Normal forms, differentiable conjugacies and elementary bifurcations of maps

TL;DR

This paper enhances classical bifurcation theorems for maps by introducing extended normal forms with extra terms, ensuring differentiable conjugacy to the original maps on basins of attraction and repulsion.

Contribution

It develops extended normal forms with algebraically determined coefficients that are differentiably conjugate to the original maps, improving the standard bifurcation analysis.

Findings

Extended normal forms include additional terms for better conjugacy.

Normal forms are differentiably conjugate to original maps on key basins.

The approach refines bifurcation analysis beyond topological equivalence.

Abstract

We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with coefficients determined by algebraic equations. These extended normal forms are differentiably conjugate to the original maps on basins of attraction and repulsion of fixed points or periodic orbits. This reflects common assumptions about the additional information in normal forms despite standard bifurcation theorems being formulated only in terms of topological equivalence.