pith. sign in

arxiv: 2407.04397 · v2 · pith:BFDLHY2Enew · submitted 2024-07-05 · 🌊 nlin.CD

Efficient detection of chaos through the computation of the Generalized Alignment Index (GALI) by the multi-particle method

Bertin Many Manda , Malcolm Hillebrand , Charalampos Skokos This is my paper

Pith reviewed 2026-05-25 08:43 UTC · model grok-4.3

classification 🌊 nlin.CD
keywords GALIchaos detectionmulti-particle methodHamiltonian systemsnumerical methodsFermi-Pasta-Ulam-TsingouHenon-Heilesdeviation vectors
0
0 comments X

The pith

Multi-particle method computes Generalized Alignment Index for chaos detection without variational equations.

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

The paper presents a multi-particle method (MPM) to calculate the Generalized Alignment Index (GALI), a chaos indicator, by tracking differences between nearby trajectories instead of solving variational equations. It derives leading-order error estimates for both the variational and multi-particle approaches, then validates them through detailed simulations on the two-degree-of-freedom Hénon-Heiles system and the multidimensional β-Fermi-Pasta-Ulam-Tsingou oscillator chain. The MPM with double precision (ε ≈ 10^{-16}) remains reliable when deviation vector sizes are around ε^{1/2}, renormalization times stay below 1, and relative energy errors are kept below ε^{1/2}. These findings apply to systems with many degrees of freedom. Readers care because the method removes the need to derive and integrate variational equations, opening GALI-based chaos studies to a much wider range of Hamiltonian models.

Core claim

The multi-particle method (MPM) implemented with double precision accuracy (ε ≈ 10^{-16}) performs reliably for deviation vector sizes d0 ≈ ε^{1/2}, renormalization times τ ≲ 1, and relative energy errors Er ≲ ε^{1/2}. These results hold for systems with many degrees of freedom and demonstrate that the MPM is a robust and efficient method for studying the chaotic dynamics of Hamiltonian systems by eliminating the need for variational equations.

What carries the argument

The multi-particle method (MPM), which approximates deviation vectors through finite differences between multiple nearby trajectories rather than integrating linearized variational equations.

If this is right

  • GALI-based chaos detection becomes feasible for high-dimensional Hamiltonian systems where deriving variational equations is impractical.
  • Numerical studies of chaotic dynamics can now use standard integration routines without additional linearized equations.
  • The method maintains accuracy across the tested models when deviation vector size, renormalization time, and energy error satisfy the stated bounds.
  • Exploration of chaotic behavior is enabled in a much larger class of many-body Hamiltonian systems.

Where Pith is reading between the lines

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

  • The same multi-particle idea could be tested on other deviation-vector-based indicators such as the smaller alignment index.
  • In simulations with thousands of particles the computational savings might allow longer integration times or ensemble studies.
  • The error bounds might be used to select integration step sizes automatically in adaptive codes.

Load-bearing premise

The leading-order error estimation derived for the multi-particle method accurately captures the dominant numerical errors across the tested models and parameter regimes, and the two prototypical systems are representative of general Hamiltonian systems with many degrees of freedom.

What would settle it

Run the MPM and variational GALI computations in parallel on a third Hamiltonian system with at least ten degrees of freedom, using the stated parameter ranges, and check whether the GALI values agree to within the predicted leading-order error bounds.

Figures

Figures reproduced from arXiv: 2407.04397 by Bertin Many Manda, Charalampos Skokos, Malcolm Hillebrand.

