  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.
