Suppose you have a differential equation describing the evolution of an autonomous system described by $x: \mathbb{R} \rightarrow \mathbb{R}$ (domain = time, range = some scalar quantity):
$$\dot{x} = f_{\alpha\beta\gamma}(x)$$
depending on three parameters $\alpha, \beta, \gamma$, and you are looking for solutions depending on initial values $x_0 = x(0)$.
The result will be a parametrized set of curves $x_{\alpha\beta\gamma x_0}: \mathbb{R}^+ \rightarrow \mathbb{R}$.
How can this set of curves (and its structure) be made visible in the most compelling way, e.g. by showing only (or highlighting in some kind of a heat map) curves for some significant (extremal) parameter sets $\{\alpha\beta\gamma x_0\}$.
Are there heuristics to determine significant parameter sets manually (or even automatically)? And which tools are there (e.g. WolframAlpha, MATLAB) which can be fed with parametrized equations and some predefined parameter sets, and which display in a comprehensible way the "structure" of the solution set, possibly highlighting the predefined curves? (If WolframAlpha is the answer: How to do it?)
The parametrized funtion I have in mind is
$$f_{\alpha\beta\gamma}(x) = \alpha - \beta\min\{1,\gamma x\}$$
with $\alpha, \beta, \gamma > 0, \alpha < \beta$.