> 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.

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.