Consider the linear dynamical system \[ \dot x(t) = \left[ \begin{array}{cc} 1.1 & 0.1\\ -1 & 0.2 \end{array} \right] x(t). \]
This system is autonomous.
This system is time-invariant.
Whenever $x_1(t)$ and $x_2(t)$ are positive, $x_1(t)$ is increasing.