> A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions \$$\phi^t\$$ that for any element of t ∈ T, the time, map a point of the phase space back into the phase space.

Let's make it concrete, with an example. Suppose our manifold is the unit disc:

\$M = \lbrace x,y\ |\ x^2 + y^2 \leq 1 \rbrace \$

Using the given terminology, each point of the disc is a 'phase', and the whole disc comprises the phase space.

For a fixed time \$$t\$$, the function \$$\phi^t: M \rightarrow M\$$ is the mapping that sends each point in the disc to where it ends up at time \$$t\$$.

From this we get that \$$\phi^0: M \rightarrow M\$$ must be the identity function.