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A Schwarzschild-like black hole embedded in a Dehnen dark-matter halo admits a homoclinic orbit that separates bound from plunging motion, and near-horizon particle motion becomes chaotic as halo density, scale radius, or energy increase—wh

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2026-08-03 23:54 UTC pith:AJLYDAVJ

load-bearing objection A worthwhile but currently inconsistent paper: the homoclinic analysis rests on an effective potential that does not follow from the stated metric, and the chaos section is an openly toy-model study. the 4 major comments →

arxiv 2511.03657 v2 pith:AJLYDAVJ submitted 2025-11-05 gr-qc hep-thnlin.CD

Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo: Existence of Homoclinic Orbit and Horizon-Induced Chaos

Surajit Das , Surojit Dalui , Bum-Hoon Lee , Yi-Fu Cai This is my paper
classification gr-qc hep-thnlin.CD
keywords extreme-mass-ratio inspiraldark matter halohomoclinic orbitchaosLyapunov exponentsurface gravityDehnen profileblack hole geodesics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that a massive compact object inspiraling around a supermassive black hole embedded in a Dehnen-(1,4,5/2) dark-matter halo can possess a homoclinic orbit—a phase-space separatrix between bound inspiral and plunging into the horizon—when its angular momentum lies between the ISCO and marginally-bound-orbit values. The halo's density and scale radius shift the ISCO and MBO and deform the effective potential. The paper further claims that adding near-horizon harmonic confinement (an external potential) and increasing energy, halo density, or scale radius drives the breakup of KAM tori, producing chaos, while both total and radial Lyapunov exponents stay below the surface gravity of the BH-DM spacetime, so the universal chaos bound is respected. If true, this links dark-matter halo structure to chaotic dynamics in EMRI systems and gives a theoretical handle on gravitational-wave signatures.

Core claim

For a Schwarzschild-like black hole embedded in a Dehnen-(1,4,5/2) dark-matter halo, the paper establishes, in the extreme-mass-ratio limit q=m/M<<1, the existence of a homoclinic orbit for angular momenta L in (L_ISCO, L_MBO). This orbit, associated with an unstable circular orbit at r_un, forms the boundary between bound and plunging motion. The paper then shows numerically that in Painlevé-Gullstrand coordinates, with the particle confined near the horizon by external harmonic potentials, increasing the halo density ρ_s, scale radius r_s, or particle energy leads to the destruction of invariant tori and the onset of chaos, as measured by Poincaré sections and Lyapunov exponents. Throughou

What carries the argument

The central object is the homoclinic orbit—the separatrix in the (r, p_r) phase space that emerges from the unstable circular orbit at r_un, defined by the effective potential of the BH-DM metric. The chaotic analysis is carried by the Hamiltonian in Painlevé-Gullstrand coordinates, in which the near-horizon particle is confined by two hand-imposed harmonic potentials, a(r)=½K_r(r−r_c)^2 and b(φ)=½K_φ r_H^2 φ^2; the paper then monitors Poincaré sections and Lyapunov exponents against the surface gravity κ of the combined spacetime.

Load-bearing premise

The chaos claim depends entirely on the externally imposed harmonic potentials that confine the particle near the horizon; the paper itself states these are not an inherent feature of the BH-DM spacetime, and Appendix A shows an unconfined particle would plunge or escape.

What would settle it

Integrate the exact geodesic equations for the metric f(r)=1−2M/r−32π ρ_s r_s^2 sqrt(1+r_s/r)/r without the external harmonic potentials: the unstable equilibrium at r_un should yield a smooth homoclinic loop with zero Lyapunov exponent, and nearby trajectories should either plunge or escape. If, contrary to the paper's expectation, additional bounded chaotic trajectories appear without the traps, the horizon-and-halo origin of the chaos would be supported; if no chaos appears, the positive Lyapunov exponents are artifacts of the harmonic confinement. A more quantitative check: smoothly reduce

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the paper's claim is correct, EMRI systems in galactic nuclei with Dehnen-type halos can exhibit chaotic orbital motion in the near-horizon region, affecting the phase evolution of emitted gravitational waves.
  • The location of the chaos transition depends on halo density and scale radius, offering a potential way to infer dark-matter halo parameters from observed EMRI waveforms.
  • The existence of the homoclinic orbit fixes a sharp separatrix between inspiral and plunge, which could improve the accuracy of EMRI waveform templates that currently ignore such boundaries.
  • The surface-gravity bound on chaos is respected in the BH-DM halo environment, meaning that the universal MSS bound remains valid even when extended dark-matter structures are present.
  • The results provide a theoretical basis for a companion work connecting horizon-induced chaos to gravitational-wave observables for future space-based detectors.

