Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

Abstract

Bifurcations of the Aµ class in Arnold's classification, in nonautonomous p-periodic difference equations generated by parameter depending families with p maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relative to cyclic order of compositions for any natural number µ. The main tool for the proofs is local topological conjugacy. Invariance results are essential to the proper definition of the bifurcations of class Aµ, and lower codimension bifurcations associated, using all the possible cyclic compositions of the fiber families of maps of the p-periodic difference equation. Finally, we present two actual examples of class A 3 or swallowtail bifurcation occurring in period two difference equations for which the bifurcation conditions are invariant.Date: February, 2015 -AMS: Primary:37G15; Secondary: 39A28.

Bifurcation equations for periodic orbits of implicit discrete dynamical systems

“…It is completely open and would be interesting to investigate the invariance of the bifurcation equations for periodic non-autonomous systems defined implicitly in the line of work of [24].…”