Numerical determination of the singularity order of a system of differential equations

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Abstract

We consider moving singular points of systems of ordinary differential equations. A review of Painlevé’s results on the algebraicity of these points and their relation to the Marchuk problem of determining the position and order of moving singularities by means of finite difference method is carried out. We present an implementation of a numerical method for solving this problem, proposed by N. N. Kalitkin and A. Al’shina (2005) based on the Rosenbrock complex scheme in the Sage computer algebra system, the package CROS for Sage. The main functions of this package are described and numerical examples of usage are presented for each of them. To verify the method, computer experiments are executed (1) with equations possessing the Painlevé property, for which the orders are expected to be integer; (2) dynamic Calogero system. This system, well-known as a nontrivial example of a completely integrable Hamiltonian system, in the present context is interesting due to the fact that coordinates and momenta are algebraic functions of time, and the orders of moving branching points can be calculated explicitly. Numerical experiments revealed that the applicability conditions of the method require additional stipulations related to the elimination of superconvergence points.

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About the authors

Ali Baddour

Peoples’ Friendship University of Russia (RUDN University)

Email: alibddour@gmail.com
PhD student of Department of Applied Probability and Informatics 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Mikhail D. Malykh

Peoples’ Friendship University of Russia (RUDN University)

Email: malykhmd-md@rudn.ru
Doctor of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Alexander A. Panin

M. V. Lomonosov Moscow State University

Email: a-panin@yandex.ru
Candidate of Physical and Mathematical Sciences, assistant professor of Faculty of Physics Leninskie Gory, Moscow, 119991, Russian Federation

Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University)

Email: sevastianov-la@rudn.ru
Doctor of Physical and Mathematical Sciences, professor of Department of Applied Probability and Informatics 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

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Copyright (c) 2020 Baddour A., Malykh M.D., Panin A.A., Sevastianov L.A.

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