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

This article develops a geometric method, referred to as Natural Tangent Dynamics with Recurrent Biorthonormalizations (NTDRB), that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold (associated with an equilibrium point), or exhibiting nontrivial limit sets (limit cycles and chaotic attractors). For the first class of systems the main focus relies on the identification of the slow/fast timescales, while for the latter (dynamics within nontrivial attractors) the central issue is the temporal dichotomy between unstable/stable components as described by the structure of the Lyapunov spectrum. 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 biorthononnalization 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. (c) 2005 Elsevier B.V. All rights reserved.