KAM THEOREMS

Charles François (2004). KAM THEOREMS, International Encyclopedia of Systems and Cybernetics, 2(1): 1807.

Collection	International Encyclopedia of Systems and Cybernetics
Year	2004
Vol. (num.)	2(1)
ID	◀ 1807 ▶
Object type	Discipline oriented, Methodology or model

“'KAM” is for the Russian mathematicians and physicists A.N. K'OLMOGOROV, V.I. A'RNOL'D and J. M'OSER. The theorems are about the dynamics near a periodic orbit. They show that for some Hamiltonian systems (i. e., practically homeostatic) affected by slight perturbations, there may exist numerous different initial conditions that may produce quasi-periodic movements, describable by convergent series, and corresponding to toroidal attractors: The threshold of chaos can be, or not, crossed.

ARNOL'D showed that this applies, at least in some cases when perturbations are quite small, to POINCARÉ's n-bodies problem.

For non-Hamiltonian systems, in I. PRIGOGINE's words: “KOLMOGOROV” (K-) systems exhibit a complex behavior of individual trajectories, like exponential divergence, to the effect that no finite algorithm can reliably compute the motion of phase points sensibly faster than the dynamics itself. Systems of this kind include geodesic flow on surfaces with negative curvature…“ (1986, p.5).

K-FLOWS

(Also named after KOLMOGOROV)

K-flows are characteristic of near-chaotic systems. J. CASTI writes: “Their behavior is at the limit of unpredictability, in that having even an infinite number of measurements of where the system was in the past is of no help in predicting where it will be found next. K-flows are widespread among systems in which collisions between particles dominate the dynamics” (1990, p.288).

In synthesis, the borders between ergodic systems, thermodynamically unstable ones and chaotic ones are fuzzy.

See also