Where does bifurcation collide?

In the mathematics of dynamical systems, a bifurcation is a qualitative change in the set of solutions of a system, such as a change in the number or the stability of its steady states, that happens as a control parameter passes a critical value. Henri Poincaré named the phenomenon in his 1885 study Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation, where he located points de bifurcation at which a family of equilibrium shapes splits into new branches. Guckenheimer and Holmes's 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields set out the modern classification of these branchings and their normal forms. Here the referent is an abstract change in a system's solution structure at a parameter threshold, not a phase of a lawsuit, a division in a person's awareness, or a place where a river splits.[1][2]

Unit

One qualitative change in the solution structure of a system as a parameter crosses a critical value

Recovery means

Not applicable

Measured by

Located by continuation methods and eigenvalue analysis that track how fixed points and their stability change with a parameter

In feminist sociology, the bifurcation of consciousness is the split a person lives between the abstract, textual world of ruling institutions and the concrete, embodied world of everyday local life. Dorothy Smith introduced the idea in her 1974 article Women's Perspective as a Radical Critique of Sociology and developed it in her 1987 book The Everyday World as Problematic: A Feminist Sociology, arguing that women in particular were required to inhabit both worlds at once and to translate lived experience into the categories of formal organisation. Here the referent is a divided standpoint within a knowing person, described and interpreted rather than read off a scale, and it has nothing in common with a branching of equations, a phase of a trial, or a river that forks.[3][4]

Unit

One divided standpoint held by a person moving between institutional and everyday worlds

Recovery means

Not applicable

Measured by

Traced through institutional ethnography and accounts of everyday experience rather than a numeric scale

In river science, a bifurcation is a node where a single channel divides into two downstream branches, sending its water and sediment along diverging paths, as in a braided reach or a delta. Bolla Pittaluga, Repetto, and Tubino's 2003 study Channel bifurcation in braided rivers: Equilibrium configurations and stability modelled how discharge and sediment partition between the two branches and when that partition stays balanced or tips toward one branch. Here the referent is a physical fork in a flowing channel, surveyed and gauged on the ground, and it is distinct from a change in a system's solutions, a stage of a lawsuit, or a split in a person's awareness.[5]

Unit

One channel node where flow divides into two branches, by the share of water and sediment sent to each

Recovery means

Not applicable

Measured by

Surveyed by channel gauging and mapping of how discharge and sediment split, and reproduced in morphodynamic models and laboratory flumes