Chaos in Schwarzschild Spacetime : The Motion of a Spinning Particle

Kei-ichi Maeda; Shingo Suzuki

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Chaos in Schwarzschild Spacetime : The Motion of a Spinning Particle

Shingo Suzuki, Kei-ichi Maeda

General Relativity and Quantum Cosmology · Submitted 9 Apr 1996 · 40 references

Pith's verdict: A real first: spin-orbit coupling alone makes a test-particle orbit around Schwarzschild chaotic, with a clean four-type potential classification — but the chaos lives at spins where the body would be super-extremal.

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CitationKei-ichi Maeda, Shingo Suzuki. "Chaos in Schwarzschild Spacetime : The Motion of a Spinning Particle." arXiv:gr-qc/9604020 (1996). https://pith.science/paper/R7AXF6R5

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Editor's note

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Worth your time, with one caveat that matters more for the framing than for the math. The result is genuine: a spinless particle in Schwarzschild is integrable, and Suzuki and Maeda show that adding spin-orbit coupling through the Papapetrou-Dixon equations breaks the tori. The Poincaré maps at J = 4 mu M go from nested closed curves at S = 0.4-0.6 mu M to a spreading sea of chaos by S = 1.0-1.4 mu M, and the Lyapunov exponent jumps sharply past S_cr ~ 0.64 mu M. The four-type effective-potential classification (B1/B2/U1/U2) is the part I'd actually steal: B2, with two saddle points bifurcating off the equatorial plane, is where the chaos lives, and the heteroclinic orbit connecting the two hyperbolic fixed points is a plausible mechanism. The numerics look careful — Bulirsch-Stoer, constraint residuals under 1e-11 after 10^3 orbits, Sano-Sawada estimator rather than a geodesic-deviation formula that wouldn't apply here. They also correctly note the 'effective potential' is only a boundary contour, not a true potential, since the spin direction is a dynamical variable. That's the right kind of honesty and it's not common enough in this literature. Two soft spots. First, the Lyapunov exponents of orbits (1) and (3), which look regular in the Poincaré maps, also converge to positive values. The authors flag this themselves and offer two explanations — insufficient time series, or non-integrability invisible in the section. Either way the quantitative diagnostic is weaker than the torus-breaking, and the qualitative claim rests on the maps. Second, and this is where I'd push back on the reader's framing: the problem is not that the pole-dipole truncation fails. The expansion parameter is S/(mu r) ~ M/r, which is order unity at worst, so the equations are probably fine. The problem is that S ~ mu M with mu << M means S/mu^2 >> 1, i.e. a super-extremal body. The authors say so in section III.A and argue by analogy to comparable-mass binaries, which is an extrapolation, not a justification. So the math is likely right and the astrophysical relevance is the open question. Minor: the figure numbering slips — the text cites Fig. 7 for the J = 3.81 Poincaré map while the caption has it as the Lyapunov-vs-spin plot. Send it out. This is the reference for spin-orbit chaos in Schwarzschild and the caveats are addressable.

Referee report

Significance

If correct, this is a clean and physically well-motivated result: a spinless test particle in Schwarzschild is integrable, so any chaos is attributable purely to spin-orbit coupling, and the paper isolates a mechanism (saddle-point bifurcation off the equatorial plane) rather than merely reporting a positive exponent. The effective-potential classification is a genuinely useful organizing device that yields a falsifiable criterion (chaos only in B2) and a concrete threshold. The numerical hygiene is good: constraint residuals below 10^-11 after 10^3 orbital periods, an explicit initial-data construction, and a stated Lyapunov time of a few orbital periods. The paper is also candid about the unphysical spin values required for a test particle and about the incompleteness of the heteroclinic-tangle argument.

