# ds.dynamical systems – Lyapunov vectors along a trajectory

I have the equation:
$$dot{x}_i = F_i(x) tag{1}$$
with $$xin mathbb{R}^n$$. To deal with the Lyapunov exponents, we write the equation for small displacements $$delta x_i$$:
$$dot{delta x}_i = sum_j frac{partial}{partial x_j} F_i(x) delta x_j tag{2}$$
The rate of increase of the vectors is related to the Lyapunov exponent. Here I assume that the system is Lyapunov regular.

The definition of “Lyapunov vector” that I saw is the following. 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:
$$M = lim_{tto +infty} frac{log Y Y^T}{t} tag{3}$$
According to this definition, the Lyapunov exponents and vectors are the eigenvalues and eigenvectors of $$M$$.

I tried to investigate how the Lyapunov vectors depend on the starting point $$x$$, taking two points $$x_A$$ and $$x_B$$ along a trajectory: $$x_A=x(t=0)$$ and $$x_B=x(t=tau)$$.

I calculate $$M$$ in the two points:
$$M(x_A) = lim_{tto +infty} frac{log Y(x_A,t) Y^T(x_A,t)}{t} tag{4}$$
and:
$$M(x_B) = lim_{tto +infty} frac{log Y(x_B,t) Y^T(x_B,t)}{t} tag{5}$$
Since $$Y$$ is a cocycle:
$$Y(x_A,t) = Y(x_B, t-tau) Y(x_A, tau)$$
Then:
$$M(x_A) = lim_{tto +infty} frac{log Y(x_B, t-tau) Y(x_A, tau) Y^T(x_A, tau) Y^T(x_B, t-tau)}{t} tag{6}$$
If the $$Y$$s commuted, we would write the logarithm of the products as the sum of logarithms of the factors, and thus get $$M(x_A)=M(x_B)$$ (Eq. 6 would give the same limit as Eq. 5, since $$tau$$ is constant), i.e. $$M$$ would be constant along a trajectory. However, they do not commute, so maybe $$M$$ changes along the trajectory.

My question is: Is this correct? Actually, according to a previous answer I got on MO, it is believed that $$M$$ changes if we evaluate it starting from $$x_A$$ or $$x_B$$ along the same trajectory. Moreover, it seems that it is believed that the eigenvectors of $$M$$ evolve along a trajectory according to Eq. (2). Is this correct? If so, how can we see it from Eq. (6)?