Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems

Abstract

For an autonomous system of \(N\) nonlinear ordinary differential equations considered on a semi-infinite interval \({{T}_{0}} \leqslant t < \infty \) and having a (pseudo)hyperbolic equilibrium point, the paper considers an \(n\)-dimensional \((0 < n < N)\) stable solution manifold, or a manifold of conditional Lyapunov stability, which, for each sufficiently large \(t\), exists in the phase space of the system’s variables in the neighborhood of its saddle point. A smooth separatrix saddle surface for such a system is described by solving a singular Lyapunov-type problem for a system of quasilinear first-order partial differential equations with degeneracy in the initial data. An application of the results to the correct formulation of boundary conditions at infinity and their transfer to the end point for an autonomous system of nonlinear equations is given, and the use of this approach in some applied problems is indicated.