Major objections (4)

[§III.A, Fig. 6 and Fig. 7] The authors report that orbits (1) and (3), whose Poincaré sections show unbroken tori, nevertheless yield positive Lyapunov exponents, and they offer two explanations without resolving which applies. This matters because Fig. 7, the λ-versus-S curve presented as quantitative confirmation of S_cr, uses the same estimator. If the estimator returns positive λ for regular orbits, the rise in Fig. 7 after S_cr cannot be read as the onset of chaos. A concrete test would be to apply the Sano–Sawada estimator to a spinless geodesic, where integrability requires λ = 0, and to orbit (1) with several values of ǫ, m and Δt and longer time series, reporting convergence. In addition, the estimator is applied to a 4D projection (ρ, z, v_ρ, v_z) of a flow in which the spin orientation is itself dynamical; a projection of a higher-dimensional flow can inflate the apparent exponent, and this should be checked explicitly.
[§II.C, Eqs. (2.30)–(2.32)] The boundary of the allowed region is defined as the curve where p_r and p_θ vanish simultaneously. That is a stronger condition than the boundary of the projection of the orbit onto the (r,θ) plane, which is where either p_r or p_θ vanishes. The assertion that ``the particle with energy E can move in the region of the r-θ plane such that V² < E²'' therefore needs justification, or at least numerical verification. A concrete test would be to take the chaotic orbit (2) of Fig. 5, record its turning points in r and in θ, and check whether they lie on the V contour. If the true allowed region is larger than the one bounded by V, the four-type classification and the value of S_cr read off Fig. 3 may shift.
[§III.A and Fig. 3] The value S_cr ≈ 0.64 μM is read off the boundary between B1 and B2 in the (J,S) plane, i.e., it is a property of the potential, and its identification with the onset of chaos is empirical. The Poincaré maps sample only six values of S at fixed J = 4 μM, and the initial conditions (E, r0, α) differ between panels. The spin angle α is not given in the Fig. 4 caption or in the surrounding text, so the maps cannot be reproduced. Please tabulate α for every run and show a scan in E and r0 at S just below and just above S_cr, so that the threshold is not an artifact of the chosen slice through initial-condition space.
[§III.A, physical realizability] The authors acknowledge that S ~ μM with μ << M implies S/μ² ~ M/μ >> 1, i.e., a super-extremal body, and argue by analogy with comparable-mass binaries. That analogy is an extrapolation: it replaces the fixed Schwarzschild background and the test-particle limit with a comparable-mass system for which the background is not Schwarzschild. What remains unresolved is a stated domain of validity. The abstract's closing claim that this ``may become important in some relativistic astrophysical phenomena'' should be tied to a specific regime, and the S = 1.4 μM case, which the authors themselves call unrealistic, should be labelled explicitly as a mathematical demonstration rather than a physical one.

Minor objections (7)

[§III.B] The figure cross-references are off by one. The text says ``Fig.7 shows the Poincaré map of five orbits with J = 3.81 μM'', but the Poincaré classification is Fig. 8 and Fig. 7 is the λ-versus-S plot; similarly, ``Its behavior is shown in Fig.8'' should refer to Fig. 9. The sentence also says ``five orbits'' while listing four types.
[Fig. 8 caption] The caption reads ``E = 0923μ'', which is missing a decimal point and should presumably read E = 0.923μ.
[§II.B] The statement that for α outside [π/2, 3π/2] ``the particle cannot get into any relativistic region and chaos never occurs, as we will see later'' is not in fact shown later in the paper; either supply the demonstration or drop the forward reference.
[§II.C] The phrase ``the type (U2) still appears with physically meaningful values of S and J'' is loose, since S ~ 0.987 μM with μ << M still violates S ≤ μ² for any Kerr source; ``physically meaningful'' should be replaced by a precise statement of what is meant.
[§II.B] The assertion that S_{φr} never changes its signature during the evolution is used to fix the choice between V(+) and V(−), but no argument or numerical check is given; a one-line proof or a statement of the numerical verification would help.
[§III.A] The sentence ``This is the first example of chaotic behavior in the motion of a test particle in a Schwarzschild spacetime'' should be qualified, since refs. [12] and [13] concern perturbed Schwarzschild spacetimes; the priority claim is for a spinning test particle in exact Schwarzschild.
[References] Ref. [6] is listed as ``to be published'' and is used in the introduction to support the local-instability criterion for chaos; the status of this reference should be updated or the reliance on it softened.

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The paper

The paper claims

A spinning test particle orbiting a Schwarzschild black hole can move chaotically once its spin exceeds a critical value, because spin-orbit coupling breaks the integrability that keeps a spinless orbit regular.

