Singular Bifurcations : a Regularization Theory

TL;DR

This paper introduces a regularization theory for singular bifurcations in nonlinear systems, supported by an exact swimmer model, addressing a longstanding gap in bifurcation analysis.

Contribution

It develops a general regularization framework for singular bifurcations and provides explicit examples, expanding classical bifurcation theory.

Findings

Exact swimmer model exhibits a singular bifurcation normal form.

Regularization theory effectively handles singular bifurcations.

Fills a longstanding gap in bifurcation analysis.

Abstract

Several nonlinear and nonequilibrium driven as well as active systems (e.g. microswimmers) show bifurcations from one state to another (for example a transition from a non motile to motile state for microswimmers) when some control parameter reaches a critical value. Bifurcation analysis relies either on a regular perturbative expansion close to the critical point, or on a direct numerical simulation. While many systems exhibit a regular bifurcation such as a pitchfork one, other systems undergo a singular bifurcation not falling in the classical nomenclature, in that the bifurcation normal form is not analytic. We present a swimmer model which offers an exact solution showing a singular normal form, and serves as a guide for the general theory. We provide an adequate general regularization theory that allows us to handle properly the limit of singular bifurcations, and provide several explicit examples of normal forms of singular bifurcations. This study fills a longstanding gap in bifurcations theory.