International Encyclopedia of Systems and Cybernetics

"The property of a system, or some K part of it in the phase space X such as if, starting from any initial position in K, fluctuations as small as wished end up generating macroscopic divergences" (after A. DOUADY, 1992, p.17).

RUELLE and TAKENS established that, in chaotic systems "two trajectories of the attractor, initially as close to each other as wanted, finally always diverge from each other. This divergence increases even, on average, exponentially with time".

… "Thus the trajectory actually followed by the system depends crucially from its starting point." (P BERGÉ, Y. POMEAU & C. VIDAL, 1984, p.111).

This is a basic difference between stable and unstable systems.

In conservative systems, the trajectories remain confined within closely determined limits, characterized by an asymptotic attractor or a periodic one (stability in the sense of LAGRANGE). The behavior of such systems can be predicted.

However, they are not the rule, but well the exception. Conservative systems depends on no more than two initial and independent conditions, in which case a chaotic behavior cannot appear.

On the contrary, with three or more incommensurable initial conditions (or degrees of freedom, or order parameters), chaos appears, characterized by alternatively growing and contracting deformations of the trajectories, the deformations being limited by the most extreme values that can be taken by the physical variables. Thus, global determinism is not lost, but becomes concealed behind apparently incoherent short term variations. In a sense, the system seems to lose the precise "memory" of its past, retaining only a kind of statistical (or ergodic) "memory". This is related to the reciprocal influence of the various conditions on each other and to the impossibility of instantaneous transmission of their effects, at a distance. The global determinism does not wholly disappear, but at any instant, becomes locally fragmented and, due to time lags can never be fully recovered.