Attractor

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 $x_n=x_0^{2^n}\to 0$; 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.

Attraction in a dynamical system means convergence of nearby forward trajectories toward a specified invariant state or set. Attraction and Lyapunov stability are separate properties; together they give the usual local asymptotic stability of an equilibrium.

A strange attractor, in the geometric convention, is an attractor whose structure is irregular or fractal rather than a smooth point, cycle or torus. The geometry concerns the set of possible long-term states. Chaos concerns evolution on that set. Some authors include chaos in the meaning of strange, so the convention must be stated.

The separation matters because the same plot cannot answer every question. A box-counting dimension calculation asks how many boxes are needed to cover a set as their size decreases. A Lyapunov exponent calculation follows the growth of a perturbation along a trajectory. These measure different features, and neither number can be substituted for the other.

Strange nonchaotic attractors provide the explicit warning: geometrically complicated attraction can occur without a positive Lyapunov exponent. The original construction used systems driven by two incommensurate frequencies.^[ref] Thus fractal-looking geometry alone is insufficient evidence for a chosen definition of chaos.

A strange nonchaotic attractor combines geometrically irregular attraction with no positive Lyapunov exponent for the dynamics under consideration. Systems driven at two incommensurate frequencies supplied the original examples.^[ref]

One useful framework is a skew productThe base is an irrational rotation of the circle. An attracting invariant graph of a skew product obeys $h(\theta +\omega )=F_{\theta }(h(\theta ))$; the geometric question concerns how irregular that graph is. For a differentiable one-dimensional fiber map, the fiber Lyapunov exponent along it is, when the limit exists,A negative value describes average contraction of perturbations within the fiber. The base rotation has a neutral exponent, so a negative fiber exponent does not mean that every exponent of the full forced system is negative. Under appropriate boundedness and integrability assumptions for this triangular derivative system, the full largest exponent is zero rather than positive.

These equations specify a framework, not a claim that every choice of $F_{\theta }$ produces a strange attractor. For instance, $F_{\theta }(x)=ax$ with $|a|<1$ has the smooth attracting graph $h=0$. A negative fiber exponent alone therefore establishes neither geometric strangeness nor a discontinuous graph.