Is the initial value problem of an ODE considered as a dynamic system?
A dynamic system is defined as
In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a set and Φ is a function $ \Phi: U \subset T \times M \to M $ with $ I(x) = \{ t \in T : (t,x) \in U \}\,$ $ \Phi(0,x) = x\,$ $ \Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, for \, t_1, t_2, t_1 + t_2 \in I(x)\, $
For 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) $
I don't think $\Phi$ satisfy $\Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, for \, t_1, t_2, t_1 + t_2 \in I(x)\, $. For example, when $v(t,x) = f(t)$, $\Phi(t,\boldsymbol{{x}}_0) = \int_0^t f(s) ds + x_0$.
But I often heard of the initial value problem of ODE and dynamic system together. If the former is not an example of the latter, I wonder why they are mentioned together?
Thanks!
There is also a related question posted before here https://math.stackexchange.com/q/215998/1281.