Bifurcations, Schwarzian derivatives and Feigenbaum constant revisited

TL;DR

This paper demonstrates that the Feigenbaum delta constant is more universal than previously thought, extending its applicability to broader classes of functions and bifurcations in one-parameter endomorphism families.

Contribution

It introduces the parenthesis permeability hypothesis and relaxes Singer's conditions, broadening the scope of systems where the Feigenbaum constant applies.

Findings

Feigenbaum delta constant is more universal across bifurcation types.

The conjecture applies to functions with multiple maxima and positive Schwarzian derivatives.

Universality of the Feigenbaum constant is supported in more complex dynamical systems.

Abstract

The main purpose is to show that Feigenbaum delta constant is much more universal than believed. The paper is mainly devoted to period-doubling processes in families in one parameter of endomorphisms of the interval and consider generalizations of the Feigenbaum delta constant. We formulate the so-called parenthesis permeability hypothesis, a conjecture that holds for all types of bifurcation (i.e. for flip, fold, pitchfork and transcritical bifurcations, which states that under some conditions two or three different functions may have exactly the same bifurcation points. We propose a conjecture that considerably relaxes David Singer conditions for endomorphism families to generate at most one stable orbit, showing that Feigenbaum constant appears also in some classes of functions that have more than one maximum and have positive Schwarzian in at least one sub-interval. This version contains more arguments in favor of an even greater Feiganbaum constant universality.