The word bifurcation entered modern usage courtesy of the French mathematician Jules Henri Poincare with the publication of his work, L’Équilibre d’une masse fluide animée d’un mouvement de rotation, as part of the Acta Mathematica, and first appeared in September 1885. Poincare is known as the last true great universalist in mathematics-he was skilled in most areas of math and is known today for his formulation of what was kown for a century as the Poincare Conjecture before it was solved in 2005. Poincare coined the word bifurcation to describe the divergence of two states from a common smooth state: a bifurcation occurs when a small change made to the parameter values of a system causes a sudden qualitative or topological change in its behaviour.
The word bifurcation comes directly from the Latin word furca meaning fork, attaching the prefix bi- meaning two. It should be noted that the word furca also had metaphorical and figurative uses in Roman times and the Romans had the word tridens (corresponding directly to the English word trident) which looked more like a modern fork than the two pronged affair used by Romans. In defense of Poincare, he was seeking a strict mathematical definition-he needed to limit the sense to two options as a starting point-he wanted to be sure that the furca imagined was limited to two options and not three.
Image of Poincare in the public domain. Image of Roman fork of unknown provenance.