An attractor of a dynamical system is a region in state space toward which the system tends to evolve for a range of initial conditions, called the attractor's basin.
variants of attractors

Attractors can be points, sets of points, trajectories, manifolds, or fractals.

Attractors can be sticky (the system tends to remain in the attractor region once it enters) or transitory (the system tends to pass through the region but not necessarily stay within it).

Attractors can exist on different levels of abstraction, e.g. an LLM simulation containing an agentic simulacrum may have an attractor towards the realization of the agent's abstract goals, but not any particular verbatim strings.

An attractor whose basin covers only a subset of possible states is called a local attractor, and one whose basin spans the entire state space is a global attractor.