I'm currently covering Jan de Vries' Elements of Topological Dynamics and in it he gives a brief introduction to the field through the lens of classical mechanics. He defines the state of a mechanical system in the following way:
In order to generalize his definition of state, as used in this paragraph, say to a physical system consisting of $n$ objects $p_1, \ldots, p_n$, is it right to say that a system's state, at a particular time $t$, can be represented by a tuple $$(x_1(t), \ldots, x_n(t), \dot x_1(t), \ldots, \dot x_n(t))$$ where $x_i(t)$ and $\dot x_i(t)$ are the position and velocity vectors, respectively, of object $p_i$ at time $t$?
Here, if I've understood things correctly, the first $n$ components give information on the systems configuration while the latter $n$ components give information on the system's velocity (by giving the velocity of each of its constituents).
If I've understood the above notion of state correctly (and please tell me if I haven't), I'm a little confused at the way he introduces the notion of a law of motion. He briefly introduced it in the former picture, but he finishes here in this image:
Since he uses the notation $x(t) = (x_1(t), \ldots, x_n(t))$ to denote the state of a system, I'm assuming some of the components of this tuple are velocities and not everything is simply configuration. So, if $n = 2k$ (for $k$ the number of objects in our system), the family of autonomous ordinary differential equations he gives in $(5)$ can be written as $$\dot x_i(t) = F(x_1(t), \ldots, x_k(t), \dot x_1(t), \ldots, \dot x_i(t), \ldots, \dot x_k(t))$$
So, $\dot x_i$ is the output of a function with $\dot x_i$ itself as one of the input variables. Is this correct or have I mixed up the notation?
You are correct. One small nitpick is that in general position will not be a vector intrinsically, but velocity will: in the context of dynamics on smooth manifolds a point in the manifold $M$ will represent configuration and a point in the manifold $TM$ (that is, a pair consisting of a point $x$ in $M$ and a vector $v\in T_xM$) will represent state. This is essentially what is meant with the second paragraph on p.3.
I should also mention that in the excerpt you have cited, there is no specific reference as to how many point particles are being considered; it seems the author is considering one point particle moving in an $n$-dimensional space. Classically if there are $N$ point particles the configuration space is taken to be $3N$ dimensional (as the motion is assumed to be in three dimensions), and the phase space is taken to be $6N$ dimensional (one additional dimension for the velocity in one positional dimension).
Further, one often restrict attention to a certain subspace where no two point particles are exactly at the same position, and indeed in some fields "configuration space" is used in this more restricted sense; see e.g. wikipedia.