . The unit circle is shown in black. A limit cycle is highlighted in red, illustrating its behavior as it interacts with the discontinuity. The phase portrait demonstrates how the trajectories behave in the regions inside and outside the unit circle, emphasizing the dynamics of the limit cycle and its interaction … view at source ↗

**Figure 6.** Figure 6: This figure illustrates the effect of M¨obius transformations on periodic orbits in the complex plane. Both the original periodic orbit and the transformed orbit are shown in red. M¨obius transformations not only map periodic orbits to other periodic orbits but also preserve their period. Remark 17. Theorem 16 provides a complete correspondence between the dynamics of piecewise holomorphic systems with ci… view at source ↗

**Figure 7.** Figure 7: The M¨obius transformation ϕ −1 (w) = w + i iw + 1 maps the system with switching manifold R (left) to the system with switching manifold S 1 (right). The limit cycle |w| = 2 (red) is mapped to the algebraic curve 3zz¯ − 5i(z−z¯)+3 = 0, which in real coordinates is the circle x 2+(y+5/3)2 = (4/3)2 (red) view at source ↗

read the original abstract

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise polynomial holomorphic systems we obtain lower bounds on the number of limit cycles via second-order averaging and, for low degrees, via Lyapunov quantities. For piecewise antiholomorphic systems we prove upper bounds: at most $3$ limit cycles in the linear case and $10$ in the quadratic case. We also prove a rigidity theorem: when both components admit classical holomorphic normal forms at the origin no crossing limit cycles exist. Finally, we construct explicit algebraic limit cycles in the circular context, providing, as far as we know the first such examples in the literature.

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

The paper gives upper bounds of 3 and 10 limit cycles for linear and quadratic piecewise antiholomorphic systems on circular manifolds plus the first explicit algebraic examples, all via Möbius reduction to the straight-line case.

read the letter

Hi, the punchline here is that the paper establishes upper bounds of 3 and 10 limit cycles for linear and quadratic piecewise antiholomorphic systems on circular manifolds, along with the first explicit algebraic examples, all via a Möbius equivalence to the straight-line case. It does a few things solidly. The equivalence via Möbius transformations is shown to preserve periods, stability, and algebraic structure, which allows transferring results from the better-understood linear switching case. For holomorphic piecewise systems, the lower bounds come from standard second-order averaging and Lyapunov quantities, which is straightforward. The rigidity result—no crossing limit cycles when both components have holomorphic normal forms—is a nice clean statement. And actually constructing algebraic limit cycles gives something concrete that prior work on straight lines apparently lacked. The potential issue is with the antiholomorphic upper bounds. The stress test points out that Möbius maps are holomorphic, so applying one to an antiholomorphic field might not yield another antiholomorphic field. If the transformed system doesn't keep the piecewise antiholomorphic property or the specific jump conditions that the bounds rely on, then the counting arguments may not apply directly. The abstract claims the equivalence works for algebraic structure, but without seeing the details on how they adapt the Lyapunov quantities or averaging for the antiholomorphic case after transformation, it's hard to be sure the bounds hold. This seems like the main place where a referee would dig in. Overall, this is aimed at researchers in piecewise dynamical systems, particularly those dealing with complex or holomorphic vector fields and non-smooth switching. Someone looking for specific bounds or examples in this narrow setting would get value from the constructions. It should go to peer review. The results are specific enough and build on existing techniques, even if the key reduction step needs verification. Cheers

Referee Report

2 major / 2 minor

Summary. The paper studies limit cycles in piecewise complex systems with switching manifold S^1. Using Möbius transformations, it establishes an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise polynomial holomorphic systems, lower bounds are obtained via second-order averaging and Lyapunov quantities for low degrees. For piecewise antiholomorphic systems, upper bounds are proven: at most 3 limit cycles in the linear case and 10 in the quadratic case. A rigidity theorem states that when both components admit classical holomorphic normal forms at the origin, no crossing limit cycles exist. Explicit algebraic limit cycles are constructed in the circular context, claimed as the first such examples.

