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arxiv: 2507.02716 · v2 · submitted 2025-07-03 · 🌀 gr-qc · hep-th

A Note on Chaos in Hayward Black Holes with String Fluids

Aditya Singh , Ashes Modak , Binata Panda This is my paper

Pith reviewed 2026-05-19 06:10 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Hayward black holesstring fluidsthermodynamic chaosMelnikov methodLyapunov exponentAdS black holesregular black holesphase space instability
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The pith

Charge is required for chaos under temporal perturbations in the thermodynamics of Hayward black holes with string fluids, but spatial perturbations produce chaos with or without charge.

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

The paper studies thermodynamic chaos in Hayward AdS black holes surrounded by string fluids by applying Melnikov's method to the system's Hamiltonian dynamics under small periodic perturbations. For perturbations that vary in time, such as thermal quenches, the equation of state yields a condition that chaos appears only when electric charge is present. Spatial perturbations, by contrast, trigger chaos independently of charge. The authors also compute the Lyapunov exponent and show that both the string fluid density and the Hayward regularization parameter strongly influence its magnitude, thereby affecting the strength of the instability.

Core claim

From the equation of state of the black hole, a general condition is established indicating that under temporal perturbations, the existence of charge is an essential prerequisite for chaos. However, regardless of the presence of charge, spatial perturbations result in chaotic behavior. The string fluid density and the Hayward regularization parameter have a considerable effect on the amplitude of the Lyapunov exponent.

What carries the argument

Melnikov's integral evaluated on the perturbed Hamiltonian dynamics of the thermodynamic phase space, which detects the breaking of homoclinic orbits that signals the onset of chaos.

If this is right

  • Chaos in the thermodynamic description appears only when charge supplies the necessary homoclinic structure for temporal driving.
  • Spatial driving destabilizes the system even in uncharged Hayward black holes with string fluids.
  • The magnitude of the Lyapunov exponent can be tuned by adjusting the string fluid density or the regularization parameter.
  • Regular geometry corrections and matter sources together control the threshold and strength of thermodynamic instability.

Where Pith is reading between the lines

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

  • The distinction between temporal and spatial perturbations may correspond to different physical processes such as sudden temperature changes versus spatial inhomogeneities in accretion flows.
  • If the charge dependence survives in more realistic models, it could link thermodynamic chaos to the presence of electromagnetic fields near black holes.
  • The control of Lyapunov amplitude by the Hayward parameter suggests that singularity resolution can suppress or enhance chaotic mixing in the phase space.

Load-bearing premise

The unperturbed thermodynamic system must possess a homoclinic orbit so that Melnikov's integral can be used to detect chaos under small perturbations.

What would settle it

Numerical integration of the perturbed equations of state showing that the Lyapunov exponent remains zero below a charge-dependent critical amplitude for temporal perturbations but becomes positive for any amplitude in the spatial case.

Figures

Figures reproduced from arXiv: 2507.02716 by Aditya Singh, Ashes Modak, Binata Panda.

Figure 1
Figure 1. Figure 1: The behavior of P¯ − v isotherm for fixed values of temperature T0 = 0.036, string fluid parameter a = 0.4 and charge q = 0.4. The isotherm features three distinct regions: a stable small black hole branch at low spe￾cific volume, an unstable intermediate branch where  ∂P¯ ∂v  > 0, and a stable large black hole branch at high volume. The intermediate region, known as the spinodal region, corresponds to t… view at source ↗
Figure 3
Figure 3. Figure 3: To study the behavior of γ critical, we plot γc with charge q and the string fluid parameter a, as shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) The unperturbed orbit at perturbation amplitude [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Phase portrait in the velocity–displacement plane for the perturbed case with perturbation ampli [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) The behavior of γc with charge q for fixed values of a = 0.1 and T0 = 0.1, v = 0.5, ε = 0.005, µ0 = 0.01, A = 0.05, ω = 0.005, s = 0.001. (b) The behavior of γc with string fluid parameter a for fixed values of q = 0.08, T0 = 0.2, v = 0.5, ε = 0.005, µ0 = 0.01, A = 0.05, ω = 0.005, s = 0.001. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Unperturbed phase portrait (v ′ − v) in supercritical regime with fixed values of q = 0.4, a = 0.4, T0 = 0.036, B = 0.0024810. Spatial regime: case II When the ambient pressure is lower than the reference pressure P0 (i.e. B < P0) and the system evolves along a subcritical isotherm (T0 < Tcr), the orbit in the (v ′–v) phase plane connects the saddle point at v = v1 back to itself, forming a closed trajecto… view at source ↗
Figure 6
Figure 6. Figure 6: Unperturbed phase portrait (v ′ − v) in subcritical regime with fixed values of q = 0.4, a = 0.4, T0 = 0.036, B = 0.000901. Spatial regime: case III When the ambient pressure equals the reference pressure P0 (i.e., B = P0) and the system is considered along a subcritical isotherm (T0 < Tcr), the orbit in the (v ′–v) phase plane connects the two saddle points located at v = v1 and v = v3. This type of traje… view at source ↗
Figure 7
Figure 7. Figure 7: Unperturbed phase portrait (v ′ − v) in critical regime with fixed values of q = 0.4, a = 0.4, T0 = 0.036, B = 0.0023365 A d 2v dx2 + P¯(v, T0) + ε cos(px) v(x) = B. (4.4) For any subcritical temperature T0 < Tcr, the dynamical system governed by eq. (4.4) admits three fixed points. These correspond to the intersection points between the isotherm T = T0 and the reference pressure line P¯ = B in the P¯–v di… view at source ↗
read the original abstract

