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

Abstract

Bifurcations of the class $A_{\mu}$, 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 relatively to cyclic order of compositions for any natural number $\mu$. The main tool for the proofs is the local topological conjugacy. Invariance results are essential for proper definition of bifurcations of the class $A_{\mu}$ and associated lower codimension bifurcations, using all possible cyclic compositions of fiber families of maps of the $p$-periodic difference equation. Finally, we present two examples of the class $A_{3}$ or swallowtail bifurcation occurring in period two difference equations for which bifurcation conditions are invariant.