Consider the differential equations
$$ \frac{\partial\varphi_{j}}{\partial t}(t,x,\epsilon) = F^{j}(t,\varphi_{j}(t,x,\epsilon),\epsilon), \quad\text{for}\quad j=0,1,\cdots, M $$
with the initial values,
\begin{align} \varphi_{0}(0,x,\epsilon)&=x,\\ \varphi_{j}(\alpha_{j}(x,\epsilon),x,\epsilon)&=\varphi_{j-1}(\alpha_{j}(x,\epsilon),x,\epsilon), \quad\text{for}\quad j=1,2,\cdots,M. \end{align}
Here $F^{j}:S^{1}\times \mathbb{R}^{d}\times (\epsilon_{0},\epsilon_{0})\rightarrow \mathbb{R}^{d} $, for $j=0,1,\cdots, M$ are $C^{r}$ functions and $\alpha_{j}$ represents a time variable.
I want to show inductively that $\alpha_{j}(x,\epsilon)$ and $\varphi_{j}$ are $C^{r}$ functions, for $j=0,1,\cdots, M$. I know that using the differential dependence of ODEs on the initial conditions (Theorem), we can show that $\varphi_{j}$ are $C^{r}$ functions for $j=0,1,\cdots, M$, however I don't know if we can use this Theorem in order to show the same assertion for $\alpha_{j}$. So I would be grateful if someone can help me clarify this question.