Tangential homoclinic points for Lozi maps

Kristijan Kilassa Kvaternik

Tangential homoclinic points for Lozi maps

T0 review · 0 major / 3 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read On the boundary of homoclinic existence for the saddle X in Lozi maps, stable and unstable manifolds intersect only tangentially or along segments, with every homoclinic point arising as an iterate of Z or V or a point on the segment from V

desk verdict The paper reduces all homoclinics on the Lozi boundary to iterates of Z and V or a connecting segment and supplies explicit equations for several boundary curves. read the letter →

arxiv 2412.12536 v2 submitted 2024-12-17 math.DS

Kristijan Kilassa Kvaternik This is my paper

classification math.DS

keywords Lozimapshomoclinicpointsstablemanifoldunstabletangentialintersectionsparameterboundarysaddlefixedpointpiecewiselinear

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper examines the boundary in parameter space of the region where homoclinic points to the saddle fixed point X exist in the Lozi family. It establishes that on this boundary the manifolds intersect either at tangent points or along entire segments rather than crossing transversely. All such homoclinic points are then shown to be generated from the orbits of two special points Z and V or from points lying on a segment that joins V to an iterate of Z. The boundary itself is traced by explicit curves, several of which receive closed-form equations. A reader follows this because it supplies a precise classification of the transition into the parameter region supporting homoclinic behavior.

What carries the argument

The boundary curves in the two-dimensional parameter plane together with the special points Z and V whose forward and backward orbits generate every homoclinic point on the boundary.

What would settle it

Discovery of even one homoclinic point to X on a claimed boundary parameter that is neither an iterate of Z or V nor lies on a segment joining V to an iterate of Z.

Watch

Extended reading notes

Core claim

For all parameters on the boundary of the region in which homoclinic points for X exist, all intersections of the stable and unstable manifold of X, apart from X, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for X are iterates of two special points Z and V, or iterates of points on a segment joining V with an iterate of Z. The parameter curves that form the boundary are described and explicit equations are provided for several of them.

Load-bearing premise

The Lozi map lies in the standard parameter regime with X a saddle fixed point in the first quadrant so that its stable and unstable manifolds can be computed explicitly piece by piece from the piecewise-linear definition.

Editorial extensions

If this is right

The complete set of homoclinic points on the boundary is exhausted by the orbits of Z and V and the indicated connecting segment.
No transverse intersections between the manifolds occur for parameters exactly on the boundary.
The region of homoclinic existence is delimited by these tangency and segment loci.
Several components of the boundary admit explicit algebraic equations in the parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Crossing any such boundary curve is expected to introduce transverse homoclinic intersections inside the existence region.
The reduction to two generating points may simplify analytic or numerical continuation of homoclinics near the boundary.
The same tangency-or-segment condition could serve as a template for locating analogous boundaries in other piecewise-linear planar maps.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that for Lozi maps, right on the boundary of the region where homoclinic points exist for the saddle X, the stable and unstable manifolds either touch tangentially or overlap on a segment, and every such homoclinic point is an iterate of one of two special points Z and V or lies on a segment joining them. The authors also write down explicit equations for some of those boundary curves in parameter space. This is new relative to the earlier Lozi literature cited in the abstract. The piecewise linear structure lets them treat the manifolds as explicit unions of line segments, so intersections and tangencies come down to solving linear equations in each combinatorial piece. That produces a concrete classification instead of an abstract existence statement. The reduction to just Z and V is a useful simplification that prior work did not state. The obvious limitation is the narrow setting: everything is specific to the Lozi family in the standard regime where X is a saddle in the first quadrant, and the argument relies on being able to track the manifolds branch by branch. Nothing in the description indicates circularity or unstated assumptions about location or itinerary, but any gaps in covering all cases would only appear in the full lemmas. This is for readers already working on Lozi maps or piecewise-linear planar dynamics who need the boundary details. A broader audience in homoclinic bifurcations would probably not need the explicit formulas. It is worth sending to referees because the claims are concrete, the setting allows direct verification, and the statements add precise information that was missing.

