Finals are coming up and my (graduate-level) Ordinary Differential Equations professor has gone off script for about half the class. Normally, we use the text "Nonlinear Ordinary Differential Equations" by Roger Grimshaw. However, when beginning the section of the course on autonomous systems, the professor has been going off of his own notes rather than the textbook, and I can't find any resources for the material. It appears to be a somewhat theoretical approach to autonomous systems in $\mathbb{R}^n$ rather than plane autonomous systems (in $\mathbb{R}^2$) as is covered in Grimshaw and most other texts. The general autonomous ODE material I know of is from my Dynamics classes, covering mostly bifurcations and types of attractors in a more computational way using perturbative methods. To give some broad strokes of the kinds of things the professor covered in this class:
1) If $\phi(t)$ is a solution, then $\phi(t-\tau)$ is a solution for all $\tau \in \mathbb{R}$.
2) If $\Gamma_1$ and $\Gamma_2$ are phase curves, then either they don't intersect or they conincide.
3) Any phase curve is a point, a closed curve, or a non-self-intersecting curve
4) Rectification Theorem: There is a smooth change of coordinates from $x$ to $y(x)$ such that $\frac{dy_i}{dt} = 0$ for $i$ from $1$ to $n-1$ and $\frac{dy_n}{dt} = 1$ in a neighborhood of a given (non-fixed) point.
5) If $x(t;y)$ solves $x' = f(x)$ with $x(0)=y$, then $\log\det\frac{\partial x}{\partial y}(t;y) = \text{div}f(x(t;y))$
6) Liouville's Theorem: If $D_t$ is the time evolution of a domain in phase space with volume (area/length/measure) $V_t$, then $\frac{dV_t}{dt} = \int_{D_t} \text{div}f(x)dx$. Corollary: Hamiltonian systems conserve phase volume since divergence is $0$.
7) Lie derivative of function with respect to vector field (derivative along trajectories) is invariant w.r.t. smooth change of variables
8) First integrals are functions which are constant along trajectories (Lie derivative is 0), and every non-fixed point has a neighborhood with $n-1$ independent first integrals.
9) The actions $g^{t_0}(x_0)= \phi(t_0;x_0)$, where $\phi(t_0;x_0)$ is a solution to $x'=f(x)$ with $x(0) = x_0$ at time $t_0$, form a group under composition.
These are the main things that were covered. I can locate some (but very few) of these ideas in other resources, but mostly I'm coming up with nothing. It seems like this should all be covered in a single text, and I would be interested in hearing about which texts could give me some more insight on this view of autonomous systems.