Figure 1
Figure 1. Figure 1: Theoretical reliable regions for the computation of the GALI of order [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 2D HH system. The PSS of the HH model [Eq. (38)] at x1 = 0 for H = 0.125. For each point (x2, p2) of the PSS, the remaining coordinate p1 is found such that p1 > 0. are simultaneously integrated by the ABA864 integrator using the so-called ‘tangent map method’ [27, 28]. In addition, we perform double precision computations, i.e., ε ≈ 10−16, as this is the default numeric type of almost all modern computati… view at source ↗
Figure 4
Figure 4. Figure 4: 2D HH system. Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 2D HH system. Dependence of the GALIs on the norm d0 of the initial deviation vectors. Practical reliable regions of the computation of the (a) GALI2, (b) GALI3 and (c) GALI4 for the HH model using the VM (blue triangles) and the MPM (red circles). The presented GALI values have been averaged over 1000 sets of initial orthogonal tangent vectors of norm d0 = ∥w0∥ = ∥δ0∥ about the reference chaotic orbit wit… view at source ↗
Figure 6
Figure 6. Figure 6: 2D HH system. Time evolution of the (a) GALI2, (b) GALI3 and (c) GALI4 for the regular orbit of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: 2D HH system. Dependence of the upper bound of the time evo￾lution of the relative energy error Er (41) on the integration time step h, for the chaotic orbit of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 2D HH system. Regions of different dynamical behavior on the x1 = 0, p1 > 0 PSS of the HH system (38) with H = 0.125, identified through (a) the MLE values at t = 1000, and (b) the GALI2 values at t = 500. (c) Regions of different values of the time t needed for the GALI3 to become less than 10−8 for the same set of initial conditions as in panels (a) and (b). In all computations h = 0.01, τ = 1 and d0 = 1… view at source ↗
Figure 10
Figure 10. Figure 10: β-FPUT chain. The time evolution of the ftmLE λ1 for two perturbations of the SPO1 of the β-FPUT model with N = 5 oscillators: (a) A regular orbit for energy H5 = 5, and (b) a chaotic orbit with energy H5 = 10, whose initial conditions are given in Eq. (B.1) and Eq. (B.2) respectively. Note that in both panels the solid blue curves correspond to results computed using the VM, while red dotted and green da… view at source ↗
Figure 11
Figure 11. Figure 11: β-FPUT chain. Time evolution of several computed GALIs for the regular [(a) and (b)] and the chaotic [(c) and (d)] orbits of [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: β-FPUT chain. Reliable regions for the computation of the (a) GALI2, (b) GALI4, (c)GALI6, and (d) GALI8 of the chaotic orbit of the β-FPUT lattice considered in [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
read the original abstract

We present a method for the computation of the Generalized Alignment Index (GALI), a fast and effective chaos indicator, using a multi-particle approach that avoids variational equations. We show that this approach is robust and accurate by deriving a leading-order error estimation for both the variational (VM) and the multi-particle (MPM) methods, which we validate by performing extensive numerical simulations on two prototypical models: the two degrees of freedom H\'enon-Heiles system and the multidimensional $\beta$-Fermi-Pasta-Ulam-Tsingou chain of oscillators. The dependence of the accuracy of the GALI on control parameters such as the renormalization time, the integration time step and the deviation vector size is studied in detail. We test the MPM implemented with double precision accuracy ($\varepsilon \approx 10^{-16}$) and find that it performs reliably for deviation vector sizes $d_0\approx \varepsilon^{1/2}$, renormalization times $\tau \lesssim 1$, and relative energy errors $E_r \lesssim \varepsilon^{1/2}$. These results hold for systems with many degrees of freedom and demonstrate that the MPM is a robust and efficient method for studying the chaotic dynamics of Hamiltonian systems. Our work makes it possible to explore chaotic dynamics with the GALI in a vast number of systems by eliminating the need for variational equations.

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

2 major / 2 minor

Summary. The manuscript introduces a multi-particle method (MPM) for computing the Generalized Alignment Index (GALI) without solving variational equations. It derives leading-order error estimates for both the variational method (VM) and MPM, validates the approach via extensive numerical simulations on the 2DOF Hénon-Heiles system and the multidimensional β-Fermi-Pasta-Ulam-Tsingou chain, studies dependence on renormalization time, integration step, and deviation vector size, and recommends reliable parameter windows (d0 ≈ ε^{1/2}, τ ≲ 1, Er ≲ ε^{1/2}) for double-precision implementations, claiming these hold for systems with many degrees of freedom.

