The nature and localization of critical parameter sets called bifurcations is a central issue in nonlinear dynamical system theory. Codimension‐1 bifurcations form hypersurfaces in parameter space. Some bifurcations of higher codimension can be identified as intersections of these surfaces. These bifurcations of higher codimension bare insights about the global dynamics of the system. Here we describe an algorithm that combines adaptive triangulation with the approach of generalized models and theory of complex systems to compute and visualize such bifurcations in a very efficient way.The method enables us to gain extensive insights in the local and global dynamics not only in one special system but in whole classes of systems. To illustrate these capabilities we present an example of a generalized eco‐epidemic model.