A contemporary linear representation theory for ordinary differential equations: probabilistic evolutions and related approximants for unidimensional autonomous systems

In this paper, we build on our previous research on probabilistic foundations of dynamical systems and introduce a theory of linear representation for ordinary differential equations. The theory is developed for explicit ODEs and can be further extended to cover implicit cases. In this report, we investigate the case of a canonical single unknown autonomous system. First we construct a linear representation to get an infinite linear ODE set with a constant coefficient matrix which can be transformed into an upper triangular form. Then we find its approximate truncated solutions. We describe a number of properties of the theory using this framework. The companion of this paper expands this canonical approach to cover multidimensional cases using the theory of folded arrays which is another line of research established by our research group.