Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.
A generalization of the well-known Wilson-Cowan model of excitatory and inhibitory interactions in localized neuronal populations is presented, by taking into consideration distributed time delays. A stability and bifurcation analysis is undertaken for the generalized model, with respect to two characteristic parameters of the system. The stability region in the characteristic parameter plane is determined and a comparison is given for several types of delay kernels. It is shown that if a weak Gamma delay kernel is considered, as in the original Wilson-Cowan model without timecoarse graining, the resulting stability domain is unbounded, while in the case of a discrete timedelay, the stability domain is bounded. This fact reveals an essential difference between the two scenarios, reflecting the importance of a careful choice of delay kernels in the mathematical model. Numerical simulations are presented to substantiate the theoretical results. Important differences are also highlighted by comparing the generalized model with the original Wilson-Cowan model without time delays.
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