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arxiv: 2606.17780 · v1 · pith:E6ZVC2SJnew · submitted 2026-06-16 · 🧮 math.DS

A symmetric mechanism for symmetry-breaking in oscillator networks with strong nonlinear coupling

Theodore Vo , Yangyang Wang This is my paper

Pith reviewed 2026-06-26 22:45 UTC · model grok-4.3

classification 🧮 math.DS
keywords symmetry-breakingoscillator networksnonlinear couplingcanard dynamicsfolded nodegeometric singular perturbation theoryblow-up techniquemixed-mode oscillations
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The pith

Symmetry-breaking in oscillator networks with strong nonlinear coupling originates from canard dynamics of a symmetric folded node.

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

This paper shows that in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling, symmetry-breaking arises from the canard dynamics of a folded node on the symmetry axis. A reader would care because this reveals a geometric mechanism that allows symmetric systems to generate asymmetric dynamics such as mixed-mode oscillations. Using geometric singular perturbation theory and the blow-up technique on a normal form, the analysis identifies that the fold curve is orthogonal to the symmetry axis at this node, a single primary maximal canard lies on the axis serving as the rotation center, and the number of rotations around it is what breaks the symmetry. The results are applied to a eukaryotic cell cycle model exhibiting periodic and quasi-periodic symmetry-breaking attractors.

Core claim

The symmetry-breaking originates from the canard dynamics of a folded node that lies on the axis of symmetry. Applying geometric singular perturbation theory and blow-up to the normal form reveals that the fold curve is orthogonal to the axis at the symmetric folded node, there is only one primary maximal canard on the axis which is the axis of rotation for twisting solutions, and the number of rotations is the key local diagnostic feature that breaks the symmetry. This mechanism generates small-amplitude oscillations in mixed-mode dynamics of the cell cycle model.

What carries the argument

The symmetric folded node and its associated maximal canard, which induces twisting of solutions whose number of rotations breaks symmetry.

Load-bearing premise

The blow-up technique applied to the normal form of the strongly nonlinearly coupled system correctly captures the local twisting behavior near the symmetric folded node.

What would settle it

A simulation or experiment where the symmetric folded node is removed or perturbed such that the fold curve is no longer orthogonal to the axis, resulting in no symmetry-breaking despite the symmetric setup.

Figures

Figures reproduced from arXiv: 2606.17780 by Theodore Vo, Yangyang Wang.

