Attractory

Attractory is a term used in dynamical systems to denote a set or region in the system’s phase space toward which trajectories tend to evolve over time. In practice, an attractory is the long-term destination for a broad class of initial conditions. It is invariant under the dynamics: if a trajectory starts on the attractory, it remains there, and nearby trajectories converge toward it as time progresses. The collection of initial conditions that lead to a given attractory is called its basin of attraction.

Attractories can take several forms. Fixed-point attractors correspond to steady states where the system stops changing. Limit cycles are closed orbits representing periodic motion. Tori can support quasi-periodic motion on a torus. Strange attractors exhibit chaotic dynamics with fractal structure and are associated with sensitive dependence on initial conditions. Attractories can be local, influencing only a portion of the state space, or global, attracting a large region. Basins of attraction may have smooth boundaries or complex, interwoven boundaries that produce intricate patterns.

Examples include the logistic map, where varying a parameter can produce a stable fixed point, a stable two-cycle, or chaotic attractors; and the Lorenz system, which has a well-known strange attractor illustrating chaotic long-term behavior. In applied settings, attractories help describe the qualitative long-term behavior of systems ranging from physics and biology to economics.

Note that attractory is closely related to attractor and is sometimes used interchangeably, though attractor is the more common term in contemporary literature. In mathematical contexts, attractors may be described as invariant, attracting sets that capture the asymptotic behavior of dynamical systems.