Where Pith is reading between the lines

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

  • The homoclinic-orbit existence result is geodesic and independent of the external harmonic potentials, so that part of the claim stands on the metric itself; the chaotic regime, however, is conditional on the traps—without them, as the paper's Appendix A shows, the particle would plunge or escape rather than remain confined.
  • If the hand-imposed harmonic potentials are read as proxies for generic near-horizon perturbations (magnetic fields, accretion disks, or scalar fields), the bounded-chaos result may generalize to those physical environments, but that extension is not established here.
  • A natural testable extension is to apply the Melnikov method to the homoclinic orbit with the harmonic perturbation, which would yield an analytic criterion for transversal homoclinic intersections and thus an independent prediction for the onset of chaos.
  • The paper's companion work on gravitational-wave signatures could look for a distinctive dephasing or burst when an inspiraling orbit crosses the homoclinic separatrix; that observable would distinguish chaos-induced modulations from ordinary adiabatic inspiral.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the motion of a massive test particle around a Schwarzschild-like black hole embedded in a Dehnen-(1,4,5/2) dark-matter halo. In the EMRI limit q=10^-5, the authors construct the radial effective potential, locate the ISCO and MBO, and identify a homoclinic orbit for L_ISCO < L < L_MBO that separates bound from plunging geodesics. The second half of the paper places the particle in Painlevé-Gullstrand coordinates and adds externally imposed harmonic potentials a(r) and b(phi). Numerical integration of the resulting equations yields Poincaré sections, orbital plots, and Lyapunov exponents, which the authors use to claim that increasing the halo density, scale radius, or particle energy drives the system from regular to chaotic motion while keeping both the total and radial Lyapunov exponents below the surface-gravity (MSS) bound.

Significance. If established, the homoclinic-orbit result would provide a concrete separatix in a realistic BH-DM halo spacetime and extend earlier near-horizon chaos studies to astrophysical environments. The paper has several strengths: it derives the metric from the Dehnen density profile, uses the conserved energy and angular momentum explicitly, presents numerical tables for homoclinic parameters, and is transparent about the fact that the harmonic potentials are externally imposed. However, the quantitative claims are currently undermined by an apparent inconsistency between the metric and the effective potential, and the chaos section describes a toy model whose connection to the BH-DM halo spacetime is not established. These issues must be resolved before the astrophysical conclusions can be accepted.

major comments (4)
  1. [Sec. 3, Eq. (3.5)] The effective potential (3.5) does not follow from the metric (2.5). Starting from Eq. (3.3) with f(r) = 1 - 2M/r - 32πρ_s r_s^2/r sqrt(1+r_s/r), the substitution x=2M/r gives the DM contribution (4M^2/L^2) f_DM (m^2 + L^2/r^2) = -16πρ_s r_s^2 x/M sqrt(1+x r_s/(2M)) (x^2 + 4M^2 m^2/L^2). The printed term in Eq. (3.5), -32πρ_s r_s^2 sqrt(1+x r_s/(2M)) (x^2 + 4M^2 m^2/L^2), is what one would obtain if f_DM had no 1/r factor. Since Eq. (3.5) is used for the ISCO/MBO calculations and for Tables 1-2, all homoclinic parameters and the claimed L_ISCO < L < L_MBO window are not yet shown to hold for the spacetime (2.5). Please correct Eq. (3.5) (or the metric) and recompute; the same inconsistency appears in Eq. (3.13), where the 1/r factor present in Eq. (3.12) is dropped.
  2. [Sec. 4.1, Eq. (4.4)] The chaos analysis uses hand-imposed harmonic potentials a(r)=1/2 K_r (r-r_c)^2 and b(phi)=1/2 K_phi r_H^2 phi^2, with K_r=100, K_phi=25, and r_c=3.65 chosen rather than derived. The paper explicitly calls these 'an externally imposed constraint rather than an inherent feature' of the spacetime. Therefore the Poincaré sections and Lyapunov exponents in Sec. 4.2 describe a particle in a confining trap, not a particle moving solely in the BH-DM halo geometry. The abstract's claim that increasing halo density/scale radius 'leads to chaos' in an EMRI embedded in a DM halo is not supported unless these potentials are shown to represent a realistic physical interaction. Please either derive the potentials from a physical model or clearly restrict the conclusions to this toy model and adjust the abstract and conclusions accordingly.
  3. [Sec. 4.2, Eq. (4.6)] The energy values used in the chaotic-dynamics section (E=90,95,108,115,118,122,130,132.5, and E=140-156 in Table 3) appear inconsistent with the stated EMRI scaling q=10^-5. In the geodesic section, bound energies are of order 10^-5 with M=1 and m=10^-5 (e.g., Table 1). In Eq. (4.6), with m=10^-5 and external potentials of order unity, E~100 would require momenta that are far larger than the near-horizon EMRI values used earlier. The paper does not state the units of E for the chaos section, and p_phi is computed from Eq. (4.6). This ambiguity affects all Lyapunov exponents and the comparison with kappa in Figs. 16-17. Please specify the normalization explicitly and ensure consistency with q=10^-5, or explain the rescaling used.
  4. [Sec. 2, Eq. (2.18)] The horizon radius formula (2.18) is not obviously the solution of f(r)=0 for f given in Eq. (2.5). As written, the structure of Eq. (2.18) differs from what one would obtain from the transcendental equation r - 2M - 32πρ_s r_s^2 sqrt(1+r_s/r)=0, which follows from Eq. (2.5) when the 1/r factor is included. Because r_H enters the surface gravity through Eq. (4.14) and is used in the figures and the chaos bound, the derivation of Eq. (2.18) should be shown explicitly, or the expression should be reconciled with Eq. (2.5).
minor comments (5)
  1. [Table 2, row (0.35, 4.25)] The entry '10521493' appears to be missing a decimal point; it should likely read 10.521493.
  2. [Sec. 3.2, text after Eq. (3.8)] The text refers to 'the fourth and fifth terms' in Eq. (3.5), but Eq. (3.5) as written has four terms. Please revise the wording.
  3. [Sec. 4.2.1, Figs. 7-10] The color coding of the Poincaré sections is not described in the text. Please add a note stating that different colors represent different randomly chosen initial conditions.
  4. [Sec. 4.2.4, Figs. 16-17] The quantities (λ_T^2 - κ^2) and (λ_r^2 - κ^2) are all negative and of order -0.05, with differences visible only at the level of a few 10^-6. Reporting λ/κ or the ratio (κ-λ)/κ would be more informative and would make the saturation of the bound easier to assess.
  5. [References] Reference [75] is an unpublished companion paper. If it is essential for the gravitational-wave claims in the introduction and conclusions, the connection should be described in more detail or the claims should be softened.

