A method for the complete qualitative-analysis of 2 coupled ordinary differential-equations dependent on 3 parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODES) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear nonlinearly) can be used to parametrize the relevant loci.