lets say we have a system of ordinary differential equations and want to how the trajectories starting from all possible initial states evolve, without theoretically proving any properties (e.g. stability) of equilibrium and so on. This would be good for building intuitions before trying to prove certain properties, e.g. global asymptotic stability of an equilibrium. Thus I am wondering what is usual way to observe the trajectories computationally?
i was thinking that: set a lot of initial conditions and simulate those trajectories starting from them, so I can get the phase portrait. is this the usual way? I think streamplots can help. is there any package/software specialized for doing this? if the gradient of basin of attraction can be visualized, then better.
many thanks in advance!