One property of multistep difference methods for ordinary differential equations (Q1975119)

The author considers the system of differential equations \[ \dot x=F(x) \] and the corressponding \(m\)-step difference method of the \(p\)th order \[ \sum_{i=0}^m a_i x_{k-i}=h\sum_{i=0}^m b_{i} F(x_{k-i}) \] where \(x\) is an \(s\)-dimensional vector, \(k\geq m\), \(F(x)\) is a sufficiently smooth vector function of \(x\), and the coefficients \(a_i\) and \(b_i\) satisfy the relation \[ \sum_{i=0}^m i^{(j-1)} (ia_i+jb_i)=0,\;\;j=1,\dots,p,\;\;\sum_{i=0}^m a_i=0, \] \[ \sum_{i=0}^m b_i=1,\;\;a_0\neq 0,\;\;p\leq 2m. \] If the characteristic equation \[ \rho(\lambda)=\sum_{i=0}^m a_i\lambda^{m-i}=0, \] of this method has, in addition to the root \(\lambda_1=1, l-1\) simple roots \(\lambda_2,\dots,\lambda_l\) such that \(\mid{\lambda_i}\mid=1\) and \(m-l\) roots such that \(\mid{\lambda_j}\mid<1\), than this method has stable invariant manifolds.