# Construction of dynamical systems from ODEs with initial values

From Wikipedia

Given an initial value problem $$\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})$$ $$\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0$$

The solution is an evolution function $$\boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0)$$ Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy $$\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0$$ where $\scriptstyle\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}$ is a functional from the set of evolution functions to the field of the complex numbers.

I was wondering why the domain of $\scriptstyle\mathfrak{G}$ is ${(T\times M)}^M$? I thought it was $T\times M$, since $\Phi(t, x_0) \in M$.

What is $\scriptstyle\mathfrak{G}$? How can it be determined from the initial value problem?

Thanks!

-
Each flow $\Phi: T \times M \rightarrow M$, and the domain of that functional is all such flows, and so the set of such flows can be regarded as $(T \times M)^M$. –  Christopher A. Wong Oct 18 '12 at 1:17
(1) Isn't the set of such flows $M^{T \times M}$ instead? (2) The domain of $\scriptstyle\mathfrak{G}$ doesn't look like the set of such flows, by looking at $\mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0$. –  Tim Oct 18 '12 at 1:22
Gosh, sorry, I was confusing myself with something else. –  Christopher A. Wong Oct 18 '12 at 1:36