Geometry and chaos on Riemann and Finsler manifolds

In this paper we discuss some general aspects of the so-called geometrodyna mical approach (GDA) to Chaos and present some results obtained within this framework. We firstly derive a naive but nevertheless a general geometriza tion procedure, and then specialize the discussion to the description of mo tion within the framework of two among the most representative implementati ons of the approach, namely the Jacobi and Finsler geometrodynamics. In ord er 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 understand ing of the qualitative behaviour of dynamical systems (DS's), we then compa re, from a formal point of view, the tools used within the framework of Ham iltonian dynamics to detect the presence of Chaos with the corresponding on es used within the GDA, i.e., the tangent dynamics and the geodesic deviati on equations, respectively, pointing out their general inequivalence. Moreo ver, to advance the mathematical analysis and to highlight both the peculia rities and the analogies of the methods, we work out two concrete applicati ons to the study of very different, yet typical in distinct contexts, dynam ical systems. The first is the well-known Henon-Heiles Hamiltonian, which a llows 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 t o get deeper insights into the sources of instability. We show the effectiv eness 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 Bianchi I X (BIX) cosmological model, whose peculiarity is well known as its very nat ure has been debated for a long time. Using the Finsler GDA, we obtain resu lts with a built-in invariance, which give evidence to the non-chaotic beha viour of this system, excluding any global exponential instability in the e volution of the geodesic deviation. (C) 1998 Elsevier Science Ltd. All righ ts reserved.