Singular bifurcations and regularization theory

Nonlinear sciences are present today in almost all disciplines, ranging from physics to social sciences. A major task in nonlinear science is the classification of different types of bifurcations (e.g., pitchfork and saddle-node) from a given state to another. Bifurcation analysis is traditionally based on the assumption of a regular perturbative expansion, close to the bifurcation point, in terms of a variable describing the passage of a system from one state to another. However, it is shown that a regular expansion is not the rule due to the existence of hidden singularities in many models, paving the way to a new paradigm in nonlinear science, that of singular bifurcations. The theory is first illustrated on an example borrowed from the field of active matter (phoretic microswimers), showing a singular bifurcation. We then present a universal theory on how to handle and regularize these bifurcations, bringing to light a novel facet of nonlinear sciences that has long been overlooked.