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arxiv: 2604.26447 · v1 · submitted 2026-04-29 · 🧮 math.DS

Topological Horseshoe Induced by Periodic Switching Between Non-Isochronous Planar Systems

Junfeng Cheng , Xiao-Song Yang This is my paper

Pith reviewed 2026-05-07 12:38 UTC · model grok-4.3

classification 🧮 math.DS
keywords topological horseshoeperiodic switchingnon-isochronous Hamiltonian systemsperiod functionchaotic dynamicsplanar switched systemsNewtonian systems
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The pith

Periodic switching between two non-isochronous planar Hamiltonian systems induces a topological horseshoe

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

This paper develops a criterion showing that periodic switching between two planar flows, each containing a family of closed orbits, can produce a topological horseshoe. The chaos arises because the time to traverse each orbit varies with its size, so the fixed switching interval stretches and folds trajectories in a way that creates symbolic dynamics. The authors link this variation in period to the rate at which area grows inside each orbit, producing a condition that can be verified by numerical computation. For Newtonian systems the condition simplifies further using known monotonicity properties of the period function.

Core claim

We establish a criterion for the existence of a topological horseshoe in a class of planar systems generated by periodic switching between two subsystems, each admitting a family of closed orbits, where the mechanism for chaos arises from the non-isochronicity of each subsystem. Exploiting the relationship between the period function of a Hamiltonian system and the rate of change of the area enclosed by its periodic orbits, we derive a criterion, which can be checked by numerical methods, for the existence of horseshoe in planar systems obtained by switching between two Hamiltonian subsystems. Furthermore, by invoking monotonicity results for the period function in Newtonian Hamiltonian

What carries the argument

the relation between each subsystem's period function and the derivative of its enclosed area, applied to the return map of the switched flow to produce a verifiable horseshoe condition

Load-bearing premise

The subsystems must be Hamiltonian with families of closed orbits whose periods vary with enclosed area in a way that produces a checkable inequality for the switched map.

What would settle it

A concrete pair of Newtonian systems satisfying the monotonicity condition on their period functions whose switched Poincaré map nevertheless contains no topological horseshoe detectable by symbolic dynamics.

Figures

Figures reproduced from arXiv: 2604.26447 by Junfeng Cheng, Xiao-Song Yang.

