Cusp and generalized flip bifurcations under higher degree conditions
The authors study local bifurcations generated by bi-parametric families of one-dimensional maps. It is about to define normal forms associated with codimension two bifurcations of cusp and flip types. Remind that a bifurcation is a qualitative change of a map (or an ODE) behavior under a small parameter variation. A fold bifurcation results from the merging of a stable fixed-point with an unstable one. A flip bifurcation consists in the period doubling of a fixed-point. The codimension of a bifurcation is the smallest dimension of a parameter space which contains the bifurcation in a persistent way. A normal form is the result of a series of coordinates changes leading to a simpler form of the map containing only predominant nonlinearities. Conditions for the existence of a cusp point of a fold bifurcation curve in a parameter plane, and of a codimension two flip bifurcation, are given in the general case, i.e., when these conditions depend on the map nonlinear terms of any degree. The corresponding degenerated cases are also considered. In presence of a cusp point, the bifurcation curve separates the parameter plane into two regions: one corresponding to three fixed-points, the other to only one fixed-point. In the case of a codimension two flip bifurcation, the parameter plane is divided into three regions: one related to the presence of a period two cycle and a fixed-point, one related to the presence of two period two cycles and a fixed-point, the last one with only one fixed-point. With the degenerated cusp bifurcation, the bifurcation curve splits the plane into a region without fixed-point, and a region associated with two fixed-points.
scientific article; zbMATH DE number 1879956
scientific article