arXiv subjects

T. Puzynina

Publications and source records attributed to T. Puzynina.

8 recordsLinked to original sources

arXiv2025

Numerical search for three-body periodic free-fall orbits with central symmetry

A specialized high-precision numerical search for equal-mass collisionless three-body periodic free-fall orbits with central symmetry is conducted. The search is based on Newton's method with initial approximations obtained by the grid-search method. Instead of solving the standard periodicity equation on the entire period a quarter-period equation that also characterizes the periodic orbits is solved. The number of the known orbits from the class is significantly enlarged. The linear stability of the orbits is also investigated. All of them are unstable. A discussion in relation to the efficiency of Newton's method applied with grid-search initial approximations is held.

physics.class-ph↗

arXiv2022

A database of high precision trivial choreographies for the planar three-body problem

Trivial choreographies are special periodic solutions of the planar three-body problem. In this work we use a modified Newton's method based on the continuous analog of Newton's method and a high precision arithmetic for a specialized numerical search for new trivial choreographies. As a result of the search we computed a high precision database of 462 such orbits, including 397 new ones. The initial conditions and the periods of all found solutions are given with 180 correct decimal digits. 108 of the choreographies are linearly stable, including 99 new ones. The linear stability is tested by a high precision computing of the eigenvalues of the monodromy matrices.

math.NA↗

arXiv2022

Hundreds of new satellites of figure-eight orbit computed with high precision

Satellites (topological powers) of the famous figure-eight orbit are special periodic solutions of the planar three-body problem. 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 purposeful numerical search of new satellites of the figure-eight orbit. Over 700 new satellites are found, including 76 new linearly stable ones. 7 of the newly found linearly stable satellites are choreographies. The linear stability is checked by a high precision computing of the eigenvalues of the monodromy matrices. The initial conditions of all found solutions are given with 150 correct decimal digits.

math.NA↗

arXiv2022

New families of periodic orbits for the planar three-body problem computed with high precision

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.

math.NA↗

arXiv2021

Newton's method for computing periodic orbits of the planar three-body problem

In this paper we present in detail Newton's method and its modification, based on the Continuous analog of Newton's method for computing periodic orbits of the planar three-body problem. The linear system at each step of the method is formed by solving a system of ODEs with the multiple precision Taylor series method. We consider zero angular momentum symmetric initial configuration with parallel velocities, bodies with equal masses and relatively short periods. Taking candidates for the correction method with greater return proximity as usual and correcting with the modified Newton's method, allows us to find some new topological families that are not included in the database in [SCIENCE CHINA Physics, Mechanics & Astronomy 60.12 (2017)]

nlin.CD↗

arXiv2021

On the efficient parallel computing of long term reliable trajectories for the Lorenz system

In this work we propose an efficient parallelization of multiple-precision Taylor series method with variable stepsize and fixed order. For given level of accuracy the optimal variable stepsize determines higher order of the method than in the case of optimal fixed stepsize. Although the used order of the method is greater then that in the case of fixed stepsize, and hence the computational work per step is greater, the reduced number of steps gives less overall work. Also the greater order of the method is beneficial in the sense that it increases the parallel efficiency. As a model problem we use the paradigmatic Lorenz system. With 256 CPU cores in Nestum cluster, Sofia, Bulgaria, we succeed to obtain a correct reference solution in the rather long time interval - [0,11000]. To get this solution we performed two large computations: one computation with 4566 decimal digits of precision and 5240-th order method, and second computation for verification - with 4778 decimal digits of precision and 5490-th order method.

math.NA↗

arXiv2020

Parallelizing multiple precision Taylor series method for integrating the Lorenz system

A hybrid MPI+OpenMP strategy for parallelizing multiple precision Taylor series method is proposed, realized and tested. To parallelize the algorithm we combine MPI and OpenMP parallel technologies together with GMP library (GNU miltiple precision libary) and the tiny MPIGMP library. The details of the parallelization are explained on the paradigmatic model of the Lorenz system. We succeed to obtain a correct reference solution in the rather long time interval - [0,7000]. The solution is verified by comparing the results for 2700-th order Taylor series method and precision of ~ 3374 decimal digits, and those with 2800-th order and precision of ~ 3510 decimal digits. With 192 CPU cores in Nestum cluster, Sofia, Bulgaria, the 2800-th order computation was ~ 145 hours with speedup ~ 105.

cs.MS↗

arXiv2019

OpenMP parallelization of multiple precision Taylor series method

OpenMP parallelization of multiple precision Taylor series method is proposed. A very good parallel performance scalability and parallel efficiency inside one computation node of a CPU-cluster is observed. We explain the details of the parallelization on the classical example of the Lorentz equations. The same approach can be applied straightforwardly to a large class of chaotic dynamical systems.

cs.MS↗