Oseledets' Splitting of Standard-like Maps

TL;DR

This paper investigates the relationship between local invariant manifolds, their curvature, and finite-time Lyapunov exponents in standard-like maps, revealing that hyperbolic regions contribute most to positive FTLE values.

Contribution

It establishes a simple relation between invariant manifold directions, curvature, and FTLE, providing a new analytical approach to approximate manifold splitting in standard-like maps.

Findings

Positive FTLE contributions mainly occur at hyperbolic regions with flat, transversal manifolds.

Manifold curvature and splitting angles correlate with local Lyapunov exponents.

Analytic methods suggest explicit point-wise approximation of manifold splitting.

Abstract

For the class of differentiable maps of the plane and, in particular, for standard-like maps (McMillan form), a simple relation is shown between the directions of the local invariant manifolds of a generic point and its contribution to the finite-time Lyapunov exponents (FTLE) of the associated orbit. By computing also the point-wise curvature of the manifolds, we produce a comparative study between local Lyapunov exponent, manifold's curvature and splitting angle between stable/unstable manifolds. Interestingly, the analysis of the Chirikov-Taylor standard map suggests that the positive contributions to the FTLE average mostly come from points of the orbit where the structure of the manifolds is locally hyperbolic: where the manifolds are flat and transversal, the one-step exponent is predominantly positive and large; this behaviour is intended in a purely statistical sense, since it exhibits large deviations. Such phenomenon can be understood by analytic arguments which, as a by-product, also suggest an explicit way to point-wise approximate the splitting.