Significance. If the Möbius equivalence and the derived bounds hold, the work provides concrete upper and lower bounds on limit cycles in a class of piecewise smooth complex systems, along with the first explicit algebraic examples. The approach combines standard tools (averaging, Lyapunov quantities, Möbius maps) with new constructions, which could serve as benchmarks for further research in discontinuous dynamical systems. The rigidity result also clarifies structural obstructions in the holomorphic case.

major comments (2)

[§2] §2 (Möbius equivalence): The claim that Möbius transformations preserve antiholomorphicity when mapping the circular manifold to a straight line is load-bearing for the upper-bound results. Since Möbius maps are holomorphic, their action on an antiholomorphic vector field produces a transformed field whose piecewise antiholomorphic character must be verified explicitly; the paper needs to show that the jump condition and the form of the Lyapunov quantities remain compatible with the hypotheses used to obtain the bounds of 3 (linear) and 10 (quadratic).
[§4] §4 (upper bounds for antiholomorphic systems): After the reduction to the linear case, the counting argument yielding at most 3 limit cycles relies on the transformed system satisfying the same structural assumptions as the original antiholomorphic setup. A concrete check (e.g., explicit transformation of the vector-field components and verification that the resulting discontinuity is still antiholomorphic) is required; without it the bound does not transfer.

minor comments (2)

[Abstract] Abstract: the phrase 'as far as we know the first such examples' should be replaced by a more formal statement such as 'to the best of our knowledge, the first explicit algebraic limit cycles in this setting'.
[Throughout] Notation: ensure consistent use of the symbol for the switching manifold (S^1 vs. unit circle) throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments on the Möbius equivalence and the transfer of upper bounds. We address each point below and will incorporate the requested explicit verifications into the revised version.

read point-by-point responses

Referee: [§2] §2 (Möbius equivalence): The claim that Möbius transformations preserve antiholomorphicity when mapping the circular manifold to a straight line is load-bearing for the upper-bound results. Since Möbius maps are holomorphic, their action on an antiholomorphic vector field produces a transformed field whose piecewise antiholomorphic character must be verified explicitly; the paper needs to show that the jump condition and the form of the Lyapunov quantities remain compatible with the hypotheses used to obtain the bounds of 3 (linear) and 10 (quadratic).

Authors: We agree that an explicit verification of the preservation of antiholomorphicity is necessary to make the argument fully rigorous. In the revised manuscript we will expand Section 2 with a direct computation: let φ be the Möbius map sending the unit circle to the real line. For a piecewise antiholomorphic vector field (F⁺, F⁻) with F⁺, F⁻ antiholomorphic, the transformed components are obtained by the chain rule involving φ′ and the conjugate variables. Because φ is holomorphic, the antiholomorphic dependence on the conjugate coordinate is preserved after composition with φ and φ⁻¹; the jump across the image line remains of antiholomorphic type. We will also show that the Lyapunov quantities transform by a non-vanishing analytic factor, so the algebraic conditions used for the bounds of 3 and 10 carry over unchanged. This explicit check will be added before the statement of the equivalence theorem. revision: yes
Referee: [§4] §4 (upper bounds for antiholomorphic systems): After the reduction to the linear case, the counting argument yielding at most 3 limit cycles relies on the transformed system satisfying the same structural assumptions as the original antiholomorphic setup. A concrete check (e.g., explicit transformation of the vector-field components and verification that the resulting discontinuity is still antiholomorphic) is required; without it the bound does not transfer.

Authors: We accept that the counting argument in the linear antiholomorphic case (and its quadratic extension) requires a concrete verification that the reduced system remains piecewise antiholomorphic. In the revision we will insert, immediately after the Möbius reduction in Section 4, an explicit example for the linear case: we apply the standard map φ(z) = (z − i)/(z + i) to a general linear antiholomorphic system with circular discontinuity, compute the new vector-field expressions on each side of the real line, and confirm that each piece is still antiholomorphic while the jump condition is preserved. The same verification will be sketched for the quadratic case. With these calculations in place, the structural hypotheses of the counting lemma are satisfied by the transformed system, allowing the bound of at most 3 (respectively 10) limit cycles to transfer directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent complex analysis and averaging

full rationale

The paper establishes an equivalence via Möbius transformations (standard in complex geometry) between circular and linear switching manifolds, then applies second-order averaging and Lyapunov quantities to bound limit cycles in piecewise holomorphic and antiholomorphic systems. These steps use external theorems from complex analysis and averaging theory without reducing any central claim (e.g., the 3/10 bounds or algebraic cycle constructions) to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. No equations or arguments loop back to the paper's own inputs by construction; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard results from complex analysis and averaging theory without introducing new free parameters or postulated entities.

axioms (2)

standard math Möbius transformations preserve periods, stability, and algebraic structure when mapping circular to linear discontinuities
Invoked to establish equivalence between the two classes of systems.
standard math Second-order averaging and Lyapunov quantities detect limit cycles in piecewise polynomial systems
Applied to obtain lower bounds for holomorphic cases.

pith-pipeline@v0.9.0 · 5435 in / 1339 out tokens · 80625 ms · 2026-05-07T14:08:55.235093+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references

[1]

