Coexistence of periods in a bifurcation

A particular type of order-to-chaos transition mediated by an infinite set of coexisting neutrally stable limit cycles of different periods is studied in the Varley–Gradwell–Hassell population model. We prove by an algebraic method that this kind of transition can only happen for a particular bifurcation parameter value. Previous results on the structure of the attractor at the transition point are here simplified and extended.

► The bifurcation of the Varley–Gradwell–Hassell population model is revisited.
► The structure of the attractor mediating the bifurcation to chaos is studied.
► Geometric arguments are given for the coexistence of periods in the attractor.
► An algebraic method for the localization of these bifurcations is provided.