Various closeness concepts in numerical ODE's (Q1361282)

The author considers the ordinary differential equation (ODE) \(\dot x=f(x)\) in an open neighbourhood of the \(m\)-dimensional Euclidean unit ball with sufficient smooth function \(f\). The aim of this paper is to discuss a generalization of previous results of the author when also periodic orbits are allowed. Several partial results (like continuity, surjectivity, bijectivity), examples, counterexamples and conjectures are presented. The core of the whole problem is that the standard equivalence notions of differentiable dynamics are rather incompatible with discretizations. An appropriate closeness and/or equivalence concept between the exact and the numerical solutions depends on the inner structure of the underlying differential equation.