Difference between manifold and R^n for dynamical systems I have been going through competing dynamical systems textbooks recently, and don't understand the distinction between two definitions. One of my textbooks says that a dynamical system is defined as:
$\mathbf{x}' = f(\mathbf{x})$, 
where $\mathbf f(x)$ is a vector field such that $\mathbf{f}:\mathbb{R}^{n} \to \mathbb{R}^{n}$.
My other textbook says that $\mathbf{f}$ is a mapping from $M$ (The manifold) to $TM$, the tangent bundle. 
Are the two definitions equivalent? Is it common in dynamical systems theory to just assume that $M = \mathbb{R}^n$, I'm not understanding the distinction.
Thanks.
 A: First let me address what your various textbooks say. 
Given a manifold $M$, to say that a vector field $f$ is a mapping from $M$ to $TM$ is incomplete. Let me use $T_p M$ to denote the fiber of $TM$ over the point $p \in M$, in other words $T_p M$ is the tangent space of $M$ at $p$. A vector field on a manifold $M$ is not just any old mapping from $M$ to $TM$. Instead, a vector field on $M$ is a mapping $f : M \to TM$ such that for each $p \in M$ we have $f(p) \in T_p M$, in other words $f(p)$ is a vector in the tangent space at $p$. 
In the special case where $M = \mathbb{R}^n$, if you keep all of this notation in mind, then your two textbooks are saying essentially the same thing. The tangent bundle in this case is a product $T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$, and the tangent space at each point $p\in \mathbb{R}^n$ has the form 
$$T_p \mathbb{R}^n = \{(p,v) \,|\, v \in \mathbb{R}^n\}
$$
where the vector operations on the vector space $T_p \mathbb{R}^n$ are defined by simply ignoring the $p$ coordinate, i.e. $(p,v) + (p,w) = (p,v+w)$ and similarly for scalar multiplication. Because of this, there is a canonical isomorphism between vector fields expressed as functions 
$$f : \mathbb{R}^n \to \mathbb{R}^n
$$
and vector fields expressed as functions
$$g : \mathbb{R}^n \to T \mathbb{R}^n
$$
This canonical isomorphism is given by the formula $g(p)=(p,f(p))$.
Regarding your last sentence, perhaps there may be elementary expositions of dynamical systems that restrict attention to $M=\mathbb{R}^n$, but the full theory of dynamical systems considers manifolds in all their full and general glory, and in this theory it is not sufficient to consider $\mathbb{R}^n$. Dynamical systems on spheres, on toruses, and on all kinds of manifolds are important.
I would also point out that it is misleading to say that a dynamical system on a manifold $M$ is a vector field. What is important in dynamical systems is not the vector field in particular, but its integral curves and their behavior over long time spans. 
