Error estimation in the Runge-Kutta method for systems of ODEs

Sometimes we are not able to provide analytical proof for some properties of the model. In such a case we need to carry out numerical simulations and give a rigorous argument in order to prove considered properties based on error estimation of numerical method. In my presentation I will show a technical details of such estimation for the most common numerical method - Runge-Kutta method. I will provide the estimation for one-dimensional case (based on Lotkin method - see [M. Lotkin, On the Accuracy of the Runge-Kutta Method, Math. Tables Other Aids Comput. 1951] and the generalization for the 3-dim. ODEs system.