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!
