Geometrical methods for studying systems of nonlinear differential equations of high orders, suitable in the computational plan

Question

Recently, I began to study systems of high-order nonlinear differential equations. As an example, I can cite the system of equations from this topic.

Phase portrait of n-dimensional state-space system

The problem that I encountered came from the fact that a huge amount of information on this topic is in books and published in articles. But they are extremely difficult to systematize without having sufficient experience, as well as to apply in their calculations.

I want to ask a respected public for recommendations on which side to approach the study of systems of nonlinear differential equations (stability, phase trajectories, etc.).

So that the answer to the question is not too broad, I will ask it more specifically:

1. Speaking more specifically about the study, I would like to receive meaningful answers to such questions:

   1. Stability;
   2. A qualitative study of convergence to the state of equilibrium;
   3. Does the system have closed phase curves, that is, can it return to its initial state during evolution?
   4. How is the attractor of the system, that is, the set in the phase space, to which the "majority" of trajectories tend?
   5. How do trajectories released from close points behave - do they stay close or go a considerable distance over time?

2. Are there stability criteria for a system of nonlinear differential equations of dimension $n >= 3$, determined by phase trajectories?

First of all, of course, I would like to study the stability of the system. And these may be some new tools, it will be interesting for me to study them. And then everyone knows the difficulties with the Lyapunov method ... this is the search for a suitable function.