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.
Highlights
One of the main problems that arise in numerical analysis of systems of nonlinear ordinary differential equations is the appearance of moving singular points
If a singular point of the general solution to an ordinary differential equation or a system of such equations depends on the integration constant, it is called a moving singular point [2]
Numerical experiments convincingly verify the numerical method for determining the position and order of moving singular points of ordinary differential equations, based on the Rosenbrock difference scheme
Summary
One of the main problems that arise in numerical analysis of systems of nonlinear ordinary differential equations is the appearance of moving singular points. It should be recalled that linear systems do not have such features and, the region of existence of the solution is always known in advance. On the contrary, it is never clear beforehand whether a solution is defined for all considered values of the independent variable or not. Euler noted that when approaching moving singular points of the solution of the Riccati equation, the approximate solution is increasingly deflecting from the exact one, and it has only recently been shown that, despite this, the finite difference method allows searching for the position and orders of moving singularities. In the present paper we report the implementation of this method in the computer algebra system [1], preceded by the necessary theoretical introduction
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