Index theorem for vector fields

Question

There is a theorem that is stated in non-linear systems by Hassan Khalil as follows:

Let $C$ be a simple closed curve not passing through any equilibrium point. Consider the orientation of the vector field $f(x)$ at a point $p\in C$. Letting $p$ traverse $C$ in a counterclockwise direction, the vector field the vector $f(x)$ rotates continuously and, upon returning to the original position, must have rotated an angle $2 \pi k$ for some integer $k,$ where the angle is measured counterclockwise. The integer $k$ is called the index of the closed curve $C$. If $C$ is chosen to encircle a single isolated equilibrium point $\bar{x},$ then $k$ is called the index of $\bar{x}$.

1. The index of a node, a focus or a center is +1.
2. The index of a (hyperbolic) saddle point is -1.
3. The index of a closed orbit is +1.
4. The index of a closed curve not encircling any equilibrium points is +1.
5. The index of a closed curve is equal to the sum of indices of the equilibrium points within it.

I did some research and discovered that its extension is the Poincaré-Hopf index theorem. My question is which of the above conditions still holds in higher order systems?