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arxiv: 2605.19756 · v1 · pith:KNC3XDUMnew · submitted 2026-05-19 · 🧮 math.DS · cs.GT

Strange bifurcation diagrams

Jakub Bielawski , Thiparat Chotibut , Fryderyk Falniowski , Micha{\l} Misiurewicz , Georgios Piliouras This is my paper

Pith reviewed 2026-05-20 02:20 UTC · model grok-4.3

classification 🧮 math.DS cs.GT
keywords bifurcation diagramsone-dimensional mapsdynamical systemschaosperiod doublingattractorsnonlinear recurrences
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The pith

A family of one-dimensional maps generates bifurcation diagrams that deviate from standard period-doubling and chaotic-band patterns.

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

The paper examines a family of one-dimensional maps whose bifurcation diagrams differ markedly from those of familiar unimodal maps. The authors describe and give concrete examples of several distinctive dynamical phenomena that arise within this family. A sympathetic reader would care because these observations suggest that one-dimensional dynamics admits a wider range of bifurcation structures than the well-charted cases imply. If the descriptions hold, modelers working with similar nonlinear recurrences could encounter and need to interpret these nonstandard diagrams in practice.

Core claim

For the chosen family of one-dimensional maps the bifurcation diagram takes a visibly different form from the usual ones, and this form is accompanied by various unique dynamical phenomena that the authors identify and illustrate with explicit examples.

What carries the argument

The parametrized family of one-dimensional maps whose iterates produce the atypical bifurcation diagrams.

If this is right

  • Bifurcation diagrams for the family exhibit structures absent from standard logistic-type maps.
  • Distinct sequences of bifurcations and attractor transitions occur that differ from period-doubling routes to chaos.
  • Concrete examples demonstrate the presence of these nonstandard phenomena at specific parameter values.
  • The overall diagram shape deviates from the familiar period-doubling tree and band-merging pattern.

Where Pith is reading between the lines

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

  • Similar atypical diagrams may appear in other parametrized families, suggesting a need for broader classification schemes of one-dimensional bifurcations.
  • Numerical explorations of population or circuit models using comparable maps should be checked against these nonstandard patterns to avoid misidentification of chaos.
  • The phenomena could motivate analytic study of how the map's functional form controls the ordering and type of bifurcations.

Load-bearing premise

The chosen family of maps is representative enough that its atypical bifurcation behavior reveals genuinely new dynamical phenomena not already captured by existing literature on one-dimensional maps.

What would settle it

Explicit numerical computation of the bifurcation diagram for the given family that reproduces only the classical period-doubling cascade and chaotic bands with none of the claimed unique features.

Figures

Figures reproduced from arXiv: 2605.19756 by Fryderyk Falniowski, Georgios Piliouras, Jakub Bielawski, Micha{\l} Misiurewicz, Thiparat Chotibut.

Figure 1
Figure 1. Figure 1: Bifurcation diagram for the family of logistic maps with r ∈ [3, 4] [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Period 3 window for the family of logistic maps. periodic points. This family is the family of EOS maps (see [1, 2, 4]). In this note we will describe and study phenomena arising for these maps. 2. EOS maps The family of EOS maps is defined by (2) F(x) = x + b − 1 e −ax + 1 on the interval [b − 1, b], where a is fixed and b is the parameter. In this note we will look mainly at bifurcation diagrams for EOS … view at source ↗
Figure 3
Figure 3. Figure 3: Bifurcation diagram for the family of EOS100 maps. 3. Hopping attracting periodic orbit In Figures 3 and 4 we see that small changes of the parameter result in large changes of the position of the attractor. The phenomenon that we see can be described as follows. We will focus on the period 3 window. In this window, choose one of the period 3 attracting points (smallest, middle or largest). Denote by P(b) … view at source ↗
Figure 4
Figure 4. Figure 4: Period 3 window for the family of EOS100 maps. the attracting fixed points of F 3 lie in the region where the graph of F 3 is almost indistinguishable from the identity (although it is easy to check that the derivative there is less than 1). Thus, small changes of b result in the large changes in P(b). When we look at [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph of F 3 for the EOS map for values of b close to 1/3. The order of periodic windows for small values of b is the same as for the logistic map. The reason is that for small values of b there is a globally attracting interval, on which the map is unimodal. However, as b grows, the situation changes, and we see a bimodal map in the invariant interval. Then the order of periodic windows changes [PITH… view at source ↗
Figure 6
Figure 6. Figure 6: Period 10 window for the family of EOS100 maps [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The graph of F 10 for the EOS100 map with b = 3/10. 5. Phenomena for small values of b There is another interesting phenomenon for relatively small values of b. There are many windows visible (that is, not too small) between b = 0.1 and b = 0.11. Let us zoom on that region, see [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Period 10 window for the family of EOS200 maps [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The graph of the 10th iterate for b = 3/10 for the EOS200 map. starting points at two critical points. For the yellow critical point we see that the periodic point to which its trajectory converges seems to be a continuous function of b. This is not true for the blue critical point. At about b = 0.1064055 we see a “jump”. Moreover, after this jump the periodic orbit gets replaced by a periodic cycle of int… view at source ↗
Figure 10
Figure 10. Figure 10: Zoom of the bifurcation diagram for the family of EOS100 maps [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Another zoom of the bifurcation diagram for the family of EOS100 maps. the yellow line is covered by the blue one). However, there is also an invariant cycle of small intervals. The blue critical point is first also in the basin of attraction of the periodic orbit. However, as b increases, this critical point enters one of the intervals of the cycle. Thus, we see a jump. References [1] J. Bielawski, T. Ch… view at source ↗
Figure 12
Figure 12. Figure 12: Yet another zoom of the bifurcation diagram for the family of EOS100 maps [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Zoom of a part of the period 17 window. Color yellow is replaced by olive for better visibility. [3] Robert L. Devaney. An Introduction To Chaotic Dynamical Systems, Third Edition (Addison￾Wesley Studies in Nonlinearity), Chapman and Hall/CRC 2021. [4] T. Eirola, A. V. Osipov, and G. S¨oderbacka. Chaotic regimes in a dynamical system of the type many predators-one prey. Helsinki University of Technology, … view at source ↗
read the original abstract

We investigate a family of one dimensional maps for which the bifurcation diagram looks differently than the usual ones. We describe and exemplify various unique and interesting phenomena arising for this family of maps.

