= Attractor
{wiki}
An <attractor> is an <invariant set> approached by nearby trajectories of a <dynamical system>. A useful precise convention is a compact invariant set that is <Lyapunov stable> and attracts some open neighborhood: starting sufficiently close keeps the trajectory close for all future times, and the distance to the set tends to zero. Definitions differ over the stability, uniformity and minimality conditions, so those conditions should be stated when they affect an argument.
For the <discrete dynamical system> $x_{n+1}=x_n^2$ on $\mathbb R$, the <fixed point> zero is an <attractor>. If $|x_0|<1$, then $x_n=x_0^{2^n}\to0$; if $|x_0|$ is small, every later value is at least as small. Its <basin of attraction> is exactly $(-1,1)$. The other <fixed point>, one, is invariant but repelling: its <derivative> is two, and arbitrarily close starting values can move away.
An attracting <limit cycle> is an <attractor> consisting of a whole repeating orbit. A trajectory can approach that set while continuing to move around it, so attraction to a set does not mean convergence to one stationary point. A <strange attractor> can have more complicated geometry; attracting nearby trajectories alone does not imply <chaos>.
Back to article page
