FEIGENBAUM's NUMBER
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1261 ▶ |
| Object type | Discipline oriented, Methodology or model |
FEIGENBAUM's number is related to period-doubling sequences in systems turning chaotic through the increase of parameter a in the difference equation x$_{n+1}$ = ax$_{n}$ (1-x$_{n}$), which can also be written in its quadratic form x$_{n+1}$ = ax$_{n}$ - ax$_{n}$2 , where the second term is nonlinear.
R.V. JENSEN states: “As a is increased, the long-time motion converges to period — 8, — 16, — 32, — 64… cycles, finally accumulating to a cycle of infinite period for a = a$_{inf}$~3,57… FEIGENBAUM was able to prove, using a remarquable application of the renormalization group, that the intervals over which a cycle is stable decrease at a geometric rate of ~ 4,6692016. The tremendous significance of this work is that this rate and other properties of the period-doubling bifurcation sequence are universal in the sense that they appear in the dynamics of any system which can be approximately modeled by a nonlinear map with a quadratic extremum. FEIGENBAUM's theory has subsequently been confirmed in a wide variety of physical systems such as turbulent fluids, oscillating chemical reactions, nonlinear electric circuits, and ring lasers” (1987, p.171).
It would be very interesting to see if this property conducive to the onset of chaos is also present in ecological, economic and social systems.