Geometry and chaos on Riemann and Finsler manifolds

In this paper we discuss some general aspects of the so-called geometrodynamical approach (GDA) to Chaos and present some results obtained within this framework. We firstly derive a naïve but nevertheless a general geometrization procedure, and then specialize the discussion to the description of motion within the framework of two among the most representative implementations of the approach, namely the Jacobi and Finsler geometrodynamics. In order to support the claim that the GDA is not simply a mere re-transcription of the usual dynamics, but instead can give various hints on the understanding of the qualitative behaviour of dynamical systems (DSʹs), we then compare, from a formal point of view, the tools used within the framework of Hamiltonian dynamics to detect the presence of Chaos with the corresponding ones used within the GDA, i.e., the tangent dynamics and the geodesic deviation equations, respectively, pointing out their general inequivalence. Moreover, to advance the mathematical analysis and to highlight both the peculiarities and the analogies of the methods, we work out two concrete applications to the study of very different, yet typical in distinct contexts, dynamical systems. The first is the well-known Hénon-Heiles Hamiltonian, which allows us to exploit how the Finsler GDA is well suited not only for testing the dynamical behaviour of systems with few degrees of freedom, but even to get deeper insights into the sources of instability. We show the effectiveness of the GDA, both in recovering fully satisfactory agreement with the most well-established outcomes and also in helping the understanding of the sources of Chaos. Then, in order to point out the general applicability of the method, we present the results obtained from the geometrical description of a General Relativistic DS, namely the Bianchi IX (BIX) cosmological model, whose peculiarity is well known as its very nature has been debated for a long time. Using the Finsler GDA, we obtain results with a built-in invariance, which give evidence to the non-chaotic behaviour of this system, excluding any global exponential instability in the evolution of the geodesic deviation.