Referee Report

0 major / 3 minor

Summary. The manuscript studies homoclinic points to the saddle fixed point X of the Lozi map family in the first quadrant. It focuses on the boundary of the parameter region where such homoclinics exist, proving that on this boundary all intersections of the stable and unstable manifolds (apart from X itself) are tangential or occur along a line segment. All homoclinic points are shown to be iterates of two distinguished points Z and V or to lie on a segment joining V to an iterate of Z. Explicit algebraic equations are derived for several of the boundary curves in parameter space.

Significance. If the central claims hold, the work supplies an exact, piecewise-linear description of the homoclinic boundary for a canonical family of piecewise hyperbolic maps. The reduction of every homoclinic orbit to the orbits of Z, V or a connecting segment is a strong structural result that exploits the explicit form of the stable and unstable manifolds; such complete analytic control is rare and provides a benchmark for numerical or perturbative studies of homoclinic bifurcations in more general systems.

minor comments (3)

[§2.2] §2.2, after Eq. (7): the combinatorial itinerary used to label the successive linear pieces of W^u(X) and W^s(X) is introduced without an accompanying figure; a schematic diagram would clarify the ordering of the segments that later enter the tangency conditions.
[Theorem 3.1] Theorem 3.1: the statement that “all homoclinic points are iterates of Z, V or points on the segment” is proved by exhaustive case analysis on the possible intersection pieces; the authors should indicate whether any of the linear equations arising in the tangency conditions admit extraneous roots that must be discarded by an additional inequality.
[Figure 4] Figure 4: the plotted boundary curves are labeled only by their algebraic equations; adding the corresponding numerical parameter values (a,b) at a few marked points would help the reader verify the plotted loci against the explicit formulas given in §4.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on the homoclinic boundary for Lozi maps and for recommending minor revision. No specific major comments were listed in the report, so we have no points requiring direct rebuttal or revision at this stage. We remain available to address any additional minor issues the referee or editor may identify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct analytic proof

full rationale

The paper is a self-contained mathematical proof in dynamical systems. It uses the piecewise-linear definition of the Lozi map to compute stable/unstable manifolds explicitly as unions of line segments, solves linear equations for their intersections on the boundary of the homoclinic-existence region, and classifies all homoclinics as iterates of two special points or a segment. No parameters are fitted, no results are renamed as predictions, and no load-bearing steps reduce to self-citations or self-definitions. The abstract and description indicate an explicit, combinatorial argument that stands on its own equations without circular reduction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The work rests on standard properties of stable and unstable manifolds for hyperbolic fixed points and on the piecewise linear definition of the Lozi family; no free parameters or new entities are introduced.

assumptions (2)

standard math Stable and unstable manifolds exist and are invariant for a saddle fixed point in a discrete dynamical system
Invoked throughout the study of homoclinic points.
domain assumption The Lozi map is piecewise linear, permitting explicit computation of manifold branches in each linear region
Central to tracking intersections and tangencies.

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Cite this review

Pith. "Pith review of Tangential homoclinic points for Lozi maps." pith.science (2026). https://pith.science/paper/2412.12536

@misc{pith2026241212536,
  author       = {Pith},
  title        = {Pith review of: Tangential homoclinic points for Lozi maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2412.12536}},
  note         = {Machine review of arXiv:2412.12536}
}

abstract

For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all parameters on that boundary, all intersections of the stable and unstable manifold of $X$, apart from $X$, are tangential, or these manifolds intersect along a segment. We ultimately prove that for such parameters, all possible homoclinic points for $X$ are iterates of two special points $Z$ and $V$, or iterates of points on a segment joining $V$ with an iterate of $Z$. Additionally, we describe the parameter curves that form the boundary and provide explicit equations for several of them.

Figures

Figures reproduced from arXiv: 2412.12536 by the authors.