The reading

The paper asks whether a small spinning body falling around a non-rotating black hole can move chaotically, even though a spinless particle in the same spacetime cannot, since its motion is integrable. Using the standard pole-dipole (Papapetrou-Dixon) equations with the supplementary condition $p_%s S^{%s} = 0$, the authors integrate the orbits numerically and diagnose chaos two ways: broken tori in a Poincaré section taken on the equatorial plane, and positive Lyapunov exponents computed from a time series. They find chaos setting in above a critical spin, $S_{cr} ~ 0.64 mu M$ for total angular momentum $J = 4 mu M$, with the inverse Lyapunov exponent in the most chaotic case about three orbital periods. They also build an 'effective potential' $V(r, theta; J, S)$ from the curve where both radial and meridional momentum vanish, and use its shape to sort orbits into four types; chaos appears only in type (B2), where two saddle points bifurcate off the equatorial plane. The payoff, if it holds, is that a purely relativistic spin-orbit effect can make an otherwise regular orbit unpredictable on a timescale of a few orbits, which matters for any astrophysical system where spin is dynamically relevant.

What carries the argument

The load-bearing construction is the 'effective potential' $V^{(pm)}(r, theta; J, S)$, obtained by setting $p_r = p_theta = 0$ (which also forces $S^{r theta} = 0$, pinning the spin into the meridian plane) and solving the resulting quadratic for $sinh X$ in terms of $J$, $S$, $r$ and $theta$. The particle is confined to the region where $V^2 < E^2$, so a contour map of $V$ reveals the allowed region and its saddles without ever solving the full orbit. The two branches $V^{(+)}$ and $V^{(-)}$ correspond to spin anti-parallel and parallel to the total angular momentum; the paper studies $V^{(-)}$, where the spin-orbit interaction acts repulsively and lets a stable orbit sit closer to the horizon. The classification of $V$ into B1, B2, U1 and U2 is what converts the numerical chaos result into a criterion: the existence of two off-equatorial saddle points is the structural feature that correlates with broken tori. The diagnostics carrying the evidence are the Poincaré section at $theta = pi/2$ plotting $(r, v_r)$ on downward crossings, and the Sano-Sawada algorithm for extracting a Lyapunov exponent from a sampled time series of $(rho, z, v_rho, v_z)$, needed because a spinning particle does not follow a geodesic and the geodesic deviation equation does not apply.

Extended reading notes

Core claim

The central claim is that spin-orbit coupling alone makes the motion of a spinning test particle in Schwarzschild spacetime chaotic. A spinless particle there is integrable and never chaotic, so any chaos is attributable to the spin. The authors report a critical spin below which the Poincaré map consists of closed curves and above which some tori break and the points scatter: for $J = 4 mu M$ the threshold is $S_{cr} ~ 0.64 mu M$, and the Lyapunov exponent rises steeply once $S$ passes it. The effective-potential classification explains where this happens. For large $J$ and small spin the potential has one saddle and one minimum on the equatorial plane (type B1, same as the spinless case); for large spin the single saddle splits into two saddles placed symmetrically off the equatorial plane while the minimum stays on it (type B2). For small $J$ there are no bound orbits at all: either no extremal point (U1) or one saddle on the equatorial plane that is unstable in the perpendicular direction (U2). Chaos is found only in type B2, and the authors identify a heteroclinic orbit joining the two hyperbolic fixed points of the Poincaré section as the likely mechanism, with the chaotic sea spreading around it as the energy is raised.

Load-bearing premise

The whole result rests on the pole-dipole truncation being valid at the spins where chaos appears, yet chaos needs $S$ of order $mu M$, which for a test particle with $mu$ far below $M$ means the dimensionless spin $S/mu^2$ is of order $M/mu$, far outside the regime where higher multipole moments can be dropped; the authors acknowledge this in Section III.A and argue by analogy with comparable-mass binary systems rather than justifying it for a test particle.

Editorial extensions

If this is right

Chaos in this system is a sharp threshold phenomenon: for fixed $J$, orbits are regular below $S_{cr}$ and become chaotic above it, so the spin magnitude alone can be used to sort which orbits are predictable.
The effective-potential type is a usable predictor of chaos without any numerical integration, since chaos appears only where two saddle points bifurcate off the equatorial plane (type B2).
The inverse Lyapunov exponent of about three orbital periods means two nearby spinning orbits separate measurably within a handful of rotations, so long-term trajectory prediction for a spinning body near a black hole fails on an astrophysically short timescale.
Because the effect is purely spin-orbit, it disappears in the spinless limit and in the equatorial-plane orbits where the spin is aligned with or against the orbital angular momentum, which tells observers where to look for it and where not to.
The heteroclinic orbit connecting the two hyperbolic fixed points gives a concrete geometric target: chaos should be strongest for energies near the saddle-point energy $E_{sp}$, which the authors confirm numerically.

