Tame topology and non-integrability of dynamical systems

In this paper, we study the general concept of integrability of dynamical systems in the broad sense within the framework of differential Galois theory. In particular, we investigate gradient systems that are not integrable and we recognize that, in contrast to the general view, non-integrability does not necessarily lead to chaotic behavior of the trajectories. We confirm this by investigating dynamical systems in the real case and provide an example of a non-integrable gradient system that has trajectories with finiteness properties of o-minimal type when considered as a real dynamical system.