Integration of Ordinary Differential Equations on Riemann Surfaces with Unbounded Precision

Abstract

We consider analytical systems of ODE with a real or complex time. Integration of such ODE is equivalent to an analytical continuation of a solution along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, for demonstration of how Taylor expansions can be used for integration of ODE with an arbitrary precision. We obtained some new results in these problems.