Slotine:
For autonomous systems:
For nonautonomous systems:
The second definition says there is a ball $0<R_2<R_1$, where trajectories that start in $R_1$ will converge into the smaller ball $R_2$ after period of time $T$ which is INDEPDENT of $t_0$.
If it is independent, why does this definition not revert to the autonomous system definition of asymptotic stability?
Is there an intuitive way to think about $T$? I cant see how a region of convergence. How could you get $T$ from $R_1$ and $R_2$? If $R_2<<R_1$, $T$ will be larger? Is there an example that someone has that I could test this out on?
Does $t_0$ dictate the size of R1 and therefore $R_2$? If so, then is it really independent of $t_0$?
If my initial trajectory started at t1 instead of $t_0$, my region of attraction would be smaller?
There are a lot of questions. First, Definition 3.4 and Definition 4.6 describe the same property. One is expressed in terms of stability and attractivity of the 0 equilibrium point in terms of a limit, whereas the second one is in terms of uniform stability and the $\epsilon,\delta$ formulation of a limit (see e.g. https://en.wikipedia.org/wiki/Limit_of_a_function).
Is there an intuitive way to think about $T$?
The time $T$ is just a time that is needed for the system to move its state from the first set to the other, and to remain there.
You could technically compute an estimate for $T$ using a Lyapunov function or a similar tool, but this is not usually something that is necessarily interesting nor important to do. It is also not easy. I would personally focus on other properties such as the rate of convergence, for instance.
If $R_2<<R_1$, $T$ will be larger?
Not necessarily.
Does $t_0$ dictate the size of $R_1$ and therefore $R_2$? If so, then is it really independent of $t_0$?
No, $t_0$ is just an initial time here, it has nothing to do with the initial set. Those are two different objects that are chosen independently in uniform stability.