POINCARE Section
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(2) |
| ID | ◀ 2573 ▶ |
| Object type | Discipline oriented |
A method “of reducing the study of continuous time systems (flows) to the study of an associated discrete time system (map)” (S. WIGGINS, 1988, p.67).
J. GLEICK explains this mathematical technique as follows: “The technique reduces a three-dimensional picture to two dimensions. Each time the trajectory passes through a plane, it marks a point, and gradually a minutely detailed pattern emerges” (1987, p.143).
The same technique may be applied to more complex systems, i.e., with a higher number of initial independent conditions.
WIGGINS emphasizes: “In lower dimensional problems (say dimension < 4) numerically computed POINCARÉ maps provide an insightful and striking display of the global dynamics of a system” (p.67).
In this way, it becomes possible to gather an understandable picture of any attractor, be it common or strange. POINCARÉ's method is also related to the ergodic relation.
P. BERGÉ, Y. POMEAU and Ch. VIDAL state: “… POINCARE's section leads to the replacement of the description of some evolution continuous in time by an application at discrete intervals of time.”
Moreover: “…POINCARÉ's section presents the same type of topological properties as the flow of which it is born, because of its mode of construction… If the flow passes an attractor, the structural characteristics of the same will be found in the POINCARÉ section” (1984, p.67).
Thus, this method makes quite easier the study of the different types of stabilities and instabilities.