Invariant chaos in mixmaster cosmology

We discuss mathematical aspects of determining local instability parameters by using invariant characteristics of the internal pseudo-Riemannian geometry with the Jacobi metric (in principle, for Hamiltonian dynamical systems). Analytical formulas allowing one to compute the separation rate of nearby trajectories are given and the fundamental difference between the behavior of geodesics in the Riemannian and pseudo-Riemannian spaces carrying Jacobi metrics is stressed. The formalism developed here is used as an invariant tool to detect chaos in general relativity.