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Automatic Detection of Saddle-Node-Transcritical Interactions

Dynamical systems with special structure can exhibit transcritical bifurcations of codimension one. In such systems, the interactions of transcritical bifurcations of codimension two can act as organizing centers. We consider saddle-node–transcritical interactions with either one or two zero eigenvalues and show that, using default test functions, the widely used continuation packages MatCont and AUTO classify these interactions as cusp and Bogdanov–Takens bifurcations, respectively. We propose a new test function that distinguishes these singularities and demonstrate its use in the analysis of a predator–prey-nutrient model strained by a toxicant. The details of the implementation are provided, along with test codes for MatCont.