atraktor

Atraktor, or attractor in English, is a concept in dynamical systems describing a set of states toward which a system tends to evolve from a wide range of initial conditions. Formally, it is a closed, invariant subset of the state space that has a basin of attraction: trajectories starting nearby will approach the attractor as time progresses to infinity. The attractor represents the long-term behavior of the system.

There are several common types of atraktors. Fixed-point attractors attract states to a single equilibrium. Limit cycle attractors attract states to a periodic orbit, so the long-term behavior repeats with a fixed period. Torus attractors are higher-dimensional generalizations leading to quasi-periodic motion. Strange attractors are associated with chaotic dynamics and have fractal structure, producing complex yet bounded trajectories that depend sensitively on initial conditions.

Key concepts associated with atraktors include the basin of attraction—the set of initial conditions that lead to the attractor—and stability, which describes how trajectories respond to small perturbations. Attractors are studied in both continuous-time (differential equations) and discrete-time (difference equations) systems. In dissipative systems, attractors reflect energy loss over time and often organize complex dynamics into a simpler long-term behavior.

Examples illustrate the range of attractors. The Lorenz system exhibits a famous strange attractor with chaotic dynamics. The logistic map shows fixed-point and periodic attractors depending on parameter values. Attractors also appear in physics, biology, and engineering, where they help explain how diverse initial states can converge to common, predictable patterns in the long run.

Notes on terminology: while “atraktor” is used in some languages, the English term is typically “attractor.” The concept remains central to understanding how complex systems settle into structured, repeatable behavior.