A bifurcation occurs when an attractor changes qualitatively with the smooth variation of a control parameter. Physically, bifurcations denote phase transitions from a state of equilibrium to new possible states of equilibria. (see also singularities)

In Biology, bifurcations describe transitions from a state of higher symmetry (lower complexity) to one of lower symmetry (higher complexity). This process is called symmetry breaking. Mathematically, symmetry is defined by the invariance of certain laws with respect to transformations between the corresponding reference systems of an observer. "C'est la dissymetrie, qui crée le phénomène" said Pierre Curie. 

C. Lloyd Morgan, in his Emergent Evolution, of 1923, describes emergence in much the same way, when he writes, "The emergent step, although it may seem more or less saltatory, is best regarded as a qualitative change of direction, or critical turning-point, in the course of events." According to Manuel de Landa, for Deleuze the machinic phylum is the overall set of self-organizing processes... in which a group of previously disconnected elements suddenly reaches a criticial point in which they begin to "cooperate" to form a higher entity. The notion of a machinic phylum blurs the distinction between organic and non-organic life. Phenomena of self-organization occur wheneve a bifurcation takes place in phase space: when a new attractor appears or when the system's attractors mutate in kind. 

According to Prigogine, deterministic descriptions break down at bifurcation points, and the type of fluctuation present in the system will lead to the choice of branch it will follow. (Order Out of Chaos, p. 177) Fluctuations are caused by a huge number of randomly moving particles. ( see clinamen. )

see also epigenesis.