Lorenz attractor

The Lorenz attractor in this visualization

The Lorenz equations are three ordinary differential equations that Edward Lorenz drew in 1963 from a model of convection [1]. This page keeps $\sigma = 10$ and $\beta = 8/3$ and lets you turn $\rho$, the heating. At $\rho = 28$ almost every solution ends up winding around the same bounded, butterfly-shaped set without ever settling or repeating: the Lorenz attractor.

What the pictures show

The 3D phase space, the large picture: every point is a state $(x, y, z)$. The faint grey is one long trajectory from $(1, 1, 1)$; the moving dot and its tail are the reference trajectory; the blue ring is the release point. Drag to turn the view; click its tail to choose where the next cloud is released.

The red key Launch puts a cloud of starts around the release point, each exactly $\varepsilon$ from it in its own direction, and follows them all. The colour of a point shows how far it has drifted from the reference. That colour, the glow, the trails, the dimming of the far side and the faint particles flowing before a release are all display only.

The screen at the top of the panel: above, $x(t)$ of the reference and of one start $\varepsilon$ away — two “forecasts”; the yellow line marks when their $x$ first differ by more than 1, checked every 0.005 time units (a threshold chosen here for “the forecasts have parted”, not a limit built into the system). Below, $\frac1m \sum_i \ln\bigl(d_i(t)/d_i(0)\bigr)$, the mean of the logarithm of how many times farther each point has drifted (the axis reads the factor), with the band from the 10th to the 90th percentile of the points; the “typical distance” in the readout is their geometric mean. At $\rho = 28$ a dashed line shows the slope $\lambda \approx 0.906$, the long-run average, until the distance approaches the attractor’s size.

Replay releases again from the same point with the same directions and the same random seed, using the current $\varepsilon$; the earlier curve stays, faint, for comparison. Only such a replay compares two values of $\varepsilon$ fairly.

Why it behaves like this

Stretching and folding: near the attractor, nearby states are pulled apart in one direction and squeezed together in another; the flow then folds the stretched sheet back onto itself. The cloud shows both — first a thread, then a thread folded over and over.

The Lorenz map (the window key beside $\rho$) plots each maximum of $z$ against the next one, from one long trajectory. The points lie on a thin peaked curve (Lorenz 1963, figure 4) — an approximate one-dimensional description, not a proven exact reduction. At $\rho = 28$, away from the cusp and on the same branch, the sampled slope is steeper than 1 in size, so two maxima that are close become farther apart at the next turn (two on either side of the cusp need not).

For $\rho > 1$, besides the origin, there are two fixed points $C_\pm = \bigl(\pm\sqrt{\beta(\rho - 1)},\ \pm\sqrt{\beta(\rho - 1)},\ \rho - 1\bigr)$; for $\rho < 1$ the origin attracts every solution. $C_\pm$ lose stability in a subcritical Hopf bifurcation at $\rho = \sigma(\sigma + \beta + 3)/(\sigma - \beta - 1) = 470/19 \approx 24.74$. Other values where the behaviour changes are numerical: about 13.926 (a homoclinic explosion at the origin) and about 24.06 (the chaotic attractor appears) [4, 5]. Between about 24.06 (numerical) and $470/19 \approx 24.74$ the chaotic attractor and the two stable fixed points coexist; 99.65 lies inside a window of periodic behaviour, and at 160 the solutions settle on a periodic orbit.

What the numbers are

The picture is a numerical approximation of the trajectories. Errors grow with time, so the exact position after a long time is not a reliable prediction for the given initial value; the growth of distances and long-run statistics have to be checked separately against step size, precision and sampling.

Every trajectory is integrated on your computer with the classical fourth-order Runge–Kutta method, a fixed step of 0.005 independent of the frame rate, in double precision. The cloud is integrated as differences from the reference trajectory, so even $\varepsilon = 10^{-12}$ keeps its digits.

The spreading you see is the difference between starting values under one and the same numerical integrator. On its own it does not prove that the exact equations are just as sensitive; that takes theory (for the classic parameters, Tucker’s proof [2]). The usual shadowing lemma does not apply to the Lorenz attractor [6].

$\lambda \approx 0.906$ is a statistical estimate: the long-run average rate at the classic parameters, computed independently by Sprott [3]; it is not a constant for every $\rho$. “A hundred thousand times more precision buys about $\ln 10^5 / 0.906 \approx 12.7$ more time units” is an estimate for typical exponential growth; a single release can differ a lot.

Easy to misread

• Lines that seem to cross in the picture do not: different orbits cannot cross, since a state belongs to one orbit only. What crosses is their shadow on the screen.
• The name “butterfly effect” and the attractor’s butterfly shape are two separate things: the name comes from Lorenz’s 1972 talk about a butterfly’s wings and a tornado.
• The story of Lorenz typing 0.506 instead of 0.506127 and getting a different forecast happened in 1961 with his twelve-variable weather model, not with these three equations [7].
• At $\rho = 15$ ($C_\pm$ are stable there), a cloud released from this page’s starts does not spread out; it shrinks along with the trajectory into a fixed point.

History

Lorenz published the equations in 1963 as a drastically simplified model of convection, after Saltzman’s, and showed that their solutions are nonperiodic and depend sensitively on where they start [1]. Sparrow’s 1982 book mapped how the behaviour changes with $\rho$ [4]. Whether the attractor seen on computers really exists for the exact equations stayed open until Tucker’s computer-assisted proof (announced 1999, published 2002) [2].

Related concepts

Logistic map · Double pendulum (page planned) · Strange attractor (page planned)

References

1. E. N. Lorenz, “Deterministic nonperiodic flow”, Journal of the Atmospheric Sciences 20 (1963) 130–141.
2. W. Tucker, “A rigorous ODE solver and Smale’s 14th problem”, Foundations of Computational Mathematics 2 (2002) 53–117.
3. J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press (2003).
4. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Applied Mathematical Sciences 41, Springer (1982).
5. J. L. Kaplan, J. A. Yorke, “Preturbulence: a regime observed in a fluid flow model of Lorenz”, Communications in Mathematical Physics 67 (1979) 93–108.
6. M. Komuro, “Lorenz attractors do not have the pseudo-orbit tracing property”, Journal of the Mathematical Society of Japan 37 (1985) 489–514.
7. E. N. Lorenz, The Essence of Chaos, University of Washington Press (1993).

Further reading: Wikipedia: Lorenz system; MacTutor History of Mathematics: Edward Lorenz.