Circularity Check

0 steps flagged

No circular step meets the evidentiary bar; homoclinic bounds and Lyapunov exponents are numerical outputs of the stated Hamiltonian. Score 2 reflects non-load-bearing self-citations and the admitted toy-model trapping potential, not circularity.

full rationale

Walking the derivation chain: Eq. (3.3) is the standard geodesic equation with f(r) from Eq. (2.5); the homoclinic energy in Tables 1–2 is obtained by solving dV_eff/dx=0 and then V_eff(x_un)=E_eff, so the separatrix energy is an output of the effective potential, not an input. The ISCO/MBO bounds L_ISCO<L<L_MBO are likewise computed from the extrema of the same potential; no parameter is fitted to data and no fitted quantity is renamed as a prediction. The chaos section uses explicit external potentials a(r)=1/2 K_r(r-r_c)^2 and b(phi)=1/2 K_phi r_H^2 phi^2 with K_r=100, K_phi=25, r_c=3.65. In Sec. 4.1 the paper itself labels these potentials 'an externally imposed constraint rather than an inherent feature' and states that 'alternative potential choices would modify the trajectory dynamics'; this is a model-dependence limitation, not a circular reduction, since lambda_T and lambda_r are numerically computed from that declared Hamiltonian. Refs [31,34] and the unpublished companion [75] are methodological/anticipatory self-citations, but the paper re-derives the near-horizon instability in Appendix A and computes the MSS bound from its own kappa, so those citations are not load-bearing. The skeptic-flagged mismatch between Eq. (2.5), whose DM term carries a 1/r factor, and Eq. (3.5)/(3.13), whose DM term does not, is an internal-consistency error affecting which spacetime the tables describe; it is not a case of a prediction being equal to its input by construction. No circular step meets the required quote-and-reduction standard.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The central chaos claim rests on arbitrary potential strengths and on a leading-order, A=B-approximated metric; no data fitting and no new particles, but several model choices are not independently justified.

free parameters (5)
  • K_r (radial potential strength) = 100
    Strength of the radial harmonic trap; chosen by hand in Sec. 4.2, not derived from the BH-DM system.
  • K_phi (angular potential strength) = 25
    Strength of the angular harmonic trap; chosen by hand in Sec. 4.2.
  • r_c (center of radial potential) = 3.65
    Center of the radial potential; set close to the horizon in all four cases to confine the particle near the unstable maximum.
  • Energy values for chaos runs = E=90, 95, 108, 115, 118, 122, 130, 132.5 (model units)
    Selected to illustrate regular-to-onset-to-chaotic transitions; no independent physical calibration is given.
  • Lyapunov saturation time = t*=2500
    Cutoff at which the authors declare exponential separation saturated; reported lambda values depend on this choice.
axioms (5)
  • domain assumption The combined BH-DM metric of Eq. (2.5) is a valid leading-order solution from the A(r)=B(r) approximation.
    Inherited from refs [66,77,78]; all homoclinic and chaos computations use this metric without re-deriving or validating higher-order terms.
  • ad hoc to paper External harmonic potentials a(r), b(phi) faithfully represent near-horizon physical interactions.
    Stated as an externally imposed constraint in Sec. 4.1; no physical origin is given.
  • domain assumption The MSS bound lambda_T <= kappa applies to this classical particle system.
    Adopted from refs [26,90]; the paper checks but does not derive the bound.
  • domain assumption Painlevé-Gullstrand coordinates preserve the chaos properties.
    Justified by prior work [34] and by horizon regularity, not re-derived here.
  • domain assumption Dehnen-(1,4,5/2) halo is an appropriate astrophysical environment for EMRIs.
    Motivated observationally; but the halo parameters are scanned over arbitrary ranges, not fitted to a specific galaxy.
invented entities (1)
  • External harmonic potentials a(r)=1/2 K_r (r-r_c)^2, b(phi)=1/2 K_phi r_H^2 phi^2 no independent evidence
    purpose: Confine the probe particle near the horizon so bounded chaotic motion can be studied despite the unstable maximum.
    The paper explicitly calls these externally imposed constraints; they are not derived from the DM halo or BH and carry no independent falsifiable prediction.