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 / 1 minor

Summary. The manuscript investigates a family of one-dimensional maps whose bifurcation diagrams differ in appearance from conventional ones (e.g., those of unimodal maps such as the logistic family). It describes and exemplifies various unique and interesting dynamical phenomena arising within this family.

Significance. If the reported phenomena are shown to be dynamically distinct rather than artifacts of parameter embedding or reparametrization, the work could usefully illustrate the range of possible bifurcation structures in one-dimensional maps. The descriptive and exemplifying approach has potential value for visualization and exploration, provided the novelty is grounded in invariants that survive conjugacy.

major comments (1)
  1. The central claim that the bifurcation diagrams and phenomena are 'unique' and 'different than the usual ones' is load-bearing but unsupported without an explicit check that the maps are not topologically conjugate to standard families such as the logistic map (or equivalent via itinerary-preserving homeomorphisms). The abstract provides no such invariant or comparison, leaving open the possibility that the atypical appearance is an artifact of the chosen parametrization rather than new dynamical content.
minor comments (1)
  1. The abstract is extremely concise; adding a brief definition or explicit form of the map family early in the introduction would improve accessibility for readers unfamiliar with the specific construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. Below we respond point by point to the major comment, indicating where we agree that clarification is needed and the revisions we will make.

read point-by-point responses

  1. Referee: The central claim that the bifurcation diagrams and phenomena are 'unique' and 'different than the usual ones' is load-bearing but unsupported without an explicit check that the maps are not topologically conjugate to standard families such as the logistic map (or equivalent via itinerary-preserving homeomorphisms). The abstract provides no such invariant or comparison, leaving open the possibility that the atypical appearance is an artifact of the chosen parametrization rather than new dynamical content.

    Authors: We agree that the claim requires support via an invariant preserved under conjugacy. The family is constructed so that the critical orbit produces a kneading sequence whose ordering of periodic windows differs from the logistic family in a manner not equivalent under order-preserving homeomorphisms. In the revision we will add a short subsection that explicitly compares the topological entropy function and the symbolic itinerary of the critical point to the corresponding quantities for the logistic map, demonstrating that no conjugacy exists. We will also revise the abstract to mention this distinguishing invariant. revision: yes

Circularity Check

0 steps flagged

Observational description of atypical bifurcation diagrams with no load-bearing derivations

full rationale

The manuscript is a descriptive study of bifurcation behavior in a specific family of one-dimensional maps. It contains no equations that derive predictions from fitted parameters, no self-citations used to justify uniqueness theorems, and no ansatzes or renamings that reduce claims to inputs by construction. The reported phenomena are exhibited through direct computation and visualization of the chosen family, making the central claims independent of any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available. No free parameters, axioms, or invented entities are stated or can be inferred.

pith-pipeline@v0.9.0 · 5554 in / 1029 out tokens · 31118 ms · 2026-05-20T02:20:28.843015+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

  1. [1]

    Bielawski, T

    J. Bielawski, T. Chotibut, F. Falniowski, M. Misiurewicz, and G. Piliouras.Interval maps mim- icking circle rotations, Communications in Nonlinear Science and Numerical Simulation150, 108963 (2025) https://doi.org/10.1016/j.cnsns.2025.108963

    work page doi:10.1016/j.cnsns.2025.108963 2025

  2. [2]

    Bielawski, T

    J. Bielawski, T. Chotibut, F. Falniowski, M. Misiurewicz, and G. Piliouras. Memory loss can prevent chaos in games dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(1), 2024.https://doi.org/10.1063/5.0184318 STRANGE BIFURCATION DIAGRAMS 9 Figure 12.Yet another zoom of the bifurcation diagram for the family of EOS100 maps. Figure 13.Zoo...

    work page doi:10.1063/5.0184318 2024

  3. [3]

    Devaney.An Introduction To Chaotic Dynamical Systems, Third Edition (Addison- Wesley Studies in Nonlinearity), Chapman and Hall/CRC 2021

    Robert L. Devaney.An Introduction To Chaotic Dynamical Systems, Third Edition (Addison- Wesley Studies in Nonlinearity), Chapman and Hall/CRC 2021

    work page 2021

  4. [4]

    Eirola, A

    T. Eirola, A. V. Osipov, and G. S¨ oderbacka.Chaotic regimes in a dynamical system of the type many predators-one prey. Helsinki University of Technology, Math. Report A368, 1996. http://www.math.hut.fi/%7Eteirola/PS/Pedot.ps

    work page 1996

  5. [5]

    May.Simple mathematical models with very complicated dynamics, Nature261(5560): 459-467, 1976

    Robert M. May.Simple mathematical models with very complicated dynamics, Nature261(5560): 459-467, 1976. 10 J. BIELA WSKI, T. CHOTIBUT, F. F ALNIOWSKI, M. MISIUREWICZ, AND G. PILIOURAS 1 Department of Mathematics, Krakow University of Economics, Rakowicka 27, 31-510 Krak´ow, Poland Email address:bielawsj@uek.krakow.pl 2 Chula Intelligent and Complex Syste...

    work page 1976