Invariance of bifurcation equations for high degeneracy bifurcations of   non-autonomous periodic maps
  Bifurcations of the $A_{n}$ type in Arnold's classification, in non-autonomous $p$-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 $n$. The main tool for the proofs is local topological conjugacy. Invariance results are essential to the proper definition of the bifurcations $A_{n}$, 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 $A_{3}$ or swallowtail bifurcation occurring in period two difference equations for which the bifurcation conditions are invariant. 
Henrique Manuel Oliveira
math.DS