pith-pipeline@v1.3.0-alltime-deepseek · 32569 in / 33518 out tokens · 273417 ms · 2026-08-03T23:54:12.787883+00:00 · methodology

0 comments
read the original abstract

We study the existence of homoclinic orbit and the onset of chaotic motion for a massive particle moving around a Schwarzschild-like black hole embedded in a Dehnen-(1,4,5/2) type dark matter halo, within the extreme-mass-ratio limit q=m/M<<1, where m and M are the masses of the particle and the central black hole, respectively. The presence of the halo modifies the spacetime curvature and consequently deforms the effective potential governing the particle's motion. Using the Hamiltonian formulation, we derive the conditions under which unstable circular orbit and the associated homoclinic trajectory arise, marking the separatrix between bound and plunging motion. By analyzing the effective potential and the corresponding phase-space structure, we identify the transition from regular to chaotic dynamics in the near-horizon region. Numerical analyses through Poincare sections and Lyapunov exponents calculations demonstrate that increasing the halo density, scale radius along with energy amplifies nonlinear effects which leads to chaos eventually. We demonstrate that within a dark matter halo environment, the dynamical stability of particle motion can be significantly altered without violating the universal surface gravity bound on chaos. This work provides a deeper understanding of horizon-induced chaos in astrophysically realistic environments and serves as a theoretical basis for exploring its possible imprints on gravitational wave signals in extreme-mass-ratio inspirals system.

discussion (0)

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Chaotic particle dynamics near a traversable wormhole throat

    gr-qc 2026-07 conditional novelty 6.0

    Confined test particles near a wormhole throat show high-energy chaos coexisting with surviving KAM tori, in contrast to horizon-driven chaos in black holes.

  2. Black hole spacetimes with dark matter spikes: Energy-momentum tensor and backreaction effects

    gr-qc 2025-11 conditional novelty 6.0

    A dark-matter spike built from the full orbital motion of its particles has ~50% more energy density near the black hole and produces metric deviations ~2.5 times larger than mass-only models.

  3. Chaotic behaviors of particles around the black hole with an anisotropic matter immersed in a magnetic field

    gr-qc 2026-07 unverdicted novelty 4.0

    Exact black hole solution with anisotropic matter and magnetic field shows the matter parameter reduces local chaos (Lyapunov exponent) while the magnetic field drives qualitative shifts in global chaos (Poincaré sections).

  4. Chaotic imprints of dark matter in extreme mass-ratio inspirals

    gr-qc 2026-02 reject novelty 4.0

    A toy model shows that adding a hand-made angle-dependent force to dark-matter spacetimes produces visual chaos and irregular numerical-Kludge waveforms, but the force is not a realistic dark-matter perturbation.

Reference graph

Works this paper leans on

85 extracted references · 59 linked inside Pith · cited by 4 Pith papers

  1. [12]

    Chandrasekhar,The mathematical theory of black holes (Oxford Classic Texts in the Physical Sciences)

    S. Chandrasekhar,The mathematical theory of black holes (Oxford Classic Texts in the Physical Sciences). Clarendon Press, 1998

    1998

  2. [13]

    Carter,Global structure of the kerr family of gravitational fields,Phys

    B. Carter,Global structure of the kerr family of gravitational fields,Phys. Rev.174(Oct, 1968) 1559–1571

    1968

  3. [14]

    V. P. Frolov, P. Krtous and D. Kubiznak,Black holes, hidden symmetries, and complete integrability,Living Rev. Rel.20(2017) 6, [1705.05482]

    Pith/arXiv arXiv 2017

  4. [15]

    Suzuki and K.-i

    S. Suzuki and K.-i. Maeda,Signature of chaos in gravitational waves from a spinning particle, Phys. Rev. D61(2000) 024005, [gr-qc/9910064]

    Pith/arXiv arXiv 2000

  5. [16]

    Kiuchi and K.-i

    K. Kiuchi and K.-i. Maeda,Gravitational waves from chaotic dynamical system,Phys. Rev. D 70(2004) 064036, [gr-qc/0404124]

    Pith/arXiv arXiv 2004

  6. [17]

