Tame topology and non-integrability of dynamical systems

TL;DR

This paper explores the concept of integrability in differential Galois theory, focusing on non-integrable gradient systems and their trajectories' o-minimal properties as real dynamical systems.

Contribution

It introduces a broad perspective on integrability, highlighting the o-minimal properties of non-integrable gradient systems in real dynamics.

Findings

Non-integrable gradient systems have trajectories with o-minimal finiteness properties.

Differential Galois theory provides a framework for understanding integrability.

Real dynamical systems can exhibit finiteness properties despite non-integrability.

Abstract

In this paper we study the general concept of integrability in the broad sense within the frame of differential Galois theory. We concentrate on the gradient systems which are not integrable. In spite of it, if we consider them as the real dynamical systems, they have trajectories with finiteness properties of o-minimal type.