so I am studying dynamical systems with the term autonomous and time-invariant systems a part of it.
Are all autonomous systems time-invariant or not? and are all time-invariant systems autonomous? I am a bit perplexed with the terms. I would appreciate if someone could help. Thanks
Yes, all autonomous systems are time(-shift) invariant. If you have a solution $x(t)$ of $\dot x=f(x)$, $x(0)=x_0$, then $y(t)=y(t+a)$ solves $$\dot y(t)=\dot x(t+a))=f(x(t+a))=f(y(t)),~~y(0)=x(a).$$
And if the dynamic of how the system moves from a state $x(t)$ to $x(t+\Delta t)$ is time independent, and smooth enough, the the generating vector field is also time invariant.
