Global Behavior of Solutions of Nonlinear ODEs: First Order Equations

摘要

We determine the behavior of the general solution, small or large, of nonlinear first-order ODEs in a neighborhood of an irregular singular point chosen to be infinity. We show that the solutions can be controlled in a ramified neighborhood of infinity using a finite set of asymptotic constants of motion; the asymptotic formulas can be calculated to any order by quadratures. These constants of motion enable us to obtain qualitative and accurate quantitative information on the solutions in a neighborhood of infinity, as well as to determine the position of their singularities. We discuss how the method extends to higher order equations. There are some conceptual similarities with a KAM approach, and we discuss this briefly.