Significance. If the error estimates are accurate and the method generalizes beyond the tested models, the MPM would enable efficient GALI-based chaos detection in high-dimensional Hamiltonian systems where variational equations are costly to implement, representing a practical advance for studying chaotic dynamics in many-body problems. The explicit parameter recommendations and numerical validation on two models constitute concrete strengths.

major comments (2)
  1. [Abstract] Abstract and concluding section: the claim that 'these results hold for systems with many degrees of freedom' and enable exploration 'in a vast number of systems' is load-bearing for the central conclusion, yet rests solely on validation for the 2DOF Hénon-Heiles system and a nearest-neighbor 1D β-FPUT chain; no analytic argument or cross-check on a long-range or all-to-all Hamiltonian is supplied to confirm that the bound Er ≲ ε^{1/2} and the (d0, τ) window transfer when global error accumulation differs.
  2. [Error estimation section] Error estimation section (leading-order derivation for MPM): the analysis is presented as capturing dominant numerical errors across tested regimes, but the derivation's implicit assumptions (e.g., regarding locality of interactions in deviation-vector propagation or renormalization) are not stated explicitly, creating a correctness-risk for the generality assertion over arbitrary many-DOF Hamiltonians.
minor comments (2)
  1. [Notation] The notation for relative energy error Er and deviation vector size d0 should be defined at first use with an explicit equation reference to aid readers.
  2. [Figures] Figure captions for the numerical results on the two models could more clearly indicate the number of realizations and integration tolerances used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments below, agreeing that the claims of generality require qualification and that assumptions in the error analysis should be stated explicitly. We will make targeted revisions to the abstract, conclusions, and error estimation section.

read point-by-point responses

  1. Referee: [Abstract] Abstract and concluding section: the claim that 'these results hold for systems with many degrees of freedom' and enable exploration 'in a vast number of systems' is load-bearing for the central conclusion, yet rests solely on validation for the 2DOF Hénon-Heiles system and a nearest-neighbor 1D β-FPUT chain; no analytic argument or cross-check on a long-range or all-to-all Hamiltonian is supplied to confirm that the bound Er ≲ ε^{1/2} and the (d0, τ) window transfer when global error accumulation differs.

    Authors: We agree that the numerical evidence is restricted to the two tested models and that no additional cross-check on long-range interactions is provided. The leading-order error estimates were derived from the general structure of the deviation equations and renormalization without assuming nearest-neighbor locality. Nevertheless, to avoid overstatement we will revise the abstract and concluding section to qualify the parameter recommendations as validated for the Hénon-Heiles and β-FPUT systems and expected to apply more broadly on the basis of the error analysis, while noting that verification on long-range Hamiltonians remains desirable. revision: partial

  2. Referee: [Error estimation section] Error estimation section (leading-order derivation for MPM): the analysis is presented as capturing dominant numerical errors across tested regimes, but the derivation's implicit assumptions (e.g., regarding locality of interactions in deviation-vector propagation or renormalization) are not stated explicitly, creating a correctness-risk for the generality assertion over arbitrary many-DOF Hamiltonians.

    Authors: We will revise the error estimation section to list the assumptions explicitly: (i) deviation vectors remain small enough for the leading-order Taylor expansion of the flow to hold, (ii) renormalization is performed by rescaling the Euclidean norm at fixed intervals, and (iii) the dominant error sources are local truncation error of the integrator and floating-point round-off, independent of interaction range. The derivation uses only the general form of the Hamiltonian vector field and does not invoke locality. revision: yes

Circularity Check

0 steps flagged

No circularity: error derivation and numerical validation are independent

full rationale

The paper derives a leading-order error estimation for both VM and MPM methods from first principles and validates the resulting accuracy bounds through direct numerical simulations on the Hénon-Heiles and β-FPUT models. No equation or claim reduces by construction to a fitted parameter renamed as a prediction, a self-referential definition, or a load-bearing self-citation chain; the central reliability statements for many-DOF systems rest on the explicit derivations and tests rather than on quantities defined in terms of themselves.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The claim depends on standard assumptions of numerical integration accuracy and the representativeness of the two tested models; no new entities are postulated.

free parameters (2)
  • deviation vector size d0
    Control parameter chosen approximately as ε^{1/2} for double precision to balance truncation and round-off errors.
  • renormalization time τ
    Control parameter studied and bounded by τ ≲ 1 for reliable performance.
axioms (1)
  • domain assumption Numerical integration schemes preserve the leading-order error behavior assumed in the analytic estimates.
    Invoked when deriving and validating the leading-order error formulas for both VM and MPM.

