Loosely speaking, I am looking for references on convergence properties within a family of solutions to a differential equation, with respect to varying the initial conditions. Having mostly studied basic engineering courses in differential equations I am unable to find a reference to back up a statement I want to make (and of course to confirm that it is even true). Let me try to explain.
Consider an initial value problem $$y'(t) = f(y(t)), \quad y(t_0) = y_0$$ where $t$ is a real variable and $y$ may be, say, $\mathbb C^n$-valued. By the Picard-Lindelöf theorem this problem has a unique solution on some interval $I$, provided that $f$ is Lipschitz continuous on a neighborhood of $y_0$.
Consider now a curve $\gamma: [a, b] \to \mathbb R^n$, and suppose that the whole curve is contained in an open set $D \in \mathbb R^n$ where $f$ is Lipschitz continuous. This curve generates a family $\{y_\alpha\}$ of solutions to the equation above by taking $y(t_0) = \gamma(\alpha)$ for each $\alpha \in [a, b]$.
I would now like to establish some sort of continuity notion within the family $\{y_\alpha\}$. Something like: "Given $\alpha \in [a,b]$, there exists $\epsilon > 0$ such that for every sequence $\{\alpha_n\}$ in $[a,b]$ converging to $\alpha$ the associated sequence $\{y_{\alpha_n}\}$ converges to $y_\alpha$ on the interval $[t_0 - \epsilon, t_0 + \epsilon]$".
Is (something like) this true, and what is a good reference for such a result? If it helps, we can add the following assumptions:
- Suppose $f$ is (continuously) differentiable on $D$ (rather than just Lipschitz).
- Suppose there is a number $r > 0$ such that each point on the curve $\gamma$ has a neighborhood of radius $r$ contained in $D$.
I found a satisfactory enough answer in the concept of the local flow of an autonomous system, introduced in section 4.6.1 of Ordinary Differential Equations - Analysis, Qualitative Theory and Control by Logemann and Ryan, which formalizes continuous dependence on initial conditions in a nice way.