LIMIT CYCLE (Stable) (= Attractor)
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1916 ▶ |
| Object type | Methodology or model |
- “Closed integral curve which represents a periodic regime” (M. DUBOIS et al, 1987, p.193).
This representation, introduced by H. POINCARÉ, was indeed the first example of a (non punctual) attractor.
In order to obtain it, POINCARÉ introduced the notion of transverse arc: it is an arc of curve, which is tangent to none of the points of an integral curve (meaning that it intersects many times with the latter). Considering the succession of the intersections of an integral curve with a transverse arc, he reduces the study of the curve in the plane to the study of the succession of points on a line. POINCARÉ next shows that any integral curve which does not end into a singular point is a cycle'' or a spiral asymptotically winding around the cycle'', which he thus calls a limit cycle.“ (J.L. CHABERT & A. DAHAN DALMEDICO, 1991, p.568).
The limit cycle corresponds to the VAN DER POL equation which “describes the behavior of an oscillator which is kept going, with increase of the oscillations of reduced amplitude and decrease of those of wide amplitude.” (P. BERGE et al, 1984, p.26).
Any trajectory of this kind converges towards the limit cycle, which corresponds to the type of systems tending to oscillating dynamic stability.
The basic condition for the existence of a limit cycle is a constant input of sufficient energy to the system. However, an excess of energy may throw it out of its stability range, in accordance with thermodynamics of systems far-from-equilibrium.
See also
See also: Cycle (Limit), Cycle (Stable limit)