pith-pipeline@v0.9.0 · 5786 in / 1236 out tokens · 26820 ms · 2026-05-25T08:43:53.062904+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    R. C. Hilborn, Chaos and nonlinear dynamics: an introduction for scien- tists and engineers, Oxford University Press on Demand, 2000

    work page 2000

  2. [2]

    Carpintero, N

    D. Carpintero, N. Ma ffione, L. Darriba, Lp-vicode: A program to com- pute a suite of variational chaos indicators, Astronomy and Computing 5 (2014) 19

    work page 2014

  3. [3]

    Skokos, G

    C. Skokos, G. A. Gottwald, J. Laskar, Chaos Detection and Predictability, V ol. 915 of Lecture Notes in Physics, Springer, 2016

    work page 2016

  4. [4]

    Pikovsky, A

    A. Pikovsky, A. Politi, Lyapunov exponents: a tool to explore complex dynamics, Cambridge University Press, 2016

    work page 2016

  5. [5]

    Benettin, L

    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov character- istic exponents for smooth dynamical systems and for Hamiltonian sys- tems; a method for computing all of them. Part 1: Theory, Meccanica 15 (1980) 9

    work page 1980

  6. [6]

    Benettin, L

    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov character- istic exponents for smooth dynamical systems; a method for computing all of them. Part 2: Numerical application, Meccanica 15 (1980) 21

    work page 1980

  7. [7]

    Skokos, The Lyapunov characteristic exponents and their computation, Lecture Notes in Physics 790 (2010) 63

    C. Skokos, The Lyapunov characteristic exponents and their computation, Lecture Notes in Physics 790 (2010) 63

    work page 2010

  8. [8]

    Awrejcewicz, A

    J. Awrejcewicz, A. V . Krysko, N. P. Erofeev, V . Dobriyan, M. A. Barulina, V . A. Krysko, Quantifying chaos by various computational methods. Part 1: Simple systems, Entropy 20 (2018) 175

    work page 2018

  9. [9]

    Rautenberg, M

    M. Rautenberg, M. G ¨arttner, Classical and quantum chaos in a three-mode bosonic system, Phys. Rev. A 101 (2020) 053604

    work page 2020

  10. [10]

    Szumi ´nski, A new model of variable-length coupled pendulums: from hyperchaos to superintegrability, Nonlinear Dynamics 112 (2024) 4117– 4145

    W. Szumi ´nski, A new model of variable-length coupled pendulums: from hyperchaos to superintegrability, Nonlinear Dynamics 112 (2024) 4117– 4145

    work page 2024

  11. [11]

    Szumi ´nski, A

    W. Szumi ´nski, A. J. Maciejewski, Dynamics and non-integrability of the double spring pendulum, Journal of Sound and Vibration 589 (2024) 118550

    work page 2024

  12. [12]

    Froeschl ´e, E

    C. Froeschl ´e, E. Lega, R. Gonczi, Fast Lyapunov indicators. Application to asteroidal motion, Celestial Mechanics and Dynamical Astronomy 67 (1997) 41

    work page 1997

  13. [13]

    E. Lega, M. Guzzo, C. Froeschl ´e, Theory and applications of the fast Lyapunov indicator (FLI) method, Lecture Notes in Physics 915 (2016) 35

    work page 2016

  14. [14]

    M., Sim ´o, C., Simple tools to study global dynamics in non- axisymmetric galactic potentials - I, Astronomy and Astrophysics Sup- plement Series 147 (2000) 205

    Cincotta, P. M., Sim ´o, C., Simple tools to study global dynamics in non- axisymmetric galactic potentials - I, Astronomy and Astrophysics Sup- plement Series 147 (2000) 205

    work page 2000

  15. [15]

    P. M. Cincotta, C. M. Giordano, Theory and applications of the mean exponential growth factor of nearby orbits (MEGNO) method, Lecture Notes in Physics 915 (2016) 93

    work page 2016

  16. [16]

    Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits, Journal of Physics A: Mathematical and General 34 (2001) 10029