Reading between the lines

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

If the mechanism is a heteroclinic tangle, then the same signature should appear in any spacetime where spin-orbit coupling produces a pair of off-equatorial saddles, including Kerr; the authors themselves report preliminary evidence that the critical spin drops as the black hole spin rises.
The chaotic band sits at energies just below the saddle-point energy, which suggests a clean observational proxy: a spinning body spiralling in through gravitational-wave emission will cross that band and its waveform should show a transient loss of phase coherence rather than a permanent one.
Because the effect requires spin comparable to $mu M$, a test-particle treatment is being pushed outside its comfort zone; a natural check is whether the same threshold survives when the quadrupole moment of the body is restored, which is a calculation nobody in this paper performs.
The classification by potential type is a cheap screening tool that could be applied to other metric-plus-spin problems before any orbit integration, turning a numerical search into a root-finding exercise on the saddle structure.

Original abstract

We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincar\'e map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g. $S_{cr} \sim 0.64 \mu M$ for the total angular momentum $J=4 \mu M$), where $\mu$ and $M$ are the masses of a particle and of a black hole, respectively. The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods, which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena. The ``effective potential" analysis enables us to classify the particle orbits into four types as follows. When the total angular momentum $J$ is large, some orbits are bounded and the ``effective potential"s are classified into two types: (B1) one saddle point (unstable circular orbit) and one minimal point (stable circular orbit) on the equatorial plane exist for small spin; and (B2) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin. When $J$ is small, no bound orbits exist and the potentials are classified into another two types: (U1) no extremal point is found for small spin; and (U2) one saddle point appears on the equatorial plane, which is unstable in the direction perpendicular to the equatorial plane, for large spin. The types (B1) and (U1) are the same as those for a spinless particle, but the potentials (B2) and (U2) are new types caused by spin-orbit coupling. The chaotic behavior is found only in the type (B2) potential. The ``heteroclinic orbit'', which could cause chaos, is also observed in type (B2).

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.63 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincar\'e map and the Lyapunov exponent. We find chaotic behavior for a particle with spin higher than some critical value (e.g. $S_{cr} \sim 0.64 \mu M$ for the total angular momentum $J=4 \mu M$), where $\mu$ and $M$ are the masses of a particle and of a black hole, respectively. The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods, which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena. The ``effective potential" analysis enables us to classify the particle orbits into four types as follows. When the total angular momentum $J$ is large, some orbits are bounded and the ``effective potential"s are classified into two types: (B1) one saddle point (unstable circular orbit) and one minimal point (stable circular orbit) on the equatorial plane exist for small spin; and (B2) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin. When $J$ is small, no bound orbits exist and the potentials are classified into another two types: (U1) no extremal point is found for small spin; and (U2) one saddle point appears on the equatorial plane, which is unstable in the direction perpendicular to the equatorial plane, for large spin. The types (B1) and (U1) are the same as those for a spinless particle, but the potentials (B2) and (U2) are new types caused by spin-orbit coupling. The chaotic behavior is found only in the type (B2) potential. The ``heteroclinic orbit'', which could cause chaos, is also observed in type (B2).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the motion of a spinning test particle in Schwarzschild spacetime, analyzing the Poincar\'e map and the Lyapunov exponent.	contribution 0.98	0.93	0.05	asserted 0.78
s1	We find chaotic behavior for a particle with spin higher than some critical value (e.g. $S_{cr} \sim 0.64 \mu M$ for the total angular momentum $J=4 \mu M$), where $\mu$ and $M$ are the masses of a particle and of a black hole, respectively.	result 1.00	0.96	0.68	asserted 1.00
s2	The inverse of the Lyapunov exponent in the most chaotic case is about three orbital periods, which suggests that chaos of a spinning particle may become important in some relativistic astrophysical phenomena.	interpretation 0.87	0.91	0.22	hedged possibility 0.95
s3	The ``effective potential" analysis enables us to classify the particle orbits into four types as follows.	method 0.72	0.87	0.03	asserted 0.93
s4	When the total angular momentum $J$ is large, some orbits are bounded and the ``effective potential"s are classified into two types: (B1) one saddle point (unstable circular orbit) and one minimal point (stable circular orbit) on the equatorial plane exist for small spin; and (B2) two saddle points bifurcate from the equatorial plane and one minimal point remains on the equatorial plane for large spin.	result 0.98	0.90	0.07	asserted 1.00
s5	When $J$ is small, no bound orbits exist and the potentials are classified into another two types: (U1) no extremal point is found for small spin; and (U2) one saddle point appears on the equatorial plane, which is unstable in the direction perpendicular to the equatorial plane, for large spin.	result 0.99	0.91	0.05	asserted 1.00
s6	The types (B1) and (U1) are the same as those for a spinless particle, but the potentials (B2) and (U2) are new types caused by spin-orbit coupling.	result 0.60	0.86	0.08	asserted 1.00
s7	The chaotic behavior is found only in the type (B2) potential.	result 1.00	0.95	0.27	asserted 1.00
s8	The ``heteroclinic orbit'', which could cause chaos, is also observed in type (B2).	result 0.97	0.91	0.09	hedged possibility 0.58

