Relation between numbers of stable and unstable fixed points or manifolds

Question

In 1D dynamical systems, it is well-known that in general between any two stable fixed points there is an unstable fixed point. How does this result generalize to higher dimensions? Are there general theorems that establish a connection between the number of stable fixed points versus unstable fixed points?

Of course in higher dimension the extra complication is that we could have higher dimensional manifolds defined by $\frac{dx}{dt} = 0$, e.g. lines or surfaces. Are there general results for their stability in relation to the number of stable and unstable lines / surfaces / higher dimensional manifolds?

How about systems defined not on $\mathbb{R}^n$ but on some manifold?

Answer 1

The answer is "sometimes yes". The full statement is: for some particular classes of dynamical systems there is an analogue of Morse inequalities; see, for example, this and this. For gradient systems (as far as I understand) this is just a reformulation of Morse theory for critical points.

Answer 2

As you seem to have guessed correctly, the answer in no. Really the $1$-dimensional is topological: in particular you need not have stable/unstable hyperbolic fixed points, only the directions matter.