    Bombelli and E

    L. Bombelli and E. Calzetta,Chaos around a black hole,Class. Quant. Grav.9(1992) 2573–2599

    1992

  7. [18]

    Y. Sota, S. Suzuki and K.-i. Maeda,Chaos in static axisymmetric space-times. 1: Vacuum case,Class. Quant. Grav.13(1996) 1241–1260, [gr-qc/9505036]

    Pith/arXiv arXiv 1996

  8. [19]

    W. M. Vieira and P. S. Letelier,Chaos around a Henon-Heiles inspired exact perturbation of a black hole,Phys. Rev. Lett.76(1996) 1409–1412, [gr-qc/9604037]. – 34 –

    Pith/arXiv arXiv 1996

  9. [20]

    Suzuki and K.-i

    S. Suzuki and K.-i. Maeda,Chaos in Schwarzschild space-time: The motion of a spinning particle,Phys. Rev. D55(1997) 4848–4859, [gr-qc/9604020]

    Pith/arXiv arXiv 1997

  10. [21]

    N. J. Cornish and N. E. Frankel,The Black hole and the pea,Phys. Rev. D56(1997) 1903–1907

    1997

  11. [22]

    A. P. S. de Moura and P. S. Letelier,Chaos and fractals in geodesic motions around a nonrotating black hole with an external halo,Phys. Rev. E61(2000) 6506–6516, [chao-dyn/9910035]

    Pith/arXiv arXiv 2000

  12. [23]

    M. D. Hartl,Dynamics of spinning test particles in Kerr space-time,Phys. Rev. D67(2003) 024005, [gr-qc/0210042]

    Pith/arXiv arXiv 2003

  13. [24]

    Han,Chaos and dynamics of spinning particles in Kerr spacetime,Gen

    W. Han,Chaos and dynamics of spinning particles in Kerr spacetime,Gen. Rel. Grav.40 (2008) 1831–1847, [1006.2229]

    Pith/arXiv arXiv 2008

  14. [25]

    Takahashi and H

    M. Takahashi and H. Koyama,Chaotic motion of Charged Particles in an Electromagnetic Field Surrounding a Rotating Black Hole,Astrophys. J.693(2009) 472–485, [0807.0277]

    Pith/arXiv arXiv 2009

  15. [26]

    Hashimoto and N

    K. Hashimoto and N. Tanahashi,Universality in Chaos of Particle Motion near Black Hole Horizon,Phys. Rev. D95(2017) 024007, [1610.06070]

    Pith/arXiv arXiv 2017

  16. [27]

    Li and X

    D. Li and X. Wu,Chaotic motion of neutral and charged particles in a magnetized Ernst-Schwarzschild spacetime,Eur. Phys. J. Plus134(2019) 96, [1803.02119]

    Pith/arXiv arXiv 2019

  17. [28]

    Lei, X.-H

    Y.-Q. Lei, X.-H. Ge and C. Ran,Chaos of particle motion near a black hole with quasitopological electromagnetism,Phys. Rev. D104(2021) 046020, [2008.01384]

    Pith/arXiv arXiv 2021

  18. [29]

    De Falco and W

    V. De Falco and W. Borrelli,Detection of chaos in the general relativistic Poynting-Robertson effect: Kerr equatorial plane,Phys. Rev. D103(2021) 064014, [2001.04979]

    Pith/arXiv arXiv 2021

  19. [30]

    De Falco and W

    V. De Falco and W. Borrelli,Timescales of the chaos onset in the general relativistic Poynting-Robertson effect,Phys. Rev. D(5, 2021) , [2105.00965]

    Pith/arXiv arXiv 2021

  20. [31]

    Dalui, B

    S. Dalui, B. R. Majhi and P. Mishra,Presence of horizon makes particle motion chaotic,Phys. Lett. B788(2019) 486–493, [1803.06527]

    Pith/arXiv arXiv 2019

  21. [32]

    Dalui, B

    S. Dalui, B. R. Majhi and P. Mishra,Induction of chaotic fluctuations in particle dynamics in a uniformly accelerated frame,Int. J. Mod. Phys. A35(2020) 2050081, [1904.11760]

    Pith/arXiv arXiv 2020

  22. [33]

    A. Bera, S. Dalui, S. Ghosh and E. C. Vagenas,Quantum corrections enhance chaos: Study of particle motion near a generalized Schwarzschild black hole,Phys. Lett. B829(2022) 137033, [2109.00330]

    Pith/arXiv arXiv 2022

  23. [34]

    S. Das, S. Dalui and R. Samanta,Near-horizon chaos beyond Einstein gravity,Phys. Rev. D 110(2024) 124037, [2405.09945]

    Pith/arXiv arXiv 2024

  24. [35]

    Levin and G

    J. Levin and G. Perez-Giz,Homoclinic Orbits around Spinning Black Holes. I. Exact Solution for the Kerr Separatrix,Phys. Rev. D79(2009) 124013, [0811.3814]

