Dynamical Systems on a Manifold

My textbook on dynamical systems says that a dynamical system written as:

$\mathbf{x'} = \mathbf{f(x)}$,

where the right-hand-side $\mathbf{f(x)}$ is a vector field defined on a manifold equivalent to the dimension of the system. My question is that if this is a linear system, where $\mathbf{f(x)} = A\mathbf{x}$, is this still defined on a manifold?

Thanks.

Thomas

1 Answer

Sure; in particular note that just a finite dimensional vector space is a manifold as well. Its tangent spaces are isomorphic to the original space (intuitively, the isomorphism amounts to "moving the origin" to the point where you take the tangent space).

• But, is it more restrictive then that? Since, in a general sense I thought that a vector field on a manifold is a mapping from that manifold to the tangent bundle, that's how you can talk about "vectors". But what about the case where the manifold is a vector space? Is that mapping from a vector space to the tangent bundle of the vector space? I.e., what is the tangent space of a linear vector space? – Thomas Moore Mar 2 '17 at 23:10
• @ThomasMoore The tangent space to a (real) vector space is just isomorphic to the space itself. Think of the vectors in this space as being the same as in the base space, but with translated origin. – Alfred Yerger Mar 2 '17 at 23:35
• Hi @AlfredYerger That makes sense. Thanks! – Thomas Moore Mar 2 '17 at 23:39