Figure 1
Figure 1. Figure 1: Case I, Subcase I: γ1 ∩ O2 6= ∅. Solid curves denote portions of the orbits γi , Γi (i = 1, 2), whereas dotted curves and dashed curves denote portions of the orbits γ˜1 and Γ˜ 2, respectively. Suppose there exist points A ∈ R2 and B ∈ I2, and let l : [0, 1] → R 2 be a continuous path connecting A to B, with l(0) = A and l(1) = B. Define t∗ = inf¶ t ∈ [0, 1] : l(s) ∈ γ2 ∪ I2 for all s ∈ [t, 1]© , (3.18) an… view at source ↗
Figure 2
Figure 2. Figure 2: Case I, Subcase II: γ1 ∩ I2 6= ∅. Solid curves denote portions of the orbits γi , Γi (i = 1, 2), whereas dotted curves and dashed curves denote portions of the orbits γ˜1 and γ˜2, respectively. 𝐴 𝐵 𝐶 𝛾1 𝛤2 𝛤෨2 𝛤1 𝛾2 𝛤෨1 𝒬 view at source ↗
Figure 3
Figure 3. Figure 3: Case II, Subcase I: Γ1 ∩ Γ2 6= ∅. Solid curves denote portions of the orbits γi , Γi (i = 1, 2), whereas dotted curves and dashed curves denote portions of the orbits Γ˜ 1 and Γ˜ 2, respectively. Proof of Theorem 3.5. From Theorem 3.7, we assume that the four orbits defining the curvi￾linear quadrilateral Q are γ ‹ 1, γ ‹ 2, Γf1, and Γf2 and E1(γ ‹ 1) = c1, (3.23) E2(γ ‹ 2) = c2, (3.24) E1(Γf1) = C1, (3.25… view at source ↗
Figure 4
Figure 4. Figure 4: Case II, Subcase II: Γ1 ∩ γ2 6= ∅. Solid curves denote portions of the orbits γi , Γi (i = 1, 2), whereas dotted curves and dashed curves denote portions of the orbits Γ˜ 1 and γ˜2, respectively. We denote the edges and vertices of Q as illustrated in view at source ↗
Figure 5
Figure 5. Figure 5: Edges and vertices of Q. Arrows indicate the directions of the orbits. Define T ∗ 1 := T1(c1) T1(C1) |T1(c1) − T1(C1)| . (3.38) Note that 5T ∗ 1 T1(c1) − 5T ∗ 1 T1(C1) = 5, (3.39) implies that ú 5T ∗ 1 T1(c1) ü − ú 5T ∗ 1 T1(C1) ü = 5. (3.40) It follows that whenever T1 ≥ 5T ∗ 1 , we have N1 − n1 ≥ 5. (3.41) Consider the following two compact subsets of Q¯: Q1 := ¶ (x, y) ∈ Q¯ : T1 T1(x, y) ∈ [n1 + 1, n1 +… view at source ↗
Figure 6
Figure 6. Figure 6: Edges of Q1 and Q2. We denote the edges of Q1 and Q2 as illustrated in view at source ↗
Figure 7
Figure 7. Figure 7: Preimage of Q˜ 1 and edges of Q˜ 1. The figure illustrates the case in which P1(Q1) “crosses” Q1 exactly once. (Left) Relative position of P −1 1 (Q˜ 1) and Q˜ 1. (Right) P1(Q1) “crosses” Q1 once; in this case, Q˜ 1 = P1(Q1) ∩ Q1. We denote the edges of Q‹ 1 as illustrated in view at source ↗
Figure 8
Figure 8. Figure 8: Illustration of the formation of the topological horseshoe. The figure illustrates the case in which P1(Qi) “crosses” Qi (i = 1, 2) exactly once. In this case, P1 is the identity map when restricted to the orbits containing Ei and Gi (i = 1, 2). Moreover, for each i, the points Ei and Gi lie on the same orbit. Similarly, for i, j = 1, 2, the following relation holds: P2 ◦ P1 Ä P −1 1 (Qfi) ä 7→ P −1 1 (Qfj… view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the topological horseshoe: P2 ◦ P1 Ä P −1 1 (EiFiGiHi) ä 7→ P −1 1 (EjFjGjHj ), i, j = 1, 2. Remark 3.8. We do not require Q3 1 to be different from Q1 2 . As can be seen from Figures 8 and 9, even if Q3 1 = Q1 2 , we have P −1 1 (E1F1G1H1) ∩ P −1 1 (E2F2G2H2) = ∅. (3.65) Remark 3.9. It can be verified that other types of monotonicity of the period functions Ti(c) do not affect the validity… view at source ↗
Figure 10
Figure 10. Figure 10: γi , Γi (i = 1, 2) and the curvilinear quadrilateral ABCD. Consequently, T ∗ 1 = 4π × 6π |4π − 6π| = 12π = T ∗ 2 . (5.11) According to Theorem 3.5, we choose T1 = 5T ∗ 1 = 60π, T2 = 3T ∗ 2 = 36π. (5.12) The orbit with initial condition (0, 1) over the time interval [0, 1000π] is depicted in Fig￾ure 11, while view at source ↗
Figure 11
Figure 11. Figure 11: Orbit with initial condition (0, 1) over the time interval [0, 1000π]. The circle marks the initial point, and the square marks the terminal point. 0 500 1000 1500 2000 2500 3000 t -10 0 10 x(t) 0 500 1000 1500 2000 2500 3000 t -10 0 10 y(t) view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of the state variables. 20 view at source ↗
Figure 13
Figure 13. Figure 13: γi , Γi (i = 1, 2) and the curvilinear quadrilateral ABCD. By a slight modification of Theorem 4.13, or alternatively by applying the classical results in Ref. [12], it follows that the derivative of the period function associated with each energy level c > 0 is strictly positive. For the purpose of numerical simulation, we fix A = B = 1, yielding the two subsystems ( x˙ = −y, y˙ = (1 + y)(x + αi). (5.15)… view at source ↗
Figure 14
Figure 14. Figure 14: Orbit with initial condition (0, 1.5) over the time interval [0, 80T ∗ 1 ]. The circle marks the initial point, and the square marks the ter￾minal point. In Ref. [10], it is shown that the switching system (5.14) exhibits chaotic dynamics provided that T1 > 6T1(− 1 3 , 0)T1(− 2 3 , 0) view at source ↗
Figure 15
Figure 15. Figure 15: Illustration of a semi-horseshoe that may occur. 6. Discussion In contrast to Ref. [10], we investigate a broader class of systems and establish explicit geometric criteria for the existence of the topological horseshoe. Moreover, by a more detailed analysis of the mechanism underlying the formation of the topological horseshoe, we derive sharper lower bounds on the dwell times of the subsystems (5.2). No… view at source ↗
Figure 16
Figure 16. Figure 16: An incomplete horseshoe that does not induce chaotic behavior. Several questions remain open. First, it is natural to ask whether sharper lower bounds on the dwell times can be obtained by considering higher iterates of the Poincaré map. Second, from a practical perspective, it is important to determine whether the chaotic dynamics persist when the dwell times satisfy the proposed conditions but are allow… view at source ↗
read the original abstract

We establish a criterion for the existence of a topological horseshoe in a class of planar systems generated by periodic switching between two subsystems, each admitting a family of closed orbits, where the mechanism for chaos arises from the non-isochronicity of each subsystem. Exploiting the relationship between the period function of a Hamiltonian system and the rate of change of the area enclosed by its periodic orbits, we derive a criterion, which can be checked by numerical methods, for the existence of horseshoe in planar systems obtained by switching between two Hamiltonian subsystems. Furthermore, by invoking monotonicity results for the period function in Newtonian Hamiltonian systems, we obtain an explicit and computable criterion that guarantees chaotic dynamics in planar systems generated by switching between two such subsystems.