Figure 1
Figure 1. Figure 1: Representative SSB MMO attractors in a model for the eukaryotic cell cycle (see Section 5). (a) Periodic attractor. Oscillator 1 (red) exhibits two large-amplitude spikes per period, whereas oscillator 2 (blue) exhibits one spike per period, the timing of which occurs approximately in between the two spike events of oscillator 1. (b) Quasi-periodic attractor. The number of large-amplitude spikes varies fro… view at source ↗
Figure 2
Figure 2. Figure 2: Critical manifold, S, of the layer problem in a neighbourhood of the origin (black marker) for representative values of λ1, a and b. The (blue) attracting sheet, Sa, and (red) saddle sheet, Ss, are separated by the (solid black) fold curve, SL. Also shown are the intersections of the symmetry and anti-symmetry axes, Ls and Las, with the critical manifold (dashed black curves). In this class of coupled osci… view at source ↗
Figure 3
Figure 3. Figure 3: Reduced flow (9) of the SFN in a neighbourhood of the origin. The diagonal and anti-diagonal (dashed lines) correspond to the symmetry and anti-symmetry axes, Ls and Las, respectively, onto the (u1, u2) plane. The region below (resp., above) the fold curve SL corresponds to the attracting (resp., saddle) sheet of the critical manifold. There is only one singular canard solution, which coincides with Ls. (a… view at source ↗
Figure 4
Figure 4. Figure 4: Critical manifold, S, of the normal form (5) in the (ˆv1, vˆ2, u2) projection. For each fixed ˆv1 < 0, S is cubic￾shaped with a two-fold structure. These two folds coalesce at ˆv1 = 0 at the cusp point (black marker), corresponding to the symmetric folded singularity. For each fixed ˆv1 > 0, S is no longer folded. folded singularities. We refer forward to Section 5.4 for examples of such bifurcations in a … view at source ↗
Figure 5
Figure 5. Figure 5: Critical manifold (blue and red surfaces) and reduced flow (black curves with arrows) for the system restricted to the center manifold (13) for the (a) strong SFN and (b) weak SFN. The only singular canard solution (the strong canard γ0 in (a) and the weak canard γw in (b)) is aligned with the v1-axis, which corresponds to the axis of symmetry. The funnel of the SFN is (locally) the entire attracting manif… view at source ↗
Figure 6
Figure 6. Figure 6: Dynamics of the symmetric folded node S 3 in the blown-up coordinates  x, ± p y 2 + z 2 , ε  for the (a) strong case and (b) weak case. There is a special solution Γ (Section 3.1) corresponding to the x-axis that connects an equilibrium point p1 on the equator of the hemisphere in chart K1 (Section 3.2) to another equilibrium point p3 on the equator of the hemisphere in chart K3 (Section 3.3). Solutions … view at source ↗
Figure 7
Figure 7. Figure 7: Linearly independent solutions ψ2,1 (blue) and ψ2,2 (red) for (a) µ ∈ (0, 1), (b) µ ∈ (1, 2), and (c) µ ∈ (2, 3). For the weak SFN, i.e., for λ2 < λ1 < 0, if λ2 λ1 is an integer, then one of the solutions in (21) simplifies to a Hermite polynomial. More precisely, for λ2 λ1 = 2k − 1 with k ∈ N \ {1}, ψ2,1 = 1F1(−k, 1 2 , τ 2 2 ) = (−1)k k! (2k)!H2k(τ2), where Hn(τ2) is a Hermite polynomial of degree n. Sim… view at source ↗
Figure 8
Figure 8. Figure 8: Attracting slow manifold near a strong SFN (black marker) in the 4D coupled oscillator normal form (12) for λ1 = −2, λ2 = −0.5, a = 0.1, b = −0.2, and ε = 0.025 with all other parameters set to zero. (a) The transition past the strong SFN causes the two halves of the slow manifold, Sba,ε and Sea,ε, to reverse their ‘polarities’ relative to the strong canard γ ε 0. (b) Zoom of Sba,ε near the SFN. (c) Zoom o… view at source ↗
Figure 9
Figure 9. Figure 9: Attracting slow manifold, Sea,ε = ∪iTi, of the normal form (12) near the weak SFN. Here, Ti denotes the subset of Sba,ε that exhibits i twists about γ ε w. The bold curves that separate the twist regions are the maximal canards, and were computed using the methods in Appendix C. The parameters are λ1 = −0.275, λ2 = −2, and ε = 0.001 with all other parameters set to zero. Since µ = λ2 λ1 ≈ 7.273 ∈ (7, 8), t… view at source ↗
Figure 10
Figure 10. Figure 10: Geometric structure of the cell cycle model for the standard parameters with α = 0.0592. (a) Projection of S (red and blue surfaces) into the positive (A1, A2, I1) orthant. The fold curves (L u , Ld , Lℓ , Lr ) separate the attracting sheets (S u a , Sd a, Sℓ a, Sr a) from the saddle sheet Ss. There is a weak SFN (black marker, Mn sym) on L d , a symmetric saddle equilibrium (blue marker, E) on Ss, and a … view at source ↗
Figure 11
Figure 11. Figure 11: Classification of the folded singularities of (27). (a) Eigenvalue ratio, µ, of the symmetric and asymmetric folded singularities as functions of α. Where appropriate, segments of the branches of symmetric ordinary singularities (purple curves) have been included to show the transcritical and pitchfork structures of the folded saddle-node II and III bifurcations, respectively. The symmetric folded singula… view at source ↗
Figure 12
Figure 12. Figure 12: Local geometry of (27) for the Z2-symmetric FSN bifurcations. The blue and red regions correspond to Sa and Ss, respectively. Folded singularities (black markers) lie in the intersections of the black fold curve L, dashed red {F1 = 0} contour, and dashed green {F2 = 0} contour. Ordinary singularities (blue squares) lie in {F1 = 0} ∩ {F2 = 0} ∩ {G1 = 0} ∩ {G2 = 0}, where Gk := gk(Ak, IkS) for k = 1, 2, and… view at source ↗
Figure 13
Figure 13. Figure 13: Dynamics of system (26). (a) Bifurcation diagram with respect to α. The symmetric equilibria, Esym (black curves), exist to the right of the SNIC bifurcation. The lower Esym branch undergoes a subcritical HB, from which a (green) branch of AP limit cycles emerges. The AP branch is unstable between the HB and TR bifurcations, and stable to the left of the TR bifurcation. Between the TR and SNIC points, the… view at source ↗
Figure 14
Figure 14. Figure 14: Local twisting of solutions about γ ε w for α = 0.0592. Since the SFN (black marker) is weak with µ ≈ 1.52, there are up to 3 twists. The secondary canards γ ε 1 (green) and γ ε 2 (orange) separate the subsets of S d a,ε with different numbers of twists. The star, triangle, and square markers correspond to initial conditions in the different twist ribbons. (a) Projection into the (A1, A2) plane. (b) Proje… view at source ↗
Figure 15
Figure 15. Figure 15: Geometric deconstruction of the periodic SSB (11.5 , 2 1.5 ) MMO for α = 0.0592. (a) Time series showing two full periods together with markers corresponding to key points in the geometric deconstruction. (b) Projection of the dynamics, restricted to the physically meaningful domain, into the (A1, A2) plane together with the SSB MMO attractor (blue trajectory). The geometry of the critical manifold and th… view at source ↗
Figure 16
Figure 16. Figure 16: Deconstructions of a representative periodic AP MMO (left column) for α = 0.05946 and a representative quasi-periodic SSB MMO for α = 0.05937 (right column). Top row: projections into the (A1, A2) plane. Bottom row: projections into the ( A1+A2 2 , A1−A2 2 ) plane in a small neighbourhood of γw. (a) The global return alternately projects orbits to a point p (resp., Rp) in the funnel in the lower (resp., u… view at source ↗
Figure 17
Figure 17. Figure 17: Dynamics near the SFN for α ≈ αSNIC. (a) For α = 0.062 < αSNIC, the slow nullclines (cyan and magenta curves) have no intersections. As such, there is nothing to prevent solutions from tunnelling through the SFN along the weak canard γ ε w ⊂ Ls. The attractor (thick black curve) is a quasi-periodic SSB MMO with small￾amplitude oscillations and interspike intervals generated by the canard dynamics around t… view at source ↗
Figure 18
Figure 18. Figure 18: Bifurcations of the ordinary and folded singularities of the desingularized system (27). The thin black vertical line along α = 0.0592 indicates the α value used in Figs. 10, 13(b), 14, and 15. There are branches of symmetric and asymmetric ordinary singularities, Esym and Easym, respectively, as well as branches of symmetric and asymmetric folded singularities, Msym and Masym, respectively. These branche… view at source ↗
Figure 19
Figure 19. Figure 19: Dynamics of the coupled cell model (26) for the standard parameter set. (a) Partial bifurcation diagram showing only the biologically relevant states, i.e., those with non-negative values. Stability/instability is indicated by solid/dashed curves. The symmetric equilibria, Esym (black curves), undergo Hopf bifurcations, which give rise to anti-phase (AP) limit cycles (green curve), and pitchfork bifurcati… view at source ↗
read the original abstract