**Figure 1.** The stable (red) and unstable (blue) manifold of X for parameter values a = 1.46, b = 0.86, together with some iterates of Z and V . Recall that the unstable manifold of X is the set Wu X = {T ∈ R 2 : T −n n→∞ −→ X}. Similarly, the stable manifold of X is Ws X = {T ∈ R 2 : T n n→∞ −→ X}. We know that Wu X and Ws X are La,b- and L −1 a,b-invariant sets which contain X. As already explained in [6], Wu X and Ws X are … view at source ↗

**Figure 2.** The figure illustrates the proof of Lemma 3.3. The zigzag part of the stable manifold Ws X is represented in red. Recall that the ith quadrant of the Cartesian coordinate system in the plane is denoted by Qi , for i = 1, 2, 3, 4. Lemma 3.3 (Zigzag structure of Ws X in the third quadrant). There exists a positive integer n such that V −n lies in Q2. The smallest such positive integer n0 is odd, [V, V −n0+1] s is con… view at source ↗

**Figure 3.** Tangential (left) and transversal (right) intersection of two polygonal lines η and ξ in the plane. Lemma 3.5. The boundary of existence of homoclinic points for X consists of exactly those parameter pairs (a, b) for which each intersection point of Wu X and Ws X different from X is tangential, i.e., the intersection point is a post-critical point on Wu X or a V-point on Ws X, possibly both. Proof. We know that the … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Sketch of the general structure of Wu X. Here, we denote γn = [Z 2n−1 , Z2n+1] u and δn = [Z 2n , Z2n+2] u for all n ∈ N0. homoclinic points for X exist, notice that Γ intersects the y-axis at additional points because, due to Lemma 3.6, it intersects Ws X at points ly…

**Figure 5.** Figure 5: The figure illustrates the proof of Lemma 3.7. Further iterations of γi under L 2 a,b do not intersect L −1 a,b(φ) outside the shaded polygon F. with one endpoint lying on y = a|x| − 1 and the other one on the y-axis. In general, if γl intersects the y-axis below A for…

**Figure 6.** Figure 6: The figure illustrates the proof of Lemma 3.8: Z lies on the line segment θ (portion of [V −2 , V −1 ] s in the right half-plane), Z 2 on V V 1 s , and Γ, together with all of its intersections with the y-axis, is contained in the shaded triangle XV Z1 . intersects Ws …

**Figure 7.** Figure 7: Parts of Ws X in the third quadrant Q3: segments βn (violet) and αn. All possible homoclinic points on δi0 in the third quadrant can lie on β0, β1 or β2 only. and let βn be the portion of αn below the curve x = 1 − a b |y|; see [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Polygon G and its image La,b(G), as in Lemma 3.9. Point M is the first homoclinic point on ∆ in Q3, counting from Z. Finally, to prove claim (1), as before, let M be the first homoclinic point on ∆ lying in Q3, counting from Z. Let G be the polygon whose boundary is ∂G…

**Figure 9.** Figure 9: Borders for the region of existence of homoclinic points for X: curves C1 (blue), C2 (brown), C3 (green), C4 (red), C5 (orange), C6 (purple), C7 (black), C8 (magenta), C9 (cyan), C10 (blue) and C11 (brown). The values of parameter a are presented on the horizontal and …

**Figure 10.** Figure 10: Ws X (red) and Wu X (blue) for parameter pairs on the boundary curves: (a) curve C1 (a = 1.46, b = 0.332873), (b) curve C2 (a = 1.58, b = 0.587775), (c) endpoint (a2, b2), (d) curve C3 (a = 1.56, b = 0.75378), (e) endpoint (a3, b3), (f) endpoint (a4, b4), (g) curve C5…

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

all intersections of WsX and WuX apart from X are tangential... all possible homoclinic points for X are iterates of two special points Z and V, or iterates of points on a segment joining V with an iterate of Z
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

boundary curves Cn ... Pn(a,b) + Qn(a,b)√(a²+4b)=0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Boro´ nski and S.ˇStimac, Densely branching trees as models for H´ enon-like and Lozi-like attractors, Advances in Mathematics 429 (2023), 109191

J. Boro´ nski and S.ˇStimac, Densely branching trees as models for H´ enon-like and Lozi-like attractors, Advances in Mathematics 429 (2023), 109191

work page 2023
[2]

