Dynamical system

Remarks

Clauses 1 and 2 say that $t \mapsto \Phi^t$ is a monoid homomorphism from $\mathbb{T}$ to the maps $X \to X$ under composition, that is, an action of $\mathbb{T}$ on $X$. In practice $X$ carries more structure, and $\Phi$ is required to respect it. Examples are continuous maps on a topological space, smooth maps on a manifold, or measurable maps on a measure space. With $\mathbb{T} = \R$ the system is a flow, with $\mathbb{T} = [0, \infty)$ a semiflow, and with $\mathbb{T} = \N$ or $\Z$ a discrete-time system.

Example: $\mathbb{T} = \R$, $X = \R$ and $\Phi(t, x) = e^{-t} x$, the solutions of $x' = -x$. Clause 1 holds since $e^{0} x = x$, and clause 2 holds since $e^{-s}(e^{-t} x) = e^{-(s + t)} x$.

Non-example: $\Phi(t, x) = x + t^2$ on $\R$ with $\mathbb{T} = \R$. Clause 1 holds, but $\Phi(1, \Phi(1, 0)) = 2$ while $\Phi(2, 0) = 4$, so clause 2 fails.