Acary, O

V. Acary, O. Bonnefon, and B. Brogliato.Nonsmooth modeling and simulation for switched circuits., volume 69 ofLect. Notes Electr. Eng.Dordrecht: Springer, 2011

2011
[2]

G. K. Batchelor.An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge Uni- versity Press, Cambridge, paperback edition, 1999

1999
[3]

A. F. Beardon.The Geometry of Discrete Groups, volume 91 ofGraduate Texts in Mathematics. Springer- Verlag, New York, 1983

1983
[4]

I. S. Berezin and N. P. Shidkov.Computing methods. Vols. I, II. Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965

1965
[5]

Brickman and E

L. Brickman and E. S. Thomas. Conformal equivalence of analytic flows.J. Differential Equations, 25(3):310–324, 1977

1977
[6]

Brogliato.Nonsmooth mechanics

B. Brogliato.Nonsmooth mechanics. Models, dynamics and control. Commun. Control Eng. Cham: Springer, 3rd edition edition, 2016

2016
[7]

C. A. Buzzi, A. Gasull, and J. Torregrosa. Algebraic limit cycles in piecewise linear differential systems. International Journal of Bifurcation and Chaos, 28(03):1850039, 2018

2018
[8]

A. J. Chorin and J. E. Marsden.A mathematical introduction to fluid mechanics. Springer-Verlag, New York-Heidelberg, 1979

1979
[9]

B. Coll, A. Gasull, and R. Prohens. Bifurcation of limit cycles from two families of centers.Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12(2):275–287, 2005

2005
[10]

J. B. Conway.Functions of one complex variable, volume 11 ofGraduate Texts in Mathematics. Springer- Verlag, New York-Heidelberg, 1973

1973
[11]

J. B. Conway.Functions of one complex variable, volume 11 ofGraduate Texts in Mathematics. Springer- Verlag, New York, second edition, 1978

1978
[12]

A. F. Filippov.Differential equations with discontinuous righthand sides, volume 18 ofMathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian

1988
[13]

Freire, E

E. Freire, E. Ponce, and F. Torres. Canonical discontinuous planar piecewise linear systems.SIAM Journal on Applied Dynamical Systems, 11(1):181–211, 2012

2012
[14]

Garijo, A

A. Garijo, A. Gasull, and X. Jarque. Normal forms for singularities of one dimensional holomorphic vector fields.Electron. J. Differential Equations, pages No. 122, 7, 2004

2004
[15]

Garijo, A

A. Garijo, A. Gasull, and X. Jarque. Local and global phase portrait of equation ˙z=f(z).Discrete Contin. Dyn. Syst., 17(2):309–329, 2007

2007
[16]

Garijo, A

A. Garijo, A. Gasull, and X. Jarque. Simultaneous bifurcation of limit cycles from two nests of periodic orbits.Journal of Mathematical Analysis and Applications, 341:813–824, 05 2008

2008
[17]

Gasull, G

A. Gasull, G. Rond´ on, and P. R. da Silva. On the number of limit cycles for piecewise polynomial holo- morphic systems.SIAM Journal on Applied Dynamical Systems, 23(3):2593–2622, 2024

2024
[18]

Gasull, G

A. Gasull, G. Rond´ on, and P. R. da Silva. Simultaneous bifurcation of limit cycles for piecewise holomorphic systems.Journal of Dynamics and Differential Equations, 2025. Forthcoming

2025
[19]

L. F. Gouveia, G. Rond´ on, and P. R. da Silva. Piecewise holomorphic systems.Journal of Differential Equations, 332:440–472, 2022

2022
[20]

L. F. S. Gouveia, P. R. da Silva, and G. Rond´ on. Global phase portrait and local integrability of holomor- phic systems.Qual. Theory Dyn. Syst., 22(1):Paper No. 35, 26, 2023. 30

2023
[21]

Itikawa, J

J. Itikawa, J. Llibre, and D. D. Novaes. A new result on averaging theory for a class of discontinuous planar differential systems with applications.Rev. Mat. Iberoam., 33(4):1247–1265, 2017

2017
[22]

Kunze.Non-smooth dynamical systems, volume 1744 ofLect

M. Kunze.Non-smooth dynamical systems, volume 1744 ofLect. Notes Math.Berlin: Springer, 2000. 1 Departamento de Matem´atica, Instituto de Ci ˆencias Exatas (ICEx), Universidade Federal de Minas Gerais (UFMG), Av. Pres. Ant ˆonio Carlos, 6627, Cidade Universit´aria - Pampulha, 31270-901, Belo Horizonte, MG, Brazil 2S˜ao Paulo State University (Unesp), Inst...

2000