attractoren

An attractor in dynamical systems is a set of states toward which a system tends to evolve from a wide range of initial conditions. Attractors are invariant, meaning that once trajectories enter the attractor they stay on or near it. They are associated with a basin of attraction, the region of initial states from which trajectories converge to the attractor as time progresses.

Types of attractors include stable fixed points (point attractors), where the system settles at a single state; limit cycles (periodic attractors), where trajectories converge to a repeating cycle; torus attractors, associated with quasi-periodic motion; and strange attractors, which arise in chaotic systems and have fractal structure. Attractors can be global, influencing almost all initial conditions in a region, or local, attracting only nearby states.

Mathematically, for a continuous-time system x' = f(x) or a discrete map x_{n+1} = F(x_n) on a state space X, a nonempty compact set A ⊆ X is an attractor if there exists a neighborhood U of A such that for all x in U, the distance between the trajectory and A tends to zero as time goes to infinity. The set of initial conditions that converge to A is called the basin of attraction.

Applications span physics, biology, neuroscience, and climate science, where attractors describe stable long-term behavior, periodic rhythms, or chaotic regimes. Attractors are contrasted with repellors, which repel nearby trajectories. References to examples include damped mechanical systems, oscillators with limit cycles, and the Lorenz chaotic attractor.