New Families of Periodic Orbits for the Planar
Three-Body Problem Computed with High
Precision
Ivan Hristov 1, Radoslava Hristova 1, Igor Puzynin 2, Taisya Puzynina 2,
Zarif Sharipov 2 and Zafar Tukhliev 2
1
  Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”,
Sofia, 1164, Bulgaria
2
  Meshcheryakov Laboratory of Information Technologies, JINR, Dubna, Russia


             Abstract
             In this paper we use a Modified Newton’s method based on the Continuous
             analog of Newton’s method and high precision arithmetic for a general
             numerical search of periodic orbits for the planar three-body problem. We
             consider relatively short periods and a relatively coarse search-grid. As a
             result, we found 123 periodic solutions belonging to 105 new topological
             families that are not included in the database in [Science China Physics,
             Mechanics and Astronomy 60.12 (2017)]. The extensive numerical search
             is achieved by a parallel solving of many independent tasks using many
             cores in a computational cluster.

             Keywords
             Three-body problem, periodic orbits, modified Newton’s method with high
             precision

1. Introduction
     Numerical search of periodic orbits is an important tool for further investiga-
tion and understanding of the three-body problem. In 2013 Shuvakov and Dmi-
trashinovich found 13 new topological families applying a numerical algorithm
based on the gradient descent method in the standard double precision arithmetic
[1, 2]. It is well known that the three-body problem is very sensitive on the ini-
tial conditions. This means that working with double precision strongly limits
the number of solutions that can be found. This limitation was recognized by Li
and Liao, in 2017 they applied Newton’s method with high precision arithmetic

Information Systems & Grid Technologies: Fifteenth International Conference ISGT’2022, May 27–28, 2022, Sofia, Bulgaria
EMAIL: ivanh@fmi.uni-sofia.bg (I. Hristov); radoslava@fmi.uni-sofia.bg (R. Hristova); ipuzynin@jinr.ru (I. Puzynin);
puzynina@jinr.ru (T. Puzynina); zarif@jinr.ru (Z. Sharipov); zafar@jinr.ru (Z. Tukhliev)
ORCID: 0000-0003-0718-0721 (I. Hristov); 0000-0002-4224-4500 (R. Hristova); 0000-0002-6587-808X (I. Puzynin);
0000-0001-7839-3535 (T. Puzynina); 0000-0003-4211-2556 (Z. Sharipov); 0000-0001-9848-7336 (Z. Tukhliev)

            © 2022 Copyright for this paper by its authors.
            Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
            CEUR Workshop Proceedings (CEUR-WS.org)
to find more than 600 new families of periodic orbits [3]. In this paper we use a
Modified Newton’s method based on the Continuous analog of Newton’s method
and high precision arithmetic for a general numerical search of periodic orbits
for the planar three-body problem. By considering relatively short periods and a
relatively coarse search-grid, we found 123 solutions belonging to 105 new topo-
logical families that are not included in [3].

2. Mathematical model
    We consider three bodies with equal masses. They are treated as mass points.
A planar motion of the three bodies is considered. The normalized differential
equations describing the motion of the bodies are:


                                                                  (1)


    The vectors ri, ṙi, have two components: ri = (xi,yi), ṙi = (ẋi,ẏi). The system (1)
can be written as a first order one this way:
                                                                                      (2)
     We numerically solve the problem in this first order form. Hence, we have
a vector of 12 unknown functions u(t) = (x1, y1, vx1, vy1, x2, y2, vx2, vy2, x3,
y3, vx3, vy3)T. We search for periodic planar collisionless orbits as in [1, 2, 3]:
with zero angular momentum and symmetric initial configuration with parallel
velocities:
         (x1(0), y1(0)) = (-1, 0), (x2(0), y2(0)) = (1, 0), (x3(0), y3(0)) = (0, 0)
                      (vx1(0), vy1(0)) = (vx2(0), vy2(0)) = (vx, vy)                   (3)
                   (vx3(0), vy3(0)) = -2(vx1(0), vy1(0)) = (-2vx, -2vy)
     Here the velocities vx ∈ [0, 1], vy ∈ [0, 1] are parameters. The periods of the
orbits are denoted with T. So, our goal is to find triplets (vx, vy, T) for which the
periodicity condition u(T) = u(0) is fulfilled.

3. Description of the numerical procedure
     The numerical procedure consists of three stages. During stage (I) we
compute initial approximations for the correction method (Modified Newton’s
method) by applying the grid-search method on the rectangle [0, 0.8] × [0, 0.8]
for vx, vy with a stepsize 1/2048. The candidates for correction are the triplets
(vx, vy, T), such that the return proximity P(T) has local minima on the grid for


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vx, vy and P(T) is less than 0.7: P(T) = min1<t≤To P(t) < 0.7. Here P(t) is defined
this way:




