Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes

TL;DR

This paper investigates the scaling laws near saddle-node bifurcations, revealing new phenomena and diverse scaling behaviors in both generic and non-generic cases, including $C^0$ vector fields and two-parameter families.

Contribution

It extends the understanding of saddle-node bifurcation scaling laws to non-generic cases and $C^0$ vector fields, discovering rapid changes in scaling behavior in two-parameter systems.

Findings

Different scaling laws depending on second parameter values

Discovery of a new phenomenon in two-parameter saddle-node bifurcations

Illustration using an overdamped pendulum with varying length

Abstract

The scaling of the time delay near a "bottleneck" of a generic saddle-node bifurcation is well-known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate $C^0$ vector fields. Our main result is a new phenomenon in two-parameter families having a saddle-node bifurcation upon changing the first parameter. We find distinct scalings for different values of the second parameter ranging from power laws with exponents in (0,1) to scalings given by O(1). We illustrate this rapid quantitative change of the scaling law by a an overdamped pendulum with varying length.