    C. Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits, Journal of Physics A: Mathematical and General 34 (2001) 10029

    work page 2001

  17. [17]

    Skokos, T

    C. Skokos, T. C. Bountis, C. Antonopoulos, Geometrical properties of local dynamics in Hamiltonian systems: The generalized alignment index (GALI) method, Physica D: Nonlinear Phenomena 231 (2007) 30

    work page 2007

  18. [18]

    Skokos, T

    C. Skokos, T. Manos, The smaller (SALI) and the generalized (GALI) alignment indices: Efficient methods of chaos detection, Lecture Notes in Physics 915 (2016) 129

    work page 2016

  19. [19]

    R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, A. M. Rey, Unify- ing scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model, Nature Com- munications 10 (2019) 1581

    work page 2019

  20. [20]

    Hillebrand, G

    M. Hillebrand, G. Kalosakas, A. Schwellnus, C. Skokos, Heterogeneity and chaos in the Peyrard-Bishop-Dauxois DNA model, Physical Review E 99 (2019) 022213

    work page 2019

  21. [21]

    E. E. Zotos, F. L. Dubeibe, A. F. Steklain, T. Saeed, Orbit classification in a disk galaxy model with a pseudo-Newtonian central black hole, As- tronomy & Astrophysics 643 (2020) A33

    work page 2020

  22. [22]

    Kov ´ari, B

    E. Kov ´ari, B. ´Erdi, Z. S´andor, Application of the Shannon entropy in the planar (non-restricted) four-body problem: the long-term stability of the Kepler-60 exoplanetary system, Monthly Notices of the Royal Astronom- ical Society 509 (2022) 884. 15 B. Many Manda et al. / Communications in Nonlinear Science and Numerical Simulation 00 (2025) 1–17 16 Fi...

    work page 2022

  23. [23]

    Huang, G

    Z. Huang, G. Huang, A. Hu, Application of explicit symplectic integra- tors in a magnetized deformed Schwarzschild black spacetime, The As- trophysical Journal 925 (2022) 158

    work page 2022

  24. [24]

    Ghanbari, On detecting chaos in a prey-predator model with prey’s counter-attack on juvenile predators, Chaos Solitons & Fractals 150 (2021) 111136

    B. Ghanbari, On detecting chaos in a prey-predator model with prey’s counter-attack on juvenile predators, Chaos Solitons & Fractals 150 (2021) 111136

    work page 2021

  25. [25]

    A. F. Steklain, A. Al-Ghamdi, E. E. Zotos, Using chaos indicators to de- termine vaccine influence on epidemic stabilization, Physical Review E 103 (2021) 032212

    work page 2021

  26. [26]

    Blumenthal, J

    E. Blumenthal, J. W. Rocks, P. Mehta, Phase transition to chaos in com- plex ecosystems with nonreciprocal species-resource interactions, Physi- cal Review Letters 132 (2024) 127401

    work page 2024

  27. [27]

    Skokos, E

    C. Skokos, E. Gerlach, Numerical integration of variational equations, Physical Review E 82 (2010) 036704

    work page 2010

  28. [28]

    Gerlach, S

    E. Gerlach, S. Eggl, C. Skokos, E fficient integration of the variational equations of multidimensional Hamiltonian systems: Application to the Fermi–Pasta–Ulam lattice, International Journal of Bifurcation and Chaos 22 (2012) 1250216

    work page 2012

  29. [29]

    Senyange, C

    B. Senyange, C. Skokos, Computational e fficiency of symplectic integra- tion schemes: application to multidimensional disordered Klein–Gordon lattices, The European Physical Journal Special Topics 227 (2018) 625

    work page 2018

  30. [30]

    Tancredi, A

    G. Tancredi, A. S ´anchez, F. Roig, A comparison between methods to compute Lyapunov exponents, The Astronomical Journal 121 (2001) 1171

    work page 2001

  31. [31]

    L. Mei, L. Huang, Reliability of Lyapunov characteristic exponents com- puted by the two-particle method, Computer Physics Communications 224 (2018) 108

    work page 2018

  32. [32]