In this article, we describe and analyse a novel mechanism for symmetry-breaking in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling. We show that the symmetry-breaking, surprisingly, originates from the canard dynamics of a folded node that lies on the axis of symmetry. By applying geometric singular perturbation theory and the blow-up technique to a normal form, we determine the geometric mechanisms by which the {\em symmetric folded node} induces symmetry-breaking. More specifically, we show that (i) the fold curve of the coupled system is orthogonal to the axis of symmetry at the symmetric folded node; (ii) there is only one primary maximal canard (either strong or weak, depending on parameters), which always lies on the axis of symmetry and is the axis of rotation for the twisting of solutions; and (iii) the number of rotations is the key local diagnostic feature that breaks the symmetry. Our work is closely related to that of Kristiansen and Pedersen [SIAM J. Appl. Dyn. Syst., {\bf 22} (2023)] on symmetrically coupled FitzHugh-Nagumo oscillators with strong linear inhibitory gap junctional coupling, however, we consider nonlinear coupling and we identify and study multiple sub-types of their `cusped singularities'. We demonstrate our theoretical results by applying them to a model of the eukaryotic cell cycle in which the symmetric folded node plays a key role in rhythmogenesis. More specifically, we study periodic and quasi-periodic symmetry-breaking mixed-mode oscillatory attractors of the cell cycle model. We show that the local twisting induced by the symmetric folded node is the local mechanism that both breaks the symmetry and generates the small-amplitude oscillations in the mixed-mode dynamics.