In this work, we first examine the onset of thermodynamic chaos in Hayward AdS black holes with string fluids, emphasizing the effects of temporal and spatially periodic perturbations. We apply Melnikov's approach to examine the perturbed Hamiltonian dynamics and detect the onset of chaotic behavior. In the case of temporal perturbations induced by thermal quenches, chaos occurs for perturbation amplitude $\gamma$ exceeding a critical threshold, determined by charge $q$ and the string fluid parameter. From the equation of state of the black hole, a general condition is established indicating that under temporal perturbations, the existence of charge is an essential prerequisite for chaos. However, regardless of the presence of charge, spatial perturbations result in chaotic behavior. Further next, we compute the Lyapunov exponent associated with the thermodynamic system to further quantify chaotic behavior beyond the threshold condition. We demonstrate that the string fluid density and the Hayward regularization parameter have a considerable effect on the amplitude of the Lyapunov exponent, showing the control of thermal instability by regular geometry corrections and matter sources. These results highlight the rich nonlinear dynamics arising from the interplay of geometric regularization, matter content, and phase-space instability.

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 applies Melnikov's method to the perturbed thermodynamic Hamiltonian of Hayward AdS black holes with string fluids. It claims that temporal perturbations induce chaos only when charge q is present and the perturbation amplitude gamma exceeds a threshold set by q and the string-fluid parameter; spatial perturbations produce chaos irrespective of charge. Lyapunov exponents are then computed to quantify the chaotic regime, with explicit dependence on string-fluid density and the Hayward regularization parameter.

Significance. If the central technical steps are placed on a firm footing, the work would illustrate how geometric regularization and matter sources modulate nonlinear instability in black-hole thermodynamics, providing a concrete example of charge-dependent versus charge-independent routes to chaos under different perturbation classes.

major comments (1)
  1. The Melnikov analysis presupposes the existence of a homoclinic orbit in the unperturbed (P,V) phase space. The manuscript states that the method is applied directly to the equation of state but supplies neither an explicit phase-portrait analysis nor an analytic demonstration that such an orbit exists for the Hayward–string-fluid model. This omission renders the subsequent Melnikov integral and the derived threshold conditions on q and the string parameter formally undefined.
minor comments (2)
  1. Clarify in the text whether the string-fluid parameter and perturbation amplitude gamma are treated as independent inputs or are determined post-hoc from the equation of state; the present wording leaves open the possibility of circularity in the chaos condition.
  2. The abstract asserts that 'a general condition is established' from the equation of state; an explicit statement of this condition (perhaps as an inequality involving q) would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on thermodynamic chaos in Hayward AdS black holes with string fluids. The major comment identifies a key technical prerequisite for the Melnikov analysis that was not explicitly addressed in the original submission. We respond to this point below and outline the revisions we will make.

read point-by-point responses

  1. Referee: The Melnikov analysis presupposes the existence of a homoclinic orbit in the unperturbed (P,V) phase space. The manuscript states that the method is applied directly to the equation of state but supplies neither an explicit phase-portrait analysis nor an analytic demonstration that such an orbit exists for the Hayward–string-fluid model. This omission renders the subsequent Melnikov integral and the derived threshold conditions on q and the string parameter formally undefined.

    Authors: We agree that the existence of a homoclinic orbit in the unperturbed system is essential for the validity of Melnikov's method and that our original manuscript did not supply an explicit phase-portrait analysis or analytic demonstration for the Hayward–string-fluid equation of state. In the revised version we will add a dedicated subsection that (i) presents the phase portrait of the unperturbed Hamiltonian in the (P,V) plane, (ii) identifies the saddle point at the critical volume, and (iii) analytically integrates the unperturbed equations of motion to exhibit the homoclinic orbit connecting the saddle to itself. This addition will place the subsequent Melnikov integrals and the derived threshold conditions on charge q and the string-fluid parameter on a rigorous footing. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard Melnikov analysis to given EOS without self-referential reduction.

full rationale

The paper derives a chaos threshold condition for temporal perturbations directly from the Melnikov integral evaluated on the thermodynamic Hamiltonian constructed from the Hayward-string-fluid equation of state. Charge q and the string parameter appear as explicit model inputs in that EOS and thus in the integral; the resulting inequality on perturbation amplitude gamma is a direct consequence rather than a tautological re-expression or fitted output renamed as prediction. The assumption of an unperturbed homoclinic orbit is stated as a prerequisite for applying Melnikov's method but is not derived from or equivalent to the target chaos condition itself. No self-citation chain, ansatz smuggling, or renaming of known results is used to justify the central claim. The Lyapunov exponent computation is an independent numerical quantification performed after the threshold analysis. The derivation chain remains self-contained against the external Melnikov framework and the supplied EOS.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The analysis rests on the applicability of Melnikov's method to the thermodynamic Hamiltonian and on the existence of a suitable unperturbed orbit; no new particles or forces are introduced.

free parameters (2)
  • perturbation amplitude gamma
    Critical threshold for onset of chaos is determined by q and string fluid parameter.
  • string fluid parameter
    Enters both the chaos threshold and the amplitude of the Lyapunov exponent.
axioms (1)
  • domain assumption Melnikov's approach can be applied to the perturbed Hamiltonian dynamics of the black hole thermodynamic system.
    Invoked to detect the onset of chaotic behavior from the equation of state.

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