    Pith/arXiv arXiv 2009

  25. [36]

    Perez-Giz and J

    G. Perez-Giz and J. Levin,Homoclinic Orbits around Spinning Black Holes II: The Phase Space Portrait,Phys. Rev. D79(2009) 124014, [0811.3815]

    Pith/arXiv arXiv 2009

  26. [37]

    Jeong, B.-H

    S. Jeong, B.-H. Lee, H. Lee and W. Lee,Homoclinic orbit and the violation of the chaos bound around a black hole with anisotropic matter fields,Phys. Rev. D107(2023) 104037, [2301.12198]

    Pith/arXiv arXiv 2023

  27. [38]

    Polcar and O

    L. Polcar and O. Semer´ ak,Free motion around black holes with discs or rings: Between integrability and chaos. VI. The Melnikov method,Phys. Rev. D100(2019) 103013, [1911.09790]

    Pith/arXiv arXiv 2019

  28. [39]

    Guckenheimer and P

    J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences, 42). Springer, New York, 1983. – 35 –

    1983

  29. [40]

    Wiggins,Introduction to Applied Nonlinear Dynamical Systems and Chaos

    S. Wiggins,Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 2003

    2003

  30. [41]

    S. A. Hughes,Gravitational waves from extreme mass ratio inspirals: Challenges in mapping the space-time of massive, compact objects,Class. Quant. Grav.18(2001) 4067–4074, [gr-qc/0008058]

    Pith/arXiv arXiv 2001

  31. [42]

    Babak, J

    S. Babak, J. Gair, A. Sesana, E. Barausse, C. F. Sopuerta, C. P. L. Berry et al.,Science with the space-based interferometer LISA. V: Extreme mass-ratio inspirals,Phys. Rev. D95(2017) 103012, [1703.09722]

    Pith/arXiv arXiv 2017

  32. [43]

    Amaro-Seoane,Relativistic dynamics and extreme mass ratio inspirals,Living Rev

    P. Amaro-Seoane,Relativistic dynamics and extreme mass ratio inspirals,Living Rev. Rel.21 (2018) 4, [1205.5240]. [44]LISAcollaboration, P. Amaro-Seoane et al.,Laser Interferometer Space Antenna,1702.00786. [45]TianQincollaboration, J. Mei et al.,The TianQin project: current progress on science and technology,PTEP2021(2021) 05A107, [2008.10332]

    Pith/arXiv arXiv 2018

  33. [46]

    Hu and Y.-L

    W.-R. Hu and Y.-L. Wu,The Taiji Program in Space for gravitational wave physics and the nature of gravity,Natl. Sci. Rev.4(2017) 685–686

    2017

  34. [47]

    Bertone, D

    G. Bertone, D. Hooper and J. Silk,Particle dark matter: Evidence, candidates and constraints, Phys. Rept.405(2005) 279–390, [hep-ph/0404175]

    Pith/arXiv arXiv 2005

  35. [48]

    Bovy and H.-W

    J. Bovy and H.-W. Rix,A Direct Dynamical Measurement of the Milky Way’s Disk Surface Density Profile, Disk Scale Length, and Dark Matter Profile at 4 kpc < ∼R < ∼9 kpc,Astrophys. J.779(2013) 115, [1309.0809]

    Pith/arXiv arXiv 2013

  36. [49]

    de Swart, G

    J. de Swart, G. Bertone and J. van Dongen,How Dark Matter Came to Matter,Nature Astron. 1(2017) 0059, [1703.00013]

    Pith/arXiv arXiv 2017

  37. [50]

    R. H. Wechsler and J. L. Tinker,The Connection between Galaxies and their Dark Matter Halos,Ann. Rev. Astron. Astrophys.56(2018) 435–487, [1804.03097]

    Pith/arXiv arXiv 2018

  38. [51]

    Bertone and T

    G. Bertone and T. Tait, M. P.,A new era in the search for dark matter,Nature562(2018) 51–56, [1810.01668]

    Pith/arXiv arXiv 2018

  39. [52]

    Dubinski and R

    J. Dubinski and R. G. Carlberg,The Structure of cold dark matter halos,Astrophys. J.378 (1991) 496

    1991

  40. [53]

    Burkert,The Structure of dark matter halos in dwarf galaxies,Astrophys

    A. Burkert,The Structure of dark matter halos in dwarf galaxies,Astrophys. J. Lett.447 (1995) L25, [astro-ph/9504041]

    Pith/arXiv arXiv 1995

  41. [54]

    J. F. Navarro, C. S. Frenk and S. D. M. White,The Structure of cold dark matter halos, Astrophys. J.462(1996) 563–575, [astro-ph/9508025]

    Pith/arXiv arXiv 1996

  42. [55]

    J. F. Navarro, C. S. Frenk and S. D. M. White,A Universal density profile from hierarchical clustering,Astrophys. J.490(1997) 493–508, [astro-ph/9611107]

