Geometry and Chaos on Riemann and Finsler Manifolds

Abstract

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. In order to support the claim that the GDA isn't 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 (DS's), we then compare, from a formal point of view, the tools used within the usual 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 highlight both the peculiarities of the methods and the analogies between them, 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 Henon-Heiles Hamiltonian, which allows 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 on the sources of instability. 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, whose peculiarity is well known, as its very nature has been debated since long time, namely the Bianchi IX (BIX) cosmological model. 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.