One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

Li, Xiaoming; Li, XiaoChen; Liao, Shijun

doi:10.1007/s11433-020-1624-7

Cited by 11 publications

(13 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To overcome the above-mentioned limitations/restrictions of traditional numerical algorithms (in single/double precision) for chaotic dynamical systems, Liao (2009) suggested a numerical strategy, namely the 'clean numerical simulation' (CNS). The basic idea of the CNS (Liao 2013(Liao , 2014Hu & Liao 2020;Qin & Liao 2020;Li et al 2021;Liao & Qin 2022) is to greatly decrease the background numerical noises, i.e. truncation errors and round-off errors, to such a tiny level that the influence of numerical noises can be neglected in an interval of time 0 t T c that is long enough for statistics, where T c is the so-called 'critical predictable time'.…”

Section: Introductionmentioning

confidence: 99%

“…So, different from the Taylor series method, the key point of the CNS is the so-called 'critical predictable time' T c that determines a temporal interval [0, T c ] in which the numerical simulations are 'reliable' and 'clean', since their 'false' numerical noises are much smaller than the 'true' physical solution and thus are negligible. For more details about the CNS, please refer to Liao (2009Liao ( , 2013Liao ( , 2014 and his co-authors (Hu & Liao 2020;Qin & Liao 2020;Li et al 2021;Xu et al 2021;.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

Qin

Liao²

2022

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using two-dimensional (2-D) turbulent Rayleigh–Bénard convection (RBC) as an example, we solve numerically the Navier–Stokes equations, separately, by means of a traditional algorithm with double precision (denoted RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by RKwD is a mixture of the ‘true’ physical solution and the ‘false’ numerical noises that are random and can be regarded as a kind of artificial stochastic disturbances; unfortunately, the ‘true’ physical solution is mostly at the same level as the ‘false’ numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the ‘false’ numerical noises are negligible compared with the ‘true’ physical solution, thus the CNS solution can be used as a ‘clean’ benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. In particular, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows. The shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter; however, the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus we provide rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2-D turbulent RBC considered in this paper.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

Qin

Liao²

2022

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Similarly, Li, Jing and Liao 30 further found 1349 new families of periodic planar collisionless orbits of the three-body system with only two equal masses. In 2020, starting from a known periodic orbit with three equal masses and using the CNS to integrate the governing equations, Li, Li and Liao 31 successfully obtained 135445 new periodic orbits with arbitrarily unequal masses by means of combining the numerical continuation method 32 and the Newton-Raphson method [26][27][28] , including 13315 stable ones. Therefore, in only four years, using high-performance computer and our new strategy based on the CNS, we successfully increased the family number of the known periodic orbits of three-body systems by nearly four orders of magnitude!…”

mentioning

confidence: 99%

“…3 The times of machine learning However, it is time-consuming to use the numerical continuation method 32 to find the 135445 periodic orbits (with unequal masses) reported by Li, Li and Liao 31 . Besides, these periodic orbits are in essence discrete, say, only for some specific values of m 1 and m 2 in an irregular domain (in case of m 3 = 1 since we use the mass of the 3rd body as a characteristic mass M, without loss of generality).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Three-body problem -- from Newton to supercomputer plus machine learning

Liao,

Li,

Yang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The famous three-body problem can be traced back to Newton in 1687, but quite few families of periodic orbits were found in 300 years thereafter. In this paper, we propose an effective approach and roadmap to numerically gain planar periodic orbits of three-body systems with arbitrary masses by means of machine learning based on an artificial neural network (ANN) model. Given any a known periodic orbit as a starting point, this approach can provide more and more periodic orbits (of the same family name) with variable masses, while the mass domain having periodic orbits becomes larger and larger, and the ANN model becomes wiser and wiser. Finally we have an ANN model trained by means of all obtained periodic orbits of the same fam-ily, which provides a convenient way to give accurate enough predictions of periodic orbits with arbitrary masses for physicists and astronomers. It suggests that the high-performance computer and artificial intelligence (including machine learning) should be the key to gain periodic orbits of the famous three-body problem.

show abstract

Triple collision orbits in the free-fall three-body system without binary collisions

et al. 2021

Celest Mech Dyn Astr

View full text Add to dashboard Cite

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

Cited by 11 publications

References 41 publications

Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

Large-scale influence of numerical noises as artificial stochastic disturbances on a sustained turbulence

Three-body problem -- from Newton to supercomputer plus machine learning

Triple collision orbits in the free-fall three-body system without binary collisions

Contact Info

Product

Resources

About