Let us consider the equation: $$ \dot{x}_i = F_i(x) $$ with $x\in \mathbb{R}^n$ and $i=1\dots n$, and the equation for small displacements: $$ \dot{\delta x} = \sum_j \frac{\partial}{\partial x_j} F_i(x) \delta x_j $$ I often read (and checked in practice) the following: starting from a random initial $\delta x$, for increasing time it approaches a given "regime", such that its growth is given by the largest Lyapunov exponent $\lambda_1$.

However, in principle, it should be possible to find a particular $\delta x$ such that it will evolve with a different (lower) Lyapunov exponent $\lambda_2<\lambda_1$. Such initial vector $\delta x(t=0)$ could depend on the initial $x(t=0)$.

My first question is the following: is such a vector really dependent on $x(t=0)$, or it is constant for every $x$?

The question could appear strange, but it is related to the definition of "Lyapunov vector" that I saw often. This definition is based on the following procedure. First, a matrix $Y_{i,j}(t)$ is considered, with equation: $$ \dot{Y_{i,j}}= \sum_k \frac{\partial}{\partial x_k} F_i Y_{k,j} $$ Then a matrix $M$ is defined as: $$ M = \lim_{t\to +\infty} \frac{\log Y Y^T}{t} $$ According to this definition, the Lyapunov exponents and vectors are the eigenvalues and eigenvectors of $M$. For this reason, I would say that the Lyapunov vector is a single vector for the whole trajectory, not depending on $x$. Hence my question about its relation with the above-mentioned vector field $\delta x(x)$.

Since I think that the vector field $\delta x(x)$ is not the Lyapunov exponent, I'm asking about literature about it, where the properties of such a field are studied and discussed.