What kind of paper the classifier reports this is: simulation study 0.75 · evidence from simulation or computation 0.95.

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sources read

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not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: The “effective potential”

**Figure 3.** FIG. 3: The types of the “effective potential” are classified b

**Figure 6.** FIG. 6: The Lyapunov exponent of each orbit shown in Fig.5. We

**Figure 8.** FIG. 8: The classification of the orbits for

**Figure 9.** FIG. 9: When the energy increases from that in Fig.8, the toru

Showing 5 of the paper's 9 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9604020 by the authors.

References

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Recently, it has been pointed out that deformation of a n eutron star by a tidal force may change the innermost stable orbit in a binary system (see Lai, F. A. Raiso and S. L. Shapiro, ApJ. 420, 811 (1994)). It is one of the most important eﬀects on observ ation of gravitational waves. 16 Figure Captions FIG. 1: The “eﬀective potential” V(±) on the equato...

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Linear stability matrix and Lyapunov exponents for stationary spacetime
hep-th 2026-10 accept novelty 6.0 of 10

A general framework for computing Lyapunov exponents of geodesic orbits in stationary spacetimes, extending beyond equatorial circular orbits to spherical and non-equatorial orbits.

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Gravitational-wave signatures of spinning particles around rotating hairy black holes
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For a rotating gravitational-decoupling hairy black hole, the main waveform signature of both secondary spin and hair is accumulated dephasing, and the hair's radial reach, not its amplitude, controls how distinguisha...
Analytic motions of spinning particles in Schwarzschild-(anti-)de Sitter spacetime
gr-qc 2026-09 accept novelty 6.0 of 10

Analytic solutions for all orbital types of spinning test particles in Schwarzschild-(anti-)de Sitter spacetimes, expressed via Lauricella hypergeometric functions.
Spin-Hair Induced Chaos of Spinning Test Particles in Rotating Hairy Black Holes
gr-qc 2026-05 unverdicted novelty 6.0 of 10

Spinning test particles around rotating hairy black holes show finite-time instability in localized regions of the (spin, hair-parameter) plane that reorganize the strong-field phase space compared to Kerr.
Chaotic orbital dynamics of pulsating stars around black holes surrounded by dark matter halos
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Spherically symmetric pulsations trigger homoclinic chaos in orbits around black holes enclosed by dark matter halos, unlike the vacuum Schwarzschild case.
Asymptotic gravitational-wave fluxes from a spinning test body on generic orbits around a Kerr black hole
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Gravitational-wave energy and angular-momentum fluxes are computed to first order in the secondary's spin for generic eccentric and inclined orbits around a Kerr black hole, and cross-checked against time-domain and e...
Hamilton-Jacobi equation for spinning particles near black holes
gr-qc 2019-03 conditional novelty 6.0 of 10

A perturbative solution of the Hamilton-Jacobi equation for spinning particles in Kerr spacetime gives analytic spin corrections to orbital turning points and fundamental frequencies.
Universality in Chaos of Particle Motion near Black Hole Horizon
hep-th 2016-10 conditional novelty 6.0 of 10

