Strange attractor

An attractor is a set that orbits starting near it come ever closer to and stay close to; it is called strange when it has fractal structure, showing new detail at every magnification. Some authors use the word for any chaotic attractor; here it means the fractal kind. The page uses a positive estimate of the largest Lyapunov exponent as a numerical indication of chaos ( $\lambda_1 > 0$).

Every map on this page is a rule that sends each point of the plane to a new point, applied over and over: time is discrete, and a point jumps at each step. The bright picture is a long-run density, how often long orbits land in each pixel: a statistical estimate of how the attractor is visited. The glow and the colours are display only.

Hénon (1976) built his map from three moves: bend $(x, y) \to (x,\, y + 1 - a x^2)$, squeeze $(x, y) \to (b x,\, y)$ and flip $(x, y) \to (y, x)$. Together they give $x' = 1 - a x^2 + y,\ y' = b x$. Bend and flip keep area; squeeze multiplies it by $|b|$. So at $b = 0.3$ every region's area becomes exactly 0.3 times as large at each step, and after 12 steps $0.3^{12} \approx 5.3 \times 10^{-7}$ of it is left, still positive. Iterating the trapping region indefinitely gives a limiting set of zero area. Nearby orbits separate exponentially on average while they remain close ($\lambda_1 \approx 0.42$, numerical), so a region turns into a longer and thinner band that folds back on itself, again and again: layers.

Release shows the first 12 steps of a patch of starting points. Points in its basin approach the attractor in the long run, generally without landing on it in finitely many steps, so the end of the 12 steps is not an arrival; degenerate parameters are the exception: at $u = 0$ Ikeda’s map sends every point to its fixed point $(1,\ 0)$ in one step. From Hénon’s quadrilateral at $a = 1.4,\ b = 0.3$ no point leaves, because its image lies inside it; with other parameters the patch can reach beyond the region that is attracted, and those points escape. Ikeda’s map splits into twist, shrink (area $\times u^2$) and shift; Clifford’s does not split into simple moves.

After the 12 steps the same points keep going for a few hundred more, and where they land is added up into brightness. The page compares their normalized counts with the long-run density on a coarse grid inside the current view and reports the measured difference. A small difference is a numerical observation at that resolution, not proof that the points have reached the attractor. Other starting regions can approach another attractor or escape.

Near the saddle point $P \approx (0.6314,\, 0.1894)$, the linearized map contracts one direction by $\mu_s \approx 0.156$ per step. Dive in uses the reciprocal, about $1/\mu_s \approx 6.41$, to choose successive magnifications around P. This lets you compare nearby layers at related scales. The multiplier describes the local stable direction; it does not prove that the whole picture repeats under uniform zoom. The displayed layers are finite numerical samples.

A dimension between 1 and 2 measures the layering. The Kaplan–Yorke dimension $1 + \lambda_1/|\lambda_2|$ is about 1.26 for Hénon’s attractor, and counting boxes at the scales a computer reaches gives about 1.2 to 1.26. They are two different estimates, both numerical.

Many posters titled “strange attractor” use maps like Clifford’s, which fold the plane along the curves where $\det J = 0$. The bright threads lie on the images of those folds, the critical curves: the density piles up there. For the parameters on this page $\lambda_1 > 0$, so the orbit is chaotic, but $\lambda_1 + \lambda_2 > 0$ and box counting comes close to 2 at the scales computed; these numbers show no layering, though they do not prove that the attractor fills an area. There are also strange attractors that are not chaotic (Grebogi and others, 1984).

A computer picture cannot tell a strange attractor from a cycle of very long period. Benedicks and Carleson (1991) proved that Hénon’s map has strange attractors for many parameters near $a = 2$ with small $b$; for the classic $a = 1.4,\ b = 0.3$ there is no accepted proof. Misiurewicz (1980) proved strange attractors for Lozi’s map $x' = 1 - a|x| + y,\ y' = bx$ in a range of parameters that contains $(1.75,\, 0.3)$ but not Lozi’s own $(1.7,\, 0.5)$. Tucker (2002) proved that the Lorenz attractor is one. With $b = 0$, one step puts every point on the line $y = 0$, where Hénon’s map is $x \to 1 - a x^2$; for $a > 0$ that is the logistic map in other coordinates ($a = r(r - 2)/4$, $r > 2$).

• WA03Logistic map
• WA01Lorenz attractor
• WB01Mandelbrot set

Further reading: Wikipedia: Attractor; Wikipedia: Hénon map.