On discrete systems based on the Adams and Neström methods (Q1280998)

In dealing with continuous-time systems (CTS) described by a canonical set of first order equations, \(\dot x=f(x)\), \(x\in\mathbb{R}^n\) , one is interested to study this system by its discretized version. In many applications, an \(m\)-step discretization is used: \[ x_k=x_{k-p}+h\cdot \sum^m_{i=0} b_if(x_{k-i}). \] For \(p=1\), the Adams method is obtained, whereas for \(p=2\), one gets the Neström method. One is naturally interested to know if the discretized system (DS) inherits the properties of the corresponding CTS. The paper provides some answers to this question by studying the phase space (in both cases) of the associated DS. Two propositions regarding the conditions under which the stability and asymptotic stability are preserved are proved. The imposed conditions on the mapping \(f(.)\) are very reasonable (bounded map, Lipschitz-like conditions, etc.).