    Hillebrand, S

    M. Hillebrand, S. Zimper, A. Ngapasare, M. Katsanikas, S. Wiggins, C. Skokos, Quantifying chaos using Lagrangian descriptors, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (2022) 123122

    work page 2022

  33. [33]

    Bazzani, M

    A. Bazzani, M. Giovannozzi, C. E. Montanari, G. Turchetti, Performance analysis of indicators of chaos for nonlinear dynamical systems, Phys. Rev. E 107 (2023) 064209

    work page 2023

  34. [34]

    Danca, N

    M.-F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems, International Journal of Bifurcation and Chaos 28 (2018) 1850067

    work page 2018

  35. [35]

    K. L. Johnson, Contact mechanics, Cambridge university press, 1987

    work page 1987

  36. [36]

    C. Lee, X. Wei, J. W. Kysar, J. Hone, Measurement of the elastic prop- erties and intrinsic strength of monolayer graphene, Science 321 (2008) 385

    work page 2008

  37. [37]

    Cadelano, P

    E. Cadelano, P. L. Palla, S. Giordano, L. Colombo, Nonlinear elasticity of monolayer graphene, Phys. Rev. Lett. 102 (2009) 235502

    work page 2009

  38. [38]

    Schlegel, Exploring potential energy surfaces for chemical reactions: an overview of some practical methods, Journal of Computation Chem- istry 24 (2003) 1514

    H. Schlegel, Exploring potential energy surfaces for chemical reactions: an overview of some practical methods, Journal of Computation Chem- istry 24 (2003) 1514

    work page 2003

  39. [39]

    W. R. Smith, W. Qi, Molecular simulation of chemical reaction equilib- rium by computationally efficient free energy minimization, ACS Central Science 4 (2018) 1185

    work page 2018

  40. [40]

    K. J. Naidoo, T. Bruce-Chwatt, T. Senapathi, M. Hillebrand, Multidimen- sional free energy and accelerated quantum library methods provide a gateway to glycoenzyme conformational, electronic, and reaction mecha- nisms, Accounts of Chemical Research 54 (2021) 4120

    work page 2021

  41. [41]

    Hillebrand, B

    M. Hillebrand, B. Many Manda, G. Kalosakas, E. Gerlach, C. Skokos, Chaotic dynamics of graphene and graphene nanoribbons, Chaos 30 (2020) 063150

    work page 2020

  42. [42]

    H ´enon, C

    M. H ´enon, C. Heiles, The applicability of the third integral of motion: some numerical experiments, The Astronomical Journal 69 (1964) 73

    work page 1964

  43. [43]

    Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Physics Reports 213 (1992) 271

    J. Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Physics Reports 213 (1992) 271

    work page 1992

  44. [44]

    G. P. Berman, F. M. Izrailev, The Fermi-Pasta-Ulam problem: Fifty years of progress, Chaos 15 (2005) 015104

    work page 2005

  45. [45]

    Hyper- chaos in constrained Hamiltonian system and its control

    W. Szumi ´nski, M. Przybylska, A. J. Maciejewski, Comment on “Hyper- chaos in constrained Hamiltonian system and its control” by J. Li, H. Wu and F. Mei, Nonlinear Dynamics 101 (1) (2020) 639–654

    work page 2020

  46. [46]

    S. He, K. Sun, Y . Peng, Detecting chaos in fractional-order nonlinear sys- tems using the smaller alignment index, Physics Letters A 383 (2019) 2267

    work page 2019

  47. [47]

    Ma, Z.-C

    D.-Z. Ma, Z.-C. Long, Y . Zhu, Application of indicators for chaos in chaotic circuit systems, International Journal of Bifurcation and Chaos 26 (2016) 1650182

    work page 2016

  48. [48]

    Chatar, M

    W. Chatar, M. El Ghamari, J. Kharbach, M. Benkhali, R. Masrour, A. Rezzouk, M. Ouazzani-Jamil, Detecting order and chaos by the mLE, SALI and GALI methods in three-dimensional nonlinear Yang–Mills sys- tem, International Journal of Bifurcation and Chaos 32 (2022) 2250145

    work page 2022

  49. [49]

