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According to my lecture notes:

The properties of stability or attraction do not depend on the chosen initial time $t_0$. This is a consequence of the theorem of continuous dependence with respect to the initial conditions applied to the interval $[t_1,t_0]$ if $t_1$ is another initial time.

to give a bit of context stability and the notion of region of attraction is defined in my case for autonomous equations $x' = f(x)$. This is the theorem they're making reference to:

Let $D \subseteq \mathbb{R} \times \mathbb{R}^d$ open, $f:D \to \mathbb{R}^d$ continuous such that:

$\forall (t_0,x_0) \in D.$ (P) has a unique global solution.

Then, $\Omega = \{(t;t_0,x_0) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}^d. (t_0,x_0) \in D, \alpha(t_0,x_0) < t < \omega(t_0,x_0)\}$ is open and $G$ is continuous.

Do you see how to justify the above statement?

Edit:

After reviewing the question I realize that it is easier to use the uniform convergence of solutions. Namely:

If $(t_{0n},x_{0n},\lambda_{0n}) \to (t_0,x_0,\lambda_0)$ and $\varphi_n$ are associated solutions with parameters $(t_{0n},x_{0n},\lambda_{0n})$ then for some large $n$ if we fix $I$ a compact subset of the domain of $\varphi$ the solution for parameters $(t_0,x_0,\lambda_0)$ then there is uniform convergence from $\varphi_n$ to $\varphi$ on I.

So something like this should work for stability:

enter image description here

Here $\varphi$ is the stable solution, the orange region represents the neighbourhood where solutions are guaranteed to stay withing the stability bounds, the pink curves represent the uniform convergence. I still need to formalize this reasoning.

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