Automatic Detection of Saddle-Node–Transcritical Interactions

Abstract

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.

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics

“…( 38) diverges. The saddle-node and transcritical bifurcations collide into a saddle-node transcritical (SNT) bifurcation or tricritical point [62,63]. Observe that in Fig.…”

“…The tricritical point in case A (see Fig. 3a) corresponds to a saddle-node transcritical (SNT) bifurcation -i.e., where the lines of transcritical and saddle-node bifurcations intersect without further degeneracies [62,63]. Thus, in order to tune to this transition point one needs to set two parameters in the phase diagram (w EE , w EI ), as explicitly calculated in Appendix VI B.…”