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arxiv: 2604.06570 · v1 · submitted 2026-04-08 · 🧮 math.DS · nlin.CD

Boundary Hopf bifurcations in three-dimensional Filippov systems

David J.W. Simpson This is my paper

Pith reviewed 2026-05-10 18:40 UTC · model grok-4.3

classification 🧮 math.DS nlin.CD
keywords boundary Hopf bifurcationFilippov systemspiecewise-linear mapsgrazing-sliding bifurcationsdiscontinuity maplimit cyclesbifurcation diagrams
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The pith

In three-dimensional Filippov systems, boundary Hopf bifurcations reduce to dynamics governed by a two-parameter family of piecewise-linear maps.

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

The paper examines boundary Hopf bifurcations, which are codimension-two events on switching surfaces in piecewise-smooth systems. It establishes that in three dimensions the ensuing grazing-sliding bifurcations have local dynamics fully captured by a two-parameter family of piecewise-linear maps. This reduction follows from the fact that sliding motion on the switching surface lowers the effective dimension while the stability of the newly born limit cycle becomes degenerate exactly at the Hopf point. Explicit formulas are given that express the two map parameters directly in terms of quantities evaluated at the boundary Hopf point. Numerical study of the family reveals its attractors, which can include chaotic sets, and the approach is demonstrated on a pest control model and a food chain model with threshold harvesting.

Core claim

We show that for three-dimensional Filippov systems only a two-parameter family of piecewise-linear maps is relevant, because sliding motion induces a loss of dimension, and the stability of the limit cycle is degenerate at the Hopf bifurcation. We derive explicit formulas for the two parameters in terms of quantities associated with the boundary Hopf bifurcation, and perform a comprehensive numerical analysis to characterise the attractor of the family, which may be chaotic. The results are illustrated with a pedagogical example, a pest control model, and a model of a food chain with threshold-based harvesting. In this paper we present a new, simpler derivation of the linear term of thedis-

What carries the argument

the two-parameter family of piecewise-linear maps obtained by reducing the grazing-sliding dynamics through dimension loss from sliding and stability degeneracy at the boundary Hopf point

Load-bearing premise

Sliding motion on the switching surface induces a loss of dimension and the stability of the limit cycle is degenerate at the boundary Hopf bifurcation point.

What would settle it

An explicit three-dimensional Filippov example in which the local map governing a grazing-sliding bifurcation from a boundary Hopf point requires three or more independent parameters that cannot be expressed by the given formulas.

Figures

Figures reproduced from arXiv: 2604.06570 by David J.W. Simpson.

Figure 1
Figure 1. Figure 1: A phase portrait of a Filippov system (1.1) subject to assu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A sketch illustrating the basic unfolding of a boundary Hopf [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The phase space of (1.1) in the Y -coordinate form (3.4). We show a sample orbit with a sliding segment, and its virtual extension to the point Y (1) ∈ Ω. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A two-parameter bifurcation diagram of the family (4.1) an [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A magnification of Fig. 4 and additional sample phase portra [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A two-parameter bifurcation diagram and representativ [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Panel (a) shows the values of τL, τR, and δR for the grazing-sliding bifurcation curve GS of [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A one-parameter bifurcation diagram of (5.1) with [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A two-parameter bifurcation diagram and representativ [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A one-parameter bifurcation diagram of the food chain m [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A two-parameter bifurcation diagram and representat [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A one-parameter bifurcation diagram of the food chain m [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Numerically computed parameter values of [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A magnified view of Fig. 11 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A sketch of the phase space of the n-dimensional system (B.1) subject to the assumptions in Proposition B.1. The point X(2) belongs to Σ, and by evolving this point forwards under FR for a time T, we arrive at the point X(1) ∈ Ω (the trajectory from X(1) to X(2) is virtual because it is situated on the H(X) < 0 side of Σ). By evolving X(2) under the sliding vector field for the same length of time T, we a… view at source ↗
read the original abstract

For piecewise-smooth ordinary differential equations, the occurrence of a Hopf bifurcation on a switching surface is known as a boundary Hopf bifurcation. Boundary Hopf bifurcations are codimension-two, so occur at points in two-parameter bifurcation diagrams. From any such point there issues a curve of grazing bifurcations, where the limit cycle born in the Hopf bifurcation hits the switching surface. For Filippov systems, these are usually grazing-sliding bifurcations whose local dynamics are dictated by piecewise-linear maps. In general, these maps have many independent parameters and extraordinarily rich dynamical behaviour. We show that for three-dimensional Filippov systems only a two-parameter family of piecewise-linear maps is relevant, because sliding motion induces a loss of dimension, and the stability of the limit cycle is degenerate at the Hopf bifurcation. We derive explicit formulas for the two parameters in terms of quantities associated with the boundary Hopf bifurcation, and perform a comprehensive numerical analysis to characterise the attractor of the family, which may be chaotic. The results are illustrated with a pedagogical example, a pest control model, and a model of a food chain with threshold-based harvesting. To evaluate the parameters, we use a formula for the linear term of the discontinuity map associated with grazing-sliding bifurcations. In this paper we present a new, simpler derivation of this formula for $n$-dimensional systems based on displacements from a virtual counterpart.

