Organising centres in the semi-global analysis of dynamical systems

The notion of organising centre of a bifurcation diagram is used as an ordering principle in the analysis of nonlinear problems in pure and applied dynamical systems theory. When considering a given, generic n-parameter family of dynamical systems the codimension n bifurcations are isolated in the parameter space, generating a more global array of lower codimension bifurcations. It often makes sense to add one extra parameter to the system, e.g., by varying a 'constant' coefficient. In such cases, semi-global parts of the given n-dimensional bifurcation set can be understood as generic slices in versal unfoldings of codimension n + 1 singularities. This can give great insight in the structure of the given system, as we shall illustrate in two extensive case studies.