Structure of characteristic Lyapunov vectors in anharmonic Hamiltonian lattices

TL;DR

This paper investigates the scaling properties of Lyapunov vectors in one-dimensional Hamiltonian lattices, revealing differences from dissipative systems and highlighting the artificial nature of backward Lyapunov vectors.

Contribution

It provides a detailed analysis of characteristic Lyapunov vectors in Hamiltonian lattices and introduces a memory-efficient algorithm for large system computations.

Findings

Characteristic Lyapunov vectors show nonuniversal scaling exponents.

Backward Lyapunov vectors have similar scaling exponents across models.

A 'bit reversible' algorithm enables large system analysis.

Abstract

In this work we perform a detailed study of the scaling properties of Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices: the Fermi-Pasta-Ulam and $\Phi^4$ models. In this case, characteristic (also called covariant) LVs exhibit qualitative similarities with those of dissipative lattices but the scaling exponents are different and seemingly nonuniversal. In contrast, backward LVs (obtained via Gram-Schmidt orthonormalizations) present approximately the same scaling exponent in all cases, suggesting it is an artificial exponent produced by the imposed orthogonality of these vectors. We are able to compute characteristic LVs in large systems thanks to a `bit reversible' algorithm, which completely obviates computer memory limitations.