    T. Mai, A. Dhar, O. Narayan, Equilibration and universal heat conduction in Fermi-Pasta-Ulam chains, Physical Review Lett. 98 (2007) 184301

    work page 2007

  50. [50]

    Onorato, L

    M. Onorato, L. V ozella, D. Proment, Y . V . Lvov, Route to thermalization in the α-Fermi-Pasta-Ulam system, Proceedings of the National Academy of Sciences 112 (2015) 4208

    work page 2015

  51. [51]

    Danieli, D

    C. Danieli, D. K. Campbell, S. Flach, Intermittent many-body dynamics at equilibrium, Physical Review E 95 (2017) 060202

    work page 2017

  52. [52]

    H. Zhao, Z. Wen, Y . Zhang, D. Zheng, Dynamics of solitary wave scatter- ing in the Fermi-Pasta-Ulam model, Physical Review Letters 94 (2005) 025507

    work page 2005

  53. [53]

    Flach, A

    S. Flach, A. V . Gorbach, Discrete breathers—Advances in theory and ap- plications, Physics Reports 467 (2008) 1

    work page 2008

  54. [54]

    Cretegny, T

    T. Cretegny, T. Dauxois, S. Ru ffo, A. Torcini, Localization and equipar- tition of energy in the β-FPU chain: Chaotic breathers, Physica D 121 (1998) 109

    work page 1998

  55. [55]

    H. Yan, M. Robnik, Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model. I. Density of states and spectral statis- tics, Physical Review E 109 (2024) 054210

    work page 2024

  56. [56]

    Antonopoulos, T

    C. Antonopoulos, T. Bountis, C. Skokos, Chaotic dynamics of N-degree of freedom Hamiltonian systems, International Journal of Bifurcation and Chaos 16 (2006) 1777–1793

    work page 2006

  57. [57]

    Antonopoulos, T

    C. Antonopoulos, T. Bountis, Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice, Physical Review E 73 (2006) 056206

    work page 2006

  58. [58]

    Dauxois, A

    T. Dauxois, A. Litvak-Hinenzon, R. MacKay, A. Spanoudaki, Energy lo- calisation and transfer, World Scientific, 2004

    work page 2004

  59. [59]

    Moges, T

    H. Moges, T. Manos, C. Skokos, On the behavior of the generalized alignment index (GALI) method for regular motion in multidimensional Hamiltonian systems, Nonlinear Phenomena in Complex Systems 123 (2020) 153

    work page 2020

  60. [60]

    Skokos, T

    C. Skokos, T. Bountis, C. Antonopoulos, Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi–Pasta–Ulam lattices by the generalized alignment index method, The European Phys- ical Journal Special Topics 165 (2008) 5

    work page 2008

  61. [61]

    Blanes, F

    S. Blanes, F. Casas, A. Farres, J. Laskar, J. Makazaga, A. Murua, New families of symplectic splitting methods for numerical integration in dy- namical astronomy, Applied Numerical Mathematics 68 (2013) 58

    work page 2013

  62. [62]

    Farr ´es, J

    A. Farr ´es, J. Laskar, S. Blanes, F. Casas, J. Makazaga, A. Murua, High precision symplectic integrators for the solar system, Celestial Mechanics and Dynamical Astronomy 116 (2013) 141

    work page 2013

  63. [63]

    Danieli, B

    C. Danieli, B. Many Manda, T. Mithun, C. Skokos, Computational ef- ficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions, Mathematics in Engineering 1 (2019) 447

    work page 2019

  64. [64]

    H. Fan, J. Jiang, C. Zhang, X. Wang, Y .-C. Lai, Long-term prediction of chaotic systems with machine learning, Phys. Rev. Res. 2 (2020) 012080

    work page 2020

  65. [65]

    Przybylska, W

    M. Przybylska, W. Szumi ´nski, A. J. Maciejewski, Destructive relativ- ity, Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (2023) 063156

    work page 2023

  66. [66]

    Szumi ´nski, M

    W. Szumi ´nski, M. Przybylska, A. J. Maciejewski, Chaos and integrability of relativistic homogeneous potentials in curved space, Nonlinear Dy- namics 112 (2024) 4879. 17

    work page 2024