International Encyclopedia of Systems and Cybernetics

The splitting of a trajectory into two or more new trajectories.

Bifurcations characterize states of uncertainty.

They occur in the "region of the parameter space where a state of a system becomes unstable and gives way to a pair of states in which a new order parameter has a finite value" (R. FIVAZ, 1991 , p.31).

According to this comment by FIVAZ, a bifurcation is thus characterized by the emergence of a qualitative change in the nature of an attractor. According to I. PRIGOGINE and G. NICOLlS; "… (a) bifurcation is basically a "decision making" process. But because of the multiplicity of choices at the "decision" point, a selection mechanism is necessarily involved. Now all physical systems possess such a mechanism, in the form of random fluctuations that are generated spontaneously around the deterministic evolution" (1985, p.6)

Of course, "decision making" in physical systems does not imply any human willfully meaning

PRIGOGINE also adds: "… the phenomenon of bifurcation is extremely general and … there is a wide variety of solutions that can bifurcate: multiple steady states, oscillating solutions, propagating waves, etc. Also there could occur higer-order bifurcations" (1984, p.50).

It is by crossing a critical value in the parameter space that the trajectory becomes divided into two possible trajectories, corresponding to two different qualitative behaviors.

According to R.N. ADAMS "… a number of alternative further paths" is even possible (1988, p.66) and, at such "crossroads" any small random effect may lead to nucleation of new structures.

It thus becomes impossible to predict with perfect accuracy the future behavior of the system.

The domain of possible behaviors becomes discrete, and sudden transitions (i.e. "catastrophes") may occur from one state to another quite different one.

The first bifurcation signals the onset of chaotic behavior, proper of non-conservative, or non-Hamiltonian systems, also called dissipative.

The concept of bifurcation is a significant link between:

- the theory of irreversible systems far from equilibrium (i.e. dissipative)

- the mathematical theory of catastrophes

- the theory of chaos, or topological dynamics

According to S. WIGGINS such a change is a result of "qualitative changes that occur in the orbit structure of a dynamical system as the parameters on which the dynamical system depends are varied" (1988, p.62).

K.DE GREENE states: "A bifurcation is most generally the appearance of a new solution to reaction-diffusion equations at some critical value. Systems moving toward bifurcations reflect both deterministic and probabilistic features. These systems behave deterministically between bifurcation points; however, near bifurcation points, fluctuations are essential to selection of the path the system will follow. Such systems may be structurally unstable if arbitrary small fluctuations alter the mechanisms of interaction between system elements. Evolving complex systems show successive instabilities associated with successive bifurcations that produce increasing coherence" (1988, p.290).