Linearized system for $ begin{cases} frac{d}{dt} x_1 = -x_1 + x_2 \ frac{d}{dt} x_2 = x_1 - x_2^3 end{cases} $ is not resting at rest point?

Assume there is the dynamical system

$$
begin{align}
frac{d}{dt} x_1 &= -x_1 + x_2 \
frac{d}{dt} x_2 &= x_1 – x_2^3
end{align}
$$

The system is at rest at the point $(x_1, x_2) = (1, 1)$ and the point is stable. At this point of course

$$
begin{align}
frac{d}{dt} x_1 &= 0 \
frac{d}{dt} x_2 &= 0
end{align}
$$

I want to investigate the rest point more and so I use the linear model from the Taylor series at the rest point:

$$
frac{d}{dt}x = begin{pmatrix} -1 & 1 \ 1 & -3 end{pmatrix}x
$$

I want to simulate both nonlinear and linear model. But something is strange. At the rest point I have:

$$
frac{d}{dt}x = begin{pmatrix} -1 & 1 \ 1 & -3 end{pmatrix} begin{pmatrix} 1 \ 1 end{pmatrix} = begin{pmatrix} 0 \ -2 end{pmatrix}
$$

So although the nonlinear model is at rest at $(1, 1)$ the linear model is not at rest there! So when I simulate both systems they are very different even at the start and even if the start point is very near to the rest point. Look:

The red $x_2$ trajectory is going even in the wrong direction at the start. What is the cause of the problem? Shouldn’t the linear system approximate the nonlinear system at least when it starts near the rest point?