    Pith/arXiv arXiv 1997

  43. [56]

    Hernquist,An Analytical Model for Spherical Galaxies and Bulges,Astrophys

    L. Hernquist,An Analytical Model for Spherical Galaxies and Bulges,Astrophys. J.356(1990) 359

    1990

  44. [57]

    Jaffe,A Simple model for the distribution of light in spherical galaxies,Mon

    W. Jaffe,A Simple model for the distribution of light in spherical galaxies,Mon. Not. Roy. Astron. Soc.202(1983) 995–999

    1983

  45. [58]

    Tremaine, D

    S. Tremaine, D. O. Richstone, Y.-I. Byun, A. Dressler, S. M. Faber, C. Grillmair et al.,A Family of models for spherical stellar systems,Astron. J.107(1994) 634, [astro-ph/9309044]

    Pith/arXiv arXiv 1994

  46. [59]

    A. A. Dutton and A. V. Macci` o,Cold dark matter haloes in the Planck era: evolution of structural parameters for Einasto and NFW profiles,Mon. Not. Roy. Astron. Soc.441(2014) 3359–3374, [1402.7073]. – 36 –

    Pith/arXiv arXiv 2014

  47. [60]

    A. W. Graham, D. Merritt, B. Moore, J. Diemand and B. Terzic,Empirical models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models,Astron. J.132(2006) 2685–2700, [astro-ph/0509417]

    Pith/arXiv arXiv 2006

  48. [61]

    L. A. Urena-Lopez, T. Matos and R. Becerril,Inside oscillatons,Class. Quant. Grav.19(2002) 6259–6277

    2002

  49. [62]

    Harko,Bose-Einstein condensation of dark matter solves the core/cusp problem,JCAP05 (2011) 022, [1105.2996]

    T. Harko,Bose-Einstein condensation of dark matter solves the core/cusp problem,JCAP05 (2011) 022, [1105.2996]

    Pith/arXiv arXiv 2011

  50. [63]

    K. G. Begeman, A. H. Broeils and R. H. Sanders,Extended rotation curves of spiral galaxies: Dark haloes and modified dynamics,Mon. Not. Roy. Astron. Soc.249(1991) 523

    1991

  51. [64]

    Dehnen,A family of potential-density pairs for spherical galaxies and bulges,Mon

    W. Dehnen,A family of potential-density pairs for spherical galaxies and bulges,Mon. Not. Roy. Astron. Soc.(11, 1993) 250–256

    1993

  52. [65]

    Shukirgaliyev, A

    B. Shukirgaliyev, A. Otebay, M. Sobolenko, M. Ishchenko, O. Borodina, T. Panamarev et al., Bound mass of Dehnen models with a centrally peaked star formation efficiency,A&A654 (2021) A53, [2105.09510]

    Pith/arXiv arXiv 2021

  53. [66]

    Al-Badawi, S

    A. Al-Badawi, S. Shaymatov and Y. Sekhmani,Schwarzschild black hole in galaxies surrounded by a dark matter halo,JCAP02(2025) 014, [2411.01145]

    Pith/arXiv arXiv 2025

  54. [67]

    Alloqulov, T

    M. Alloqulov, T. Xamidov, S. Shaymatov and B. Ahmedov,Gravitational waveforms from periodic orbits around a Schwarzschild black hole embedded in a Dehnen-type dark matter halo, Eur. Phys. J. C85(2025) 798, [2504.05236]

    Pith/arXiv arXiv 2025

  55. [68]

    Liang, D

    Q.-Q. Liang, D. Liu and Z.-W. Long,Quasinormal modes of Schwarzschild black holes in the Dehnen-(1, 4, 5/2) type dark matter halos,Eur. Phys. J. C85(2025) 1107, [2505.15540]

    Pith/arXiv arXiv 2025

  56. [69]

    Z. Luo, M. Tang and Z. Xu,Shadows and observational images of a Schwarzschild-like black hole surrounded by a Dehnen-type dark matter halo,JCAP10(2025) 065, [2505.20115]

    arXiv 2025

  57. [70]

    S. Rani, A. Jawad, M. Heydari-Fard and U. Zafar,Thermodynamic and shadow analysis of Dehnen type dark matter Halo corrected Schwarzschild black hole surrounded by thin disk,Eur. Phys. J. C85(2025) 677

    2025

  58. [71]

    Xamidov, S

    T. Xamidov, S. Shaymatov, Q. Wu and T. Zhu,Probing the Schwarzschild black hole immersed in a dark matter halo through astrophysical tests,Eur. Phys. J. C85(2025) 1193, [2507.13147]

    arXiv 2025

  59. [72]

    Ashoorioon, R

    A. Ashoorioon, R. Casadio, K. Jafarzade, M. B. Jahani Poshteh and O. Luongo,Gravitational radiation reaction around a static black hole surrounded by a Dehnen type dark matter halo, 2509.08569

    arXiv

  60. [73]

