Eigenfunctions of the Perron-Frobenius operator and the finite-time Lyapunov exponents in uniformly hyperbolic area-preserving maps

TL;DR

This paper investigates the eigenfunctions of the Perron-Frobenius operator in area-preserving maps, revealing their localization patterns and relation to finite-time Lyapunov exponents through numerical analysis.

Contribution

It provides a numerical validation of the Ulam method for eigenfunctions and links eigenfunction patterns to finite-time Lyapunov exponents in hyperbolic maps.

Findings

Eigenfunctions show spatial localization around regions with sparse unstable manifolds.

Eigenfunction patterns closely match distributions of maximal finite-time Lyapunov exponents.

The Ulam method effectively captures subleading eigenvalues and eigenfunctions in these systems.

Abstract

The subleading eigenvalues and associated eigenfunctions of the Perron-Frobenius operator for 2-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme believed to provide eigenvalues and eigenfunctions of the Perron-Frobenius operator, both for linear and nonlinear maps on the torus. For the nonlinear case, the second-largest eigenvalues and the associated eigenfunctions of the Perron-Frobenius operator are investigated by calculating the Fokker-Planck operator with sufficiently small diffusivity. On the basis of numerical schemes thus established, we find that eigenfunctions for the subleading eigenvalues exhibit spatially inhomogeneous patterns, especially showing localization around the region where unstable manifolds are sparsely running. Finally, such spatial patterns of the eigenfunction are shown to be very close to the distribution of the maximal finite-time Lyapunov exponents.