# Stability and attraction properties of ODE does not depend on initial condition

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:

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.