Smooth Lyapunov Manifolds for Autonomous Systems of Nonlinear Ordinary Differential Equations and Their Application to Solving Singular Boundary Value Problems
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引用次数: 0
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.
AbstractFor an autonomous system of \(N\) nonlinear ordinary differential equations considered on a semiinfinite 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 conditional Lyapunov stability, which, for each sufficient large \(t\) exist in the phase space；n < N）稳定解流形，或者说条件 Lyapunov 稳定流形，对于每个足够大的（t），该流形存在于系统鞍点附近的变量相空间中。通过求解初始数据具有退化性的准线性一阶偏微分方程系统的奇异 Lyapunov 型问题，描述了这种系统的光滑分离矩阵鞍面。文中给出了这些结果在无穷远处边界条件的正确表述及其向自治非线性方程系统终点的转移方面的应用，并指出了这种方法在一些应用问题中的应用。
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