    Bertone and D

    G. Bertone and D. Merritt,Dark matter dynamics and indirect detection,Mod. Phys. Lett. A 20(2005) 1021, [astro-ph/0504422]

    Pith/arXiv arXiv 2005

  61. [74]

    Gondolo and J

    P. Gondolo and J. Silk,Dark matter annihilation at the galactic center,Phys. Rev. Lett.83 (1999) 1719–1722, [astro-ph/9906391]

    Pith/arXiv arXiv 1999

  62. [75]

    S. Das, S. Dalui, B.-H. Lee and Y.-F. Cai,Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo II: Chaos-Induced Gravitational Waves,Manuscript in preperation(2025)

    2025

  63. [76]

    H. Mo, F. van de Bosch and S. White,Galaxy formation and evolution. Cambridge University Press, Cambridge, United Kingdom, 2010

    2010

  64. [77]

    Z. Xu, X. Hou, X. Gong and J. Wang,Black Hole Space-time In Dark Matter Halo,JCAP09 (2018) 038, [1803.00767]

    Pith/arXiv arXiv 2018

  65. [78]

    Matos and D

    T. Matos and D. Nunez,The general relativistic geometry of the Navarro - Frenk - White model,Rev. Mex. Fis.51(2005) 71–75, [astro-ph/0303594]

    Pith/arXiv arXiv 2005

  66. [79]

    Azreg-A ¨ ınou,Generating rotating regular black hole solutions without complexification, Phys

    M. Azreg-A ¨ ınou,Generating rotating regular black hole solutions without complexification, Phys. Rev. D90(2014) 064041, [1405.2569]. – 37 –

    Pith/arXiv arXiv 2014

  67. [80]

    Shakeshaft,The formation and dynamics of galaxies (International Astronomical Union Symposia, 58)

    J. Shakeshaft,The formation and dynamics of galaxies (International Astronomical Union Symposia, 58). Springer, Germany,, 1974

    1974

  68. [81]

    Matos, F

    T. Matos, F. S. Guzman and D. Nunez,Spherical scalar field halo in galaxies,Phys. Rev. D62 (2000) 061301, [astro-ph/0003398]

    Pith/arXiv arXiv 2000

  69. [82]

    Matos, D

    T. Matos, D. Nunez and R. A. Sussman,The Spacetime associated with galactic dark matter halos,Gen. Rel. Grav.37(2005) 769–779, [astro-ph/0402157]

    Pith/arXiv arXiv 2005

  70. [83]

    Padmanabhan,Gravitation: Foundations and frontiers

    T. Padmanabhan,Gravitation: Foundations and frontiers. Cambridge University Press, Cambridge, United Kingdom, 2014

    2014

  71. [84]

    Painleve,La mecanique classique et la theorie de relativite,C.R

    P. Painleve,La mecanique classique et la theorie de relativite,C.R. Hebd. Seances Acad. Sci. 173(1921) 677

    1921

  72. [85]

    Martel and E

    K. Martel and E. Poisson,Regular coordinate systems for Schwarzschild and other spherical space-times,Am. J. Phys.69(2001) 476–480, [gr-qc/0001069]

    Pith/arXiv arXiv 2001

  73. [86]

    T. W. Baumgarte and S. L. Shapiro,Numerical Relativity: Solving Einstein ’s Equations on the Computer. Cambridge University Press, Cambridge, England, 2010

    2010

  74. [87]

    S. H. Strogatz,Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Chapman and Hall/CRC, Abingdon, Oxon, 2024

    2024

  75. [88]

    H. J. St¨ ockmann,Quantum Chaos, An Introduction. Cambridge University Press, Cambridge, United Kingdom, 1999

    1999

  76. [89]

    Sandri,Numerical calculation of Lyapunov Exponents

    M. Sandri,Numerical calculation of Lyapunov Exponents. University of Verona, Italy, 1995

    1995

  77. [90]

    Maldacena, S

    J. Maldacena, S. H. Shenker and D. Stanford,A bound on chaos,JHEP08(2016) 106, [1503.01409]

    Pith/arXiv arXiv 2016

  78. [91]

    B. Gwak, N. Kan, B.-H. Lee and H. Lee,Violation of bound on chaos for charged probe in Kerr-Newman-AdS black hole,JHEP09(2022) 026, [2203.07298]

    Pith/arXiv arXiv 2022

  79. [92]

    Zhao, Y.-Z

    Q.-Q. Zhao, Y.-Z. Li and H. Lu,Static Equilibria of Charged Particles Around Charged Black Holes: Chaos Bound and Its Violations,Phys. Rev. D98(2018) 124001, [1809.04616]

    Pith/arXiv arXiv 2018

  80. [93]

    Lei and X.-H

    Y.-Q. Lei and X.-H. Ge,Circular motion of charged particles near a charged black hole,Phys. Rev. D105(2022) 084011, [2111.06089]

    Pith/arXiv arXiv 2022

Showing first 80 references.