Discrete and continuous dynamical systems

A basic process is the observation of an N-dimensional quantity x(t) in discrete time steps a + jh where j runs through the natural numbers. Naturally, one looks for the rate of change of this ‘information’ during one time step. We show that we obtain a discrete evolution equation which turns up in many fields of numerical analysis: Newtonʹs method, descent methods, numerical methods for solving initial- or boundary value problems in ODEs, as examples. We show that such a method always approaches a solution of a differential equation if the time step h is sent to zero and if we compute over a fixed finite real time interval [a, b]. We also discuss the speed of convergence in terms of the convergence order. We present a unified theory for initial- and boundary value problems