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 establishes a criterion for the existence of a topological horseshoe in planar systems generated by periodic switching between two non-isochronous Hamiltonian subsystems, each admitting a family of closed orbits. The mechanism for chaos is the non-isochronicity of the subsystems. Exploiting the link between the period function and the rate of change of enclosed area, the authors derive a numerically checkable criterion for horseshoe existence. For Newtonian Hamiltonian subsystems, they invoke monotonicity results on the period function to obtain an explicit, computable criterion guaranteeing chaotic dynamics.

Significance. If the central derivation is sound, the work offers a useful contribution to dynamical systems by supplying practical, checkable criteria for detecting topological horseshoes in switched planar systems. The connection between period functions and area variation is a strength that enables numerical verification, and the explicit Newtonian criterion could be applicable in modeling or control contexts. Credit is due for framing the problem in terms of non-isochronicity and for aiming at computable conditions rather than abstract existence proofs.

major comments (1)
  1. [Derivation of explicit criterion for Newtonian systems] In the derivation of the explicit criterion for Newtonian systems: the claim that monotonicity of T(h) (equivalently A'(h)) yields a guarantee of chaotic dynamics rests on invoking standard monotonicity theorems, but the manuscript provides no verification that the specific potentials satisfy the hypotheses of those theorems (e.g., convexity, star-shaped level sets, or positive second derivative at the center). This is load-bearing for the explicit criterion and the associated claim of guaranteed chaos in Newtonian cases.
minor comments (2)
  1. [Abstract] The abstract refers to a 'numerically checkable criterion' without indicating the specific numerical procedure or quantities being computed; adding one sentence on the method would improve clarity.
  2. Consider including at least one fully worked numerical example with explicit potentials, computed period functions, and the resulting horseshoe verification to demonstrate the criterion in action.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below and will revise the manuscript to incorporate the necessary clarifications and verifications.

read point-by-point responses

  1. Referee: In the derivation of the explicit criterion for Newtonian systems: the claim that monotonicity of T(h) (equivalently A'(h)) yields a guarantee of chaotic dynamics rests on invoking standard monotonicity theorems, but the manuscript provides no verification that the specific potentials satisfy the hypotheses of those theorems (e.g., convexity, star-shaped level sets, or positive second derivative at the center). This is load-bearing for the explicit criterion and the associated claim of guaranteed chaos in Newtonian cases.

    Authors: We agree that the manuscript invokes standard monotonicity theorems for the period function of Newtonian Hamiltonian systems without explicitly confirming that the potentials satisfy the requisite hypotheses. The explicit criterion is meant to apply precisely in those cases where monotonicity holds (as ensured by the cited theorems under conditions such as V''(0) > 0 and suitable convexity or star-shapedness of the level sets). To address this, we will revise the manuscript by adding an explicit statement of the standing assumptions on the potentials together with a brief verification (or reference to standard checks) that these assumptions are met for the Newtonian systems under consideration. This will make the scope of the guaranteed chaotic dynamics fully rigorous while preserving the computable nature of the criterion. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard Hamiltonian relations and external monotonicity results

full rationale

The paper's core derivation exploits the known relationship between the period function T(h) and the derivative of enclosed area A'(h) for Hamiltonian subsystems to obtain a checkable criterion for horseshoes under periodic switching. This relationship is a standard first-principles property of planar Hamiltonian flows and is not defined in terms of the target horseshoe result. The explicit Newtonian criterion further invokes monotonicity of T(h) from prior literature on period functions, without the paper re-deriving or fitting those monotonicity statements to its own data or outputs. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description; the central claim remains independent of the paper's own fitted quantities or prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Ledger extracted from abstract statements; full paper may add or remove items.

axioms (3)
  • domain assumption Each subsystem admits a family of closed orbits
    Stated directly as prerequisite for generating the switched system that can exhibit horseshoes.
  • domain assumption Relationship between period function of Hamiltonian system and rate of change of enclosed area
    Exploited to derive the numerical criterion for horseshoe existence.
  • domain assumption Monotonicity results for the period function in Newtonian Hamiltonian systems
    Invoked to obtain an explicit computable criterion.

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Reference graph

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