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

TL;DR

This paper proves that the conditions defining certain high degeneracy bifurcations in non-autonomous periodic maps remain invariant under cyclic permutations, using topological conjugacy, which is crucial for understanding these bifurcations.

Contribution

It establishes the invariance of degeneracy, non-degeneracy, and unfolding conditions for $A_{n}$ bifurcations under cyclic order changes in non-autonomous periodic difference equations.

Findings

Invariance of bifurcation conditions under cyclic permutations.

Application of local topological conjugacy in proofs.

Examples of $A_{3}$ bifurcation in period two difference equations.

Abstract

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.