Information AboutAttractor |
| CATEGORIES ABOUT ATTRACTOR | |
| limit sets | |
|
A Trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic or of any other type. MOTIVATION A Dynamical System is often described in terms of Differential Equations that describe its behavior for a short period of time. To determine the behavior for longer periods it is necessary to Integrate the equations, either through analytical means or through iteration, often with the aid of computers. Dynamical systems in the physical world tend to be dissipative: if it were not for some driving force, the motion would cease. (Dissipation may come from Internal Friction , Thermodynamic Losses , or loss of material, among many causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system into its typical behavior. The part of the Phase Space of the dynamical system corresponding to the typical behavior is the attracting set or '''attractor'''. Invariant sets and Limit Set s are similar to the attractor concept. An ''invariant set'' is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A ''limit set'' is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set. For example, the Damped Pendulum has two invariant points: the point of minimum height and the point of maximum height. The point is also a limit set, as trajectories converge to it; the point is not a limit set. Because of the dissipation, the point is also an attractor. If there were no dissipation, would not be an attractor. MATHEMATICAL DEFINITION
An attractor is a Subset ''A'' of the phase space such that:
|
|
|