I've been reading about dynamical systems on a plane and the stable and unstable manifolds that can exist there. As part of this I was reading the definition of a manifold (that a manifold is a function that is homeomorphic to $\mathbb{R}^n$). However, I was not clear on whether this means that a manifold from a dynamical system on the plane would be homeomorphic to $\mathbb{R}$ (since it is a curve and can be parameterized by a single variable), or if it is homeomorphic to $\mathbb{R}^2$ since the plane that it is in is $\mathbb{R}^2$. How would I go about proving what dimension of $\mathbb{R}^n$ the manifold is homeomorphic to?
To clarify: the definition of a manifold above I saw from topology related fields and was then trying to understand how this relates to manifold of dynamical systems on a plane (such as those seen here: Finding approximation to stable manifold of saddle point)
I don't think you really need to prove anything. What you've given is a definition for a manifold, not a theorem.
The more common definition that I've encountered is that a manifold is a locally Euclidean topological space. In more technical terms, this means that every point has a neighborhood which is homeomorphic to an open ball in $\mathbb{R}^n$.
If you're describing a manifold in terms of a function, then you are probably dealing with an embedding of an $n$-dimensional manifold in $\mathbb{R}^{n+1}$. Although the manifold is embedded in an $n+1$ dimensional space, the manifold itself is $n$ dimensional, and has neighborhoods homeomorphic to open $n$-balls; a 1-manifold embedded in $\mathbb{R}^2$ (i.e. a curve), has neighborhoods homeomorphic to open intervals $\mathbb{R}$, a 2-manifold embedded in $\mathbb{R}^3$ (i.e. a surface), has neighborhoods homeomorphic to an open disk in $\mathbb{R}^2$, and so on.