INITIAL CONDITIONS (Sensibility to)
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1704 ▶ |
| Object type | Discipline oriented, General information, Methodology or model |
A condition in a nonlinear system by which very slight differences in initial conditions lead to considerable ones in the later behavior of the system.
More than one century ago, H. POINCARÉ, in his study “On the three bodies problem and the equations of dynamics” (1889), showed the impossibility to determine in an absolutely rigorous way the future positions of three (or N) interacting bodies: the problem cannot be satisfactorily integrated.
The basic cause of such a situation is that a 3-bodies (or more) system is not deterministic in a linear way.
N. PEGUIRON writes on this topic:“This sensibility to initial conditions, characteristic to any long term deterministic prediction, showed by POINCARÉ, is a general case in classical mechanics: only some quite simple cases escape to this, as for example the two-bodies system” (1989,p. 11).
In more complex cases, the discrepancies, not being still important, may remain unheeded during more or less a long time. This is what allows for example for apparently rigorous prediction of the astronomical trajectories. However, at long or very long term, it has been recently demonstrated that even the solar system is not absolutely stable.
PEGUIRON comments this situation in the following way: “Indeterminism in classical mechanics does not appears in their governing equations, but through the impossibility to apply them to a system not devoid of external influences. Clearly, in this case, the isolation process displays a property, (rigorous) determinism, which is not proper of real systems” (p.11).
The sensibility to initial conditions leads to chaotic behavior, i.e. to unpredictability, which grows with time. However this does not imply the total disappearance of global determinism.
POINCAR- established the bases for the study of deterministic chaos, when he created the concepts of limit cycle, transverse arc and section normal to the trajectory, now called “POINCAR -'s section”.
The non-instantaneous transmission of local causes and the non-simultaneity of the resulting effects makes up another way to account for partial and local breaches of determinism and of the unpredictable character of the trajectories bifurcations.
See also
Sensibility to initial conditions