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

1 major / 2 minor

Summary. The manuscript describes a novel mechanism for symmetry-breaking in minimal symmetrically coupled identical slow/fast oscillator networks with strong nonlinear mutually inhibitory coupling. The symmetry-breaking originates from the canard dynamics of a folded node on the axis of symmetry. Using geometric singular perturbation theory and the blow-up technique applied to a normal form, the authors show that the fold curve is orthogonal to the symmetry axis at the symmetric folded node, that there is only one primary maximal canard (strong or weak) lying on the axis of symmetry and serving as the rotation axis for twisting solutions, and that the number of rotations is the key local diagnostic that breaks symmetry. The work extends Kristiansen and Pedersen (2023) from linear to nonlinear coupling by identifying multiple sub-types of cusped singularities. The results are demonstrated on a eukaryotic cell cycle model, where the symmetric folded node generates periodic and quasi-periodic symmetry-breaking mixed-mode oscillatory attractors.

Significance. If the central geometric claims hold, the paper provides a surprising and technically detailed explanation of how symmetry-breaking can emerge locally from canard dynamics even in fully symmetric systems. The application of blow-up to analyze twisting around the symmetric folded node, together with the concrete demonstration in the cell cycle model, strengthens the contribution. The explicit identification of sub-types of cusped singularities and the link between rotation count and symmetry-breaking are potentially useful for other oscillator networks.

major comments (1)
  1. [Normal-form derivation and blow-up analysis (geometric analysis section)] The central claim that the blow-up of the normal form correctly captures the local twisting behavior near the symmetric folded node (and that the symmetry axis remains invariant under the leading-order blown-up dynamics) is load-bearing for all three numbered results (i)–(iii) in the abstract. The manuscript should exhibit the explicit blown-up vector field after the normal-form reduction for the nonlinear coupling and confirm that no retained non-generic terms alter the canard manifold or the rotation count.
minor comments (2)
  1. [Introduction and related-work discussion] Clarify the precise definition and classification criteria for the multiple sub-types of cusped singularities introduced in the nonlinear-coupling case, and indicate how they reduce to the linear-coupling case of Kristiansen and Pedersen (2023).
  2. [Cell-cycle model section] In the cell-cycle application, state explicitly which parameter values place the system near the symmetric folded node and which diagnostic (rotation count) is used to distinguish the periodic versus quasi-periodic mixed-mode attractors.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of the blow-up analysis in supporting the central claims. We address the single major comment below.

read point-by-point responses

  1. Referee: [Normal-form derivation and blow-up analysis (geometric analysis section)] The central claim that the blow-up of the normal form correctly captures the local twisting behavior near the symmetric folded node (and that the symmetry axis remains invariant under the leading-order blown-up dynamics) is load-bearing for all three numbered results (i)–(iii) in the abstract. The manuscript should exhibit the explicit blown-up vector field after the normal-form reduction for the nonlinear coupling and confirm that no retained non-generic terms alter the canard manifold or the rotation count.

    Authors: We agree that an explicit display of the blown-up vector field strengthens the verification of the geometric claims. In the revised manuscript we will insert the full leading-order blown-up vector field obtained after the normal-form reduction for the nonlinear coupling. We will also add a short paragraph confirming that the retained non-generic terms at this order preserve the invariance of the symmetry axis, leave the primary maximal canard on the axis, and do not change the rotation count beyond the classification into sub-types of cusped singularities already given in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard GSPT and blow-up to normal form without reduction to inputs or self-citation loops

full rationale

The paper's central claims—that symmetry-breaking originates from canard dynamics of a symmetric folded node—are obtained by applying geometric singular perturbation theory and the blow-up technique to a normal form of the coupled oscillator system. This is a standard, independent analytic procedure whose outputs (fold curve orthogonality, unique primary maximal canard on the symmetry axis, rotation count as diagnostic) are not equivalent to the inputs by construction. The only citation is to external prior work by Kristiansen and Pedersen (different authors) on linear coupling; the present paper extends it to nonlinear coupling and sub-types of cusped singularities without invoking self-citations as load-bearing premises. No fitted parameters are renamed as predictions, no ansatz is smuggled via self-citation, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the applicability of geometric singular perturbation theory and the blow-up method to the normal form; these are standard domain tools rather than new axioms invented for the paper. No free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Geometric singular perturbation theory and the blow-up technique apply to the normal form of the coupled slow/fast system near the symmetric folded node
    Invoked to determine the geometric mechanisms (i), (ii), (iii) listed in the abstract

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