A particle kept just outside a black hole horizon has a maximal Lyapunov exponent equal to the horizon's surface gravity, independent of the external force and the particle mass.
Periastron advance in spinning black hole binaries: comparing effective-one-body and Numerical Relativity
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An untuned effective-one-body model reproduces the periastron advance measured in 19 numerical-relativity spinning black-hole simulations to within the simulations' error bars, and the authors give closed-form precess...
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Numerical chaos indicators applied to the Schwarzschild-Bertotti-Robinson-Bonnor-Melvin family show that chaos occurs without swirling and that electromagnetic field strengths and directions tightly restrict bound orbits.
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A Schwarzschild black hole embedded in a Dehnen dark-matter halo admits homoclinic orbits, and raising halo density, scale radius, or particle energy drives near-horizon motion from regular to chaotic while respecting...
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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).
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Using three independent holographic methods, the authors obtain matching butterfly velocities for four QCD-like models and find a universal increase with temperature and decrease with chemical potential.
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How the literature uses it

37 arXiv papers cite this in 60 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 28 sentences without a confident label

Contests it 1 paper

In the limit of circular equatorial orbits, traces of the onset of chaos were found to persist as an instability in the meridional direction [ref] [Note that there are several typos in the matrix elements used to compute the Lyapunov exponents in Ref. [ref] .].

Periastron advance in spinning black hole binaries: comparing effective-one-body and Numerical Relativity · Periastron advance and precession frequencies · contest 0.73

Builds on it 1 paper

Finally, from <ref>, and using the relation $E= V_±(r,θ, J, S)$, we obtain the expression for the potential of the spinning particle moving on the boundary of a region where $p^r=0=p^θ$ in $r-θ$ plane as [equation] The above <ref> reduces to the effective potential for Schwarzschild case [ref] in the prescribed limit $α=0$.

Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field · Effective Potential for SQBH · 0.83

Uses its method or tool 2 papers

The use of a deviation vector method and a pre saturation fitting window is closely related to earlier numerical studies of spinning particle chaos in black hole spacetimes [ref] .

Spin-Hair Induced Chaos of Spinning Test Particles in Rotating Hairy Black Holes · Deviation-vector method · 0.91
In the future, we would intend to examine the chaos of spinning particles using processes such as the Fast Lyapunov Indicator [ref] , Poincaré map [ref] , and Milnikov technique [ref] to gain a deeper understanding of the choatic orbits around the SQBH.

Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field · Concluding remarks · 0.90

Compares with it 5 papers

This behavior is qualitatively consistent with the picture emphasized by Suzuki and Maeda, namely that the sea of chaos spreads as the spin increases, and it is also in agreement with the result of Zelenka et al. for the case $S=10^-4μM$, where typical signatures of a non-integrable system already appear for EMRI-relevant spin values [ref] .

Astrophysically Realistic Secondary Spins Trigger Chaos in Schwarzschild Spacetime and Discernible Gravitational Wave Signatures · Chaotic Signatures in Phase Space · 0.87
This is similar to the observation made in [ref] for the Schwarzschild BH.

Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field · Classification of the Effective Potential and possibility of chaotic orbits · 1.00
In addition, the chaotic motion of spinning particles associated with some spherically symmetric spacetimes [ref] is not seen here due to the existence of the above-mentioned constants of motion.

Spinning test particles in the spacetime of a global monopole · General motion · 0.88

Cites it as support 1 paper

Since, the effective potential analysis have shown a good indications for a chaotic behaviour in spinning particles [ref] .

Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field · Effective Potential for SQBH · 0.98

Cites it as background 20 papers

Within the classical framework, the black hole (BH) horizon's influence on particle trajectories and inducing chaos has a rich history in these preceding works [ref] , which shows that this topic has promising ground for huge exploration.

Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo: Existence of Homoclinic Orbit and Horizon-Induced Chaos · Introduction · 0.92
For a detailed discussion on classification of $V_eff$, one can refer to Suzuki and Maeda [ref] .

Chaos motion and Periastron precession of spinning test particles moving in the vicinage of a Schwarzschild black hole surrounded by a quintessence matter field · Classification of the Effective Potential and possibility of chaotic orbits · 0.92
In particular, the Lyapunov exponent of a probe around a black hole has been studied extensively [ref] (and also [ref] ).

Violation of bound on chaos for charged probe in Kerr-Newman-AdS black hole · Introduction · 0.95

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