Studying finite-time (non)-domination in dynamical systems using Oseledec’s splitting. Application to the standard map

Abstract

Oseledec’s theorem provides the necessary conditions for the existence of the decomposition of the tangent bundle called Oseledec’s splitting and the formal definition of the Lyapunov exponents. Using the concept of t-domination of expansion or contraction rates of invariant subspaces after a time, together with the Oseledec theorem, we propose three quantifiers of t-domination and t-non-domination, which can be used to study the complex dynamics of physical systems. The interpretation of the Oseledec t-domination in terms of the finite-time Lyapunov exponents is of great advantage for this purpose. Numerical results for the quantifiers are presented using the conservative standard map in a regime with mixed phase-space dynamics. Distinct typical regions in the phase-space are chosen, two containing a completely chaotic motion and two containing regularity islands. Results show that all quantifiers recognize that regions close to hyperbolic periodic points have a larger Oseledec t-domination than regions around islands of regularity. Furthermore, Oseledec t-non-dominated regions are also dentified and are mostly close to regularity islands. We found a relation between the mean of expansion rates between the Oseledec subspaces and the mean of the angles between them.