Differential systems with semi-stable equilibria and numerical methods (Q2570651)

The stability of numerical methods for a class of nonlinear autonomous differential systems possessing a semi-stable equilibrium of the form \[ y'=f(y)\equiv J y+g(y),\tag{\(*\)} \] \(t\in[0,\infty)\), \(y\in\mathbb R^m\), \(m\geq2\), \(f:\mathbb R^m\to\mathbb R^m\), is considered. The local dynamics of (\(*\)) are discussed under a group of assumptions, called H-assumptions. The author shows that the implicit Euler method, which is unconditionally stable, is the only one of practical interest. Several examples are given to illustrate the results obtained