A limited region of the phase space (i.e. a point, or a set of points) towards which the trajectory of the system converges, tending to a steady state or periodic motion.

This is the concept of attractor according to classical dynamics.

There are two basic types of classical attractors (as opposed to strange, or chaotic, ones):

1.-The attractor may be a single point (or state) towards which the system's trajectory converges in a monotonic way. This implies, for example, a tendency to asymptotic stability. This was the original mathematical meaning of the term.

2.- The attractor may be "a closed loop, corresponding to periodic motion" (I. STEWART – 1989, p.45)

This is a characteristic of systems which undergo a cyclical or cyclomorphical activity.

The first case corresponds to any system strictly submitted to the 2nd. principle of thermodynamics, i.e. whose entropy grows steadily or, more precisely, to which a trajectory and the corresponding energy function is associated, such as its value constantly decreases with time.

The second case corresponds to systems which retain dynamic stability by maintaining their internal structures and functions within closely defined limits. Such systems tend to produce the minimum flow of entropy compatible with their absorption of energy obtained from the environment (cfr. PRIGOGINE's irreversible systems with small fluctuations as opposed to other irreversible systems, which undergo giant fluctuations throwing them out of their stability limits; and also his principle of minimum entropy production).

Attractors are sinks, which means that the attracted system is submitted to characteristic constraints, from which, in principle, it cannot escape using its own means.

In chaotic dynamics the motion becomes quasi periodic ("The geometrical picture for quasi periodic motion in quasi space is a curve that combines two or more different "circular" motions") (L. STEELS, 1990, p.45)

Hence, a variety of more and more complex attractors, which correspond to "the qualitative diversity of dissipative systems" (I. PRIGOGINE and I. STENGERS, 1992, p.69)

R. SWENSON characterizes the attractor as "The time-independent (time-asymptotic) state, or limit set, that attract initial conditions from some region around its "basin of attraction ", during a set of time-dependent processes (evolutionary behavior) as ty "(after D. RUELLE, 1981) (SWENSON, 1989, p.189)

The concept of attractor is thus connected with the idea of dynamic irreversible change.

However, there seems to be two different kinds of these:

- Adaptive change in systems endowed with organizational closure, which cannot escape the necessity to reach a final destructive stable state ("Death is equifinal" says Stafford BEER)

- Innovative change by emergence. "Evolutionary behavior" corresponds to this kind of transformations (see "attractor (emergent)".

Francis HEYLIGHEN defines the attractor as "a region of state space invariant under the dynamics (each state in the attractor is sent upon a state of the same attractor) (1990, p.496), such that it does not contain subattractors. This means that there are no fixed points inside the attractor. Each point in the attractor is sent on another point of the attractor. This accounts for a model in which there is both conservation of the distinctions between different attractors and variation within each attractor". (Ibid).

Attractors may become multidimensional. M. FARGE defines the attractor's dimension as: "the set of the points of the phases space, i.e. of all possible positions and velocities that can be visited by the system, containing the set of all solutions within the limits of a very long evolution in time" (1992, p.214)

While any attractor controls a basin or area that converges to it, this area can be very narrow. In such a case quite small disturbances may push the system out of the basin and force it to convergence toward another attractor. These jumps from one attractor to another, very far to be a purely abstract model, are the gist of many collapses affecting real structures.

1) general information

2) methodology or model

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