Does this problem have a clear and straightforward algorithmic solution or
the resolution of this problem involve the use of heuristics?
Do a numerical experiment
trial and error
trust you intuition
jerzy
The bifurcation diagrams and dynamical maps can do the trick. Just consider whole parameters' space of the system. Actually, excluding extra specific cases (e.g. hidden attractors), it is computationally adequate task. Depends on resolution, of course, but some basic knowledge about system dynamics can be obtained easily enough.
I advice to use first more rapid but less precise algorithms instead of direct LLE or Lyapunov Spectrum calculation, which are really computationally heavy methods.
You can find some simple examples here:
Conference Paper Comparing the algorithms of multiparametric bifurcation analysis
Conference Paper Bifurcation and recurrent analysis of memristive circuits
Please remember, that in discrete simulations, timestep is also a system parameter:
Data Bifurcation Diagram Behavior for Discrete models of Rossler ...
You can also apply the method of this paper where authors just used bifurcation diagrams by means of numerical simulations.
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