attractor

An attractor in a dynamical system is a set toward which the system tends to evolve from a wide range of initial conditions. It is invariant under the evolution and "attracts" nearby states as time progresses. Formally, if a system has a flow φ(t, x) with state x, a nonempty compact invariant set A is an attractor if there exists a neighborhood U of A such that for every x in U, the distance between φ(t, x) and A tends to zero as t approaches infinity. The subset of U consisting of all x whose trajectories tend to A is called the basin of attraction of A.

Attractors can take various forms. Simple examples include fixed points (steady states) and closed orbits (limit cycles). More complex attractors include invariant tori and chaotic or strange attractors, which have fractal structures and sensitive dependence on initial conditions. In discrete-time systems, the analogue is a compact invariant set A that attracts nearby points under iteration of a map.

Global attractors are a related concept in dissipative or wide-ranging systems: there exists a global attractor that attracts all bounded sets, and it is the minimal closed invariant set with this attracting property.

Basins of attraction can be intricate, and their boundaries may be fractal, particularly in systems with chaotic attractors. Attractors describe the long-term behavior of a system and are central to understanding stability, pattern formation, and the qualitative dynamics of physical, biological, and engineered processes. They are distinguished from repellors, which attract trajectories backward in time, and from transient or non-attracting sets that do not attract an open neighborhood.