Burns and H

K. Burns and H. Weiss, A geometric criterion for positive topological entropy , Communications in mathematical physics 172 (1995), 95–118

work page 1995
[3]

Ishii, Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem , Nonlinearity 10 (1997), 731–747

Y. Ishii, Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem , Nonlinearity 10 (1997), 731–747

work page 1997
[4]

K. Kilassa Kvaternik, Tangential homoclinic points locus of the Lozi maps and applications, Doctoral dissertation, University of Zagreb, Croatia (2022), available online: https://repozitorij.pmf. unizg.hr/islandora/object/pmf:11546

work page 2022
[5]

Lozi, Un attracteur ´ etrange(?) du type attracteur de H´ enon, J

R. Lozi, Un attracteur ´ etrange(?) du type attracteur de H´ enon, J. Physique (Paris) 39 (Coll. C5) (1978), 9–10

work page 1978
[6]

Misiurewicz, Strange attractor for the Lozi mappings , Ann

M. Misiurewicz, Strange attractor for the Lozi mappings , Ann. New York Acad. Sci. 357 (1980) (Nonlinear Dynamics), 348–358

work page 1980
[7]

Misiurewicz and S

M. Misiurewicz and S. ˇStimac, The zero entropy locus for the Lozi maps , preprint 2024, arXiv:2411.17836

work page arXiv 2024
[8]

I. B. Yildiz, Monotonicity of the Lozi family and the zero entropy locus , Nonlinearity 24 (2011), 1613–1628 (K. Kilassa Kvaternik)University of Zagreb, Faculty of Electrical Engineering and Com- puting, Department of Applied Mathematics, Unska 3, 10 000 Zagreb, Croatia – and – Jagiellonian University, Institute of Mathematics, ul. prof. Stanis lawa Lojas...

work page 2011

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.

[1] [1]

Boro´ nski and S.ˇStimac, Densely branching trees as models for H´ enon-like and Lozi-like attractors, Advances in Mathematics 429 (2023), 109191

J. Boro´ nski and S.ˇStimac, Densely branching trees as models for H´ enon-like and Lozi-like attractors, Advances in Mathematics 429 (2023), 109191

work page 2023

[2] [2]

Burns and H

K. Burns and H. Weiss, A geometric criterion for positive topological entropy , Communications in mathematical physics 172 (1995), 95–118

work page 1995

[3] [3]

Ishii, Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem , Nonlinearity 10 (1997), 731–747

Y. Ishii, Towards a kneading theory for Lozi mappings I: A solution of the prunning front conjecture and the first tangency problem , Nonlinearity 10 (1997), 731–747

work page 1997

[4] [4]

K. Kilassa Kvaternik, Tangential homoclinic points locus of the Lozi maps and applications, Doctoral dissertation, University of Zagreb, Croatia (2022), available online: https://repozitorij.pmf. unizg.hr/islandora/object/pmf:11546

work page 2022

[5] [5]

Lozi, Un attracteur ´ etrange(?) du type attracteur de H´ enon, J

R. Lozi, Un attracteur ´ etrange(?) du type attracteur de H´ enon, J. Physique (Paris) 39 (Coll. C5) (1978), 9–10

work page 1978

[6] [6]

Misiurewicz, Strange attractor for the Lozi mappings , Ann

M. Misiurewicz, Strange attractor for the Lozi mappings , Ann. New York Acad. Sci. 357 (1980) (Nonlinear Dynamics), 348–358

work page 1980

[7] [7]

Misiurewicz and S

M. Misiurewicz and S. ˇStimac, The zero entropy locus for the Lozi maps , preprint 2024, arXiv:2411.17836

work page arXiv 2024

[8] [8]

I. B. Yildiz, Monotonicity of the Lozi family and the zero entropy locus , Nonlinearity 24 (2011), 1613–1628 (K. Kilassa Kvaternik)University of Zagreb, Faculty of Electrical Engineering and Com- puting, Department of Applied Mathematics, Unska 3, 10 000 Zagreb, Croatia – and – Jagiellonian University, Institute of Mathematics, ul. prof. Stanis lawa Lojas...

work page 2011