On the retainment of attraction domains under discretization of continuous systems (Q1883743)

Consider an autonomous differential equation (1) \(\dot{x}=f(x)\) in \(\mathbb R^n\). It is assumed that equation ({1}) has either a stable rest point \(p\) or a stable closed trajectory \(P\). Let \(\Pi(p)\) and \(\Pi(P)\) be the corresponding basins of attraction. The author studies a discrete system \(F\) generated by a discretization of ({1}) (using the Euler or Runge-Kutta scheme). It is shown that if the discretization steps are small enough, then the system \(F\) has a fixed point close to \(p\) (or an invariant curve close to \(P\)), and strictly interior bounded parts of \(\Pi(p)\) and \(\Pi(P)\) belong to the corresponding ``discretized'' basins.