How to show a dynamical system is smooth or not smooth?

Question

Consider a a smooth dynamical system on $\mathbb{R}^n$ : $$\begin{align} \phi:\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n \end{align}$$ where $\phi$ is a smooth function satisfies

1. $\phi_{0}(X)=X$

2. $\phi_{t} \circ \phi_{s}=\phi_{t+s}$ for $t,s \in \mathbb{R}$

Consider the following 3:

1. Time Dilation: $$\begin{align} \psi(t,X) :=\phi(at,X) \end{align}$$ where $a>0$

2. Time Reversal $$\begin{align} \psi(t,X):=\phi(-t,X) \end{align}$$

3. Time Translation $$\begin{align} \psi(t,X) := \phi(t-t_{0},X) \end{align}$$ where $\phi_{-t_{0}}$ is not the identity function.

I'm asked to show that the first 2 are still smooth dynamical systems, but the third one is not. Then, if $\phi_{-t_{0}}$ is an identity function, can the third one become a smooth dynamical system?

I don't know where to start. Any help on this? Thanks.