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 claims that boundary Hopf bifurcations in three-dimensional Filippov systems reduce to a two-parameter family of piecewise-linear maps. This reduction occurs because sliding motion on the switching surface induces a loss of dimension and the limit cycle born at the Hopf point has degenerate stability. Explicit formulas for the two parameters are derived from a new, displacement-based derivation of the linear term in the discontinuity map for general n-dimensional systems. The resulting map family is then analyzed numerically to characterize its attractors (including chaotic ones), with illustrations from a pedagogical example, a pest-control model, and a food-chain model with threshold harvesting.

Significance. If the reduction holds, the result provides a substantial simplification for local dynamics near codimension-two boundary Hopf points in 3D Filippov systems, replacing high-dimensional map families with an explicitly parameterized two-parameter family whose attractors can be exhaustively classified. The new derivation of the discontinuity-map formula is a reusable technical contribution for grazing-sliding analysis in higher dimensions. The numerical characterization, including concrete examples from applied models, demonstrates that chaotic attractors are accessible within the reduced family.

major comments (2)
  1. [derivation of the map family and examples] The central reduction rests on the claim that the limit cycle has degenerate stability at the boundary Hopf point (allowing the map to be effectively two-dimensional). This is asserted in the abstract and used to justify the two-parameter family, but the manuscript does not appear to contain an explicit verification via the Jacobian or Floquet multipliers at the Hopf point in the general case or in the three examples; without this, the dimension-loss argument remains incomplete for the central claim.
  2. [discontinuity map derivation] § on the new discontinuity-map derivation: the displacement-based argument is presented as simpler for n-dimensional systems, yet the manuscript does not include a side-by-side comparison with the prior formula it replaces. Any difference in the linear term would propagate directly into the two explicit parameter formulas and therefore into all subsequent numerical results.
minor comments (2)
  1. [numerical analysis] The numerical section would benefit from explicit statements of the parameter ranges explored for the two-parameter family and the precise diagnostics (e.g., Lyapunov exponents, return-map construction) used to identify chaotic attractors.
  2. [examples] In the three examples, the computed numerical values of the two reduced parameters should be reported alongside the original vector-field coefficients so that readers can directly map each example onto the family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify gaps in explicit verification that we agree warrant attention. We have revised the manuscript accordingly to strengthen the central claims without altering the results.

read point-by-point responses

  1. Referee: [derivation of the map family and examples] The central reduction rests on the claim that the limit cycle has degenerate stability at the boundary Hopf point (allowing the map to be effectively two-dimensional). This is asserted in the abstract and used to justify the two-parameter family, but the manuscript does not appear to contain an explicit verification via the Jacobian or Floquet multipliers at the Hopf point in the general case or in the three examples; without this, the dimension-loss argument remains incomplete for the central claim.

    Authors: We agree that an explicit verification strengthens the dimension-loss argument. The degeneracy arises because the limit cycle born at the boundary Hopf point is tangent to the switching surface, producing a zero Floquet multiplier in the sliding direction. In the revised manuscript we have added a new subsection deriving the Jacobian of the Poincaré map at the Hopf point for the general n-dimensional case and confirming the zero multiplier; the same computation is carried out numerically for each of the three examples to illustrate the degeneracy. revision: yes

  2. Referee: [discontinuity map derivation] § on the new discontinuity-map derivation: the displacement-based argument is presented as simpler for n-dimensional systems, yet the manuscript does not include a side-by-side comparison with the prior formula it replaces. Any difference in the linear term would propagate directly into the two explicit parameter formulas and therefore into all subsequent numerical results.

    Authors: The referee is right that equivalence of the linear term must be confirmed. Our displacement-based derivation recovers exactly the same linear coefficient as the earlier formula (the one obtained via the standard virtual-orbit construction). The revised manuscript now contains a short appendix that places the two expressions side by side, showing term-by-term identity for the linear part; the two explicit parameter formulas and all numerical results are therefore unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central reduction to a two-parameter family rests on standard properties of sliding motion in 3D Filippov systems and the known degeneracy of limit-cycle stability at a boundary Hopf point. It supplies an independent, new derivation of the discontinuity-map linear term (via displacements from a virtual counterpart) rather than importing the formula from prior self-citation as a load-bearing premise. No step equates a claimed prediction or uniqueness result to a quantity defined or fitted inside the same manuscript; the explicit formulas and numerical examples are derived from the system's vector fields and the Hopf data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Filippov systems and codimension-two bifurcations, with the new derivation providing the main addition; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Boundary Hopf bifurcations are codimension-two events from which grazing bifurcations emanate in piecewise-smooth systems.
    Invoked to establish the starting point for the reduction and map family analysis.
  • domain assumption Sliding motion on the switching surface induces dimension loss in the local dynamics.
    Key premise enabling the reduction to two parameters in 3D Filippov systems.

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Reference graph

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