     For every grid point the system (2) with initial conditions (3) is solved
numerically up to T0=70 by using multiple precision Taylor series method
(MPTSM) [4] with order 154 and precision 134 decimal digits. During stage
(II) we apply the modified Newton’s method, which can converge or diverge.
Convergence means that we catch a periodic solution. At each iteration step of
the modified Newton’s method, a system of 36 ODEs (the original differential
equations plus the differential equations for the partial derivatives with respect
to the parameters vx and vy ) is solved. MPTSM is used again with the same
order – 154 and the same precision – 134 decimal digits. During stage (III) we
apply the classic Newton’s method in order to specify the solutions with more
correct digits (150 correct digits in this work) and verify them. We use MPTSM
with increased order and precision for solving the system of 36 ODEs. During
this stage we make two computations. The first computation is with 220-th or-
der method and 192 decimal digits of precision and the second computation is
for verification – with 264-th order method and 231 digits of precision. All the
details of the Newton’s and the modified Newton’s method for the planar three-
body problem can be seen in our recent work [5]. The GMP library (the GNU
Multiple Precision arithmetic library) [6] is used for floating point arithmetic in
our C-programs.
     The extensive computations are performed in “Nestum” cluster, Sofia, Bul-
garia [7] and “Govorun” supercomputer, Dubna, Russia [8]. The two platforms
use SLURM as a cluster management and job scheduling system (batch system).
The input data file for stage (I) consists of the set of all grid-points (vx, vy) in the
search window [0, 0.8] × [0, 0.8] with a stepsize 1/2048. The input data file for
stage (II) consists of the set of all triplets (vx, vy, T), obtained from the first stage
(all candidates for correction with the modified Newton’s method). At last, the
input data file for the third stage consists of the set of triplets (vx, vy, T) for which
the modified Newton’s method from stage (II) converges (all caught solutions).
Each grid-point (vx, vy) at stage (I) or each triplet at stage (II) and (III) can be
processed independently in parallel. In fact, we have an example of the so-called
embarrassing parallelism where no communication between tasks is needed. The
computational process consists of dividing the input data file into many files and
submitting the corresponding jobs associated with each data file to the batch sys-
tem. When the jobs are done, the output files are assembled into one output file,


                                           47
which is eventually additionally processed. Shell scripts are used to automatize
the process of file distribution, the jobs submission and to gather the results. At
all three stages of the numerical procedure these scripts are applied. We achieve
a substantial speedup, which is absolutely needed in order to solve the problem
in a foreseeable time.




Figure 1: Initial velocities for all newfound 123 solutions (black points) and 695
old solutions of Liao et al. (white points)




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4. Results
     For each found solution, we compute the free group element (its topological
family) in order to classify it [1, 2]. In addition, we give the four numbers (vx, vy,
T, T*) with 150 correct digits, where T* is the scale-invariant period. The scale-
invariant period T* is defined as T*=T |E|3/2, where E is the energy of our initial
configuration: E=-2.5 + 3(vx2 +vy2). Equal T* for two different initial conditions
means two different representations of the same solution. We found 123 solutions
belonging to 105 new topological families that are not included in [3] and are not
their satellites. The initial velocities of the 695 solutions from [3] (white points)
and the initial velocities of our 123 solutions (black points) are shown in Figure
1. As it is seen from this figure, our solutions (black points) are found in regions
where old solutions (white points) are relatively sparse. The four numbers (vx,
vy, T, T*) with 150 correct digits, the free group elements, their lengths and ani-
mations in the real space for these 123 solutions can be found in [9].
     Our numerical procedure is also able to find all 124 solutions with periods
T < 70 from [3], except for two of them. Let us mention that a finer search-grid
is used in [3] (with a stepsize 1/4000) and periods there are up to 200. The re-
sults show that the modified Newton’s method based on the Continuous analog
of Newton’s method has a great potential for search of new periodic orbits of
the three-body problem. We did additional numerical tests for the stability of the
presented solutions. These tests (although not fully rigorous) show that all new-
found solutions are most possibly unstable.

5. Acknowledgements
     We thank for the opportunity to use the computational resources of the Nes-
tum cluster, Sofia, Bulgaria and the “Govorun” supercomputer at the Meshch-
eryakov Laboratory of Information Technologies of JINR, Dubna, Russia. The
work is supported by a grant of the Plenipotentiary Representative of the Repub-
lic of Bulgaria at JINR, Dubna, Russia.

6. References
[1]   Shuvakov, Milovan, and Veljko Dmitrashinovich. “Three classes of New-
      tonian three-body planar periodic orbits.”, Physical review letters 110.11
      (2013): 114301.
[2]   Shuvakov, Milovan, and Veljko Dmitrashinovich. “A guide to hunting peri-
      odic three-body orbits.” American Journal of Physics 82.6 (2014): 609-619.
[3]   Li, XiaoMing, and ShiJun Liao. “More than six hundred new families of
      Newtonian periodic planar collisionless three-body orbits.” Science China
      Physics, Mechanics and Astronomy 60.12 (2017): 1–7.

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[4]   Barrio, Roberto, et al. “Breaking the limits: the Taylor series method.” Ap-
      plied mathematics and computation 217.20 (2011): 7940–7954.
[5]   Hristov, I., Hristova, R., Puzynin, I., et al. “Newton’s method for comput-
      ing periodic orbits of the planar three-body problem.”, arXiv preprint arX-
      iv:2111.10839 (2021).
[6]   GMP Library, URL: https://gmplib.org.
[7]   Nestum cluster, URL: http://hpc-lab.sofiatech.bg.
[8]   HybriLIT platform, URL: http://hlit.jinr.ru.
[9]   123 new solutions, URL: http://db2.fmi.uni-sofia.bg/3body123.




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