Natural tangent dynamics with recurrent biorthonormalizations: A geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets

The method is based on vector and covector dynamics in the tangent and cotangent bundles (induced by the phase-space flow), and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable/slow subspaces. The NTDRB method provides the most convenient and natural vector and covector bases for: (i) the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold; (ii) the identification of the stable and unstable components of the dynamics on a limit set; (iii) the construction of a low-dimensional reduced model reproducing the asymptotic behavior of the original systems.