The Lorenz Attractor and Order Inside Chaos, math that surprises you lesson

In 1963 the meteorologist Edward Lorenz followed three numbers that change together in a simplified model of convection. The computed state never settled into a repeating pattern, yet it kept to one two-lobed shape. How can motion be unpredictable and still stay inside a recognizable shape? This page sets up the Lorenz system and its phase space, follows two trajectories that start only $10^{-8}$ apart until they disagree completely, and separates what can be forecast, the region the states keep visiting, from what cannot, the exact route through it.

Key terms

Lorenz attractor

Definition The Lorenz attractor is the set of states approached, as time goes on, by typical solutions of the Lorenz system

\begin{aligned} \frac{dx}{dt} &= 10\,(y - x), \\[0.4em] \frac{dy}{dt} &= x\,(28 - z) - y, \\[0.4em] \frac{dz}{dt} &= xy - \tfrac{8}{3}\,z, \end{aligned}

which uses the standard parameter values $\sigma = 10$ , $\rho = 28$ and $\beta = 8/3$ that Lorenz chose in 1963. In general, an attractor of such a system is a set of states that no solution starting in it ever leaves, that every solution starting in some region around it approaches, and that contains no smaller set with these properties. The Lorenz attractor is bounded, and it has zero volume, because these equations shrink the volume of every region of states by the factor $e^{-41/3} \approx 1.2 \times 10^{-6}$ in each unit of time. Its two lobes circle around the two steady states with $x = y = \pm 6\sqrt{2}$ and $z = 27$ . A typical solution near it winds around one lobe a few times, crosses to the other, and keeps switching in a sequence that never settles into a repeating pattern, and the motion has sensitive dependence on initial conditions. So the exact route of one solution cannot be forecast far ahead from an approximate start, while the shape it stays close to is the same for all of them. Pictures of the attractor come from numerical solutions, which are evidence and not proof. Whether a true attractor with these properties exists was settled in 2002, when Warwick Tucker gave a computer-assisted proof that it does for these parameters. The shape lives in the space of states of the model, so it is not a path traced by air in a fluid.

Chaos and fractals are different ideas. Chaos is about motion, while fractal geometry is about the fine structure of sets. The Cantor set is a fractal with no motion at all, and the logistic map with $r = 4$ is chaotic on the interval $[0, 1]$ , which is not a fractal. The Lorenz attractor connects the two, since Lorenz already argued in 1963 that it consists of infinitely many sheets packed closely together. Other chaotic systems have attractors of quite different shapes.

In plain words Solutions of the equations are drawn toward one thin two-lobed shape and then move around close to it forever. Where on the shape a solution will be far in the future cannot be forecast from an approximate start, but that it will be close to the shape can.

Example Start at the state $(1, 1, 1)$ , solve the equations numerically up to time $t = 60$ with a high-accuracy method, and drop the part from $t = 0$ to $t = 10$ as the transient. The rest traces both lobes, with $x$ staying between about $-20$ and $20$ and $z$ between about $0$ and $50$ . Solving again with error tolerances ten times tighter gives a solution that agrees with the first to within $0.001$ up to $t = 25$ but differs from it by more than $1$ by $t = 35$ , after which the two computed solutions follow different routes. Both still trace the same two-lobed shape. The broad shape of the picture can be trusted, while the exact route of a computed solution is reliable only over a limited time. In a chaotic system that drift is expected, and it does not mean that the equations or the method are wrong.

Lorenz system

Definition The Lorenz system is the set of three differential equations

\begin{aligned} \frac{dx}{dt} &= \sigma\,(y - x), \\[0.4em] \frac{dy}{dt} &= x\,(\rho - z) - y, \\[0.4em] \frac{dz}{dt} &= xy - \beta z, \end{aligned}

where $\sigma$ , $\rho$ and $\beta$ are positive constants, $t$ is time in the units of the model, and $dx/dt$ is the rate at which $x$ is changing. Lorenz obtained it in 1963 by cutting a model of a fluid layer heated from below down to three variables. The variable $x$ measures the strength of the rolling convection motion, $y$ the temperature difference between rising and sinking fluid, and $z$ how far the temperature profile from bottom to top departs from a straight line. The constant $\sigma$ is the Prandtl number of the fluid, $\rho$ compares the heating with the heating at which convection starts, and $\beta$ depends on the shape of the convection cells. The products $xz$ and $xy$ make the system nonlinear, so the effects of separate changes cannot simply be added. Every starting state has exactly one solution, and it exists for all later times. The steady states, where all three rates are zero, are the origin and, when $\rho > 1$ , the two points

\begin{aligned} x = y &= \pm\sqrt{\beta(\rho - 1)}, \\[0.4em] z &= \rho - 1. \end{aligned}

The system is a simplified model, not a faithful description of real convection, but its mathematics is exact.

In plain words Three numbers describe the state of an idealized convecting fluid, and the equations say how fast each of them is changing right now. The present state fixes the whole future.

Example With the standard parameters the two steady states away from the origin have $x = y = \pm\sqrt{72} = \pm 6\sqrt{2} \approx \pm 8.485$ and $z = 27$ . At the positive one $dx/dt = 0$ because $x = y$ , and the other two rates are

\begin{aligned} \frac{dy}{dt} &= 6\sqrt{2} \cdot (28 - 27) \\[0.4em] &\quad - 6\sqrt{2} = 0, \\[0.4em] \frac{dz}{dt} &= 72 - \tfrac{8}{3} \cdot 27 = 0, \end{aligned}

using $xy = 6\sqrt{2} \cdot 6\sqrt{2} = 72$ . The origin stands for fluid at rest, and the two other points for steady convection rolling one way or the other. At $\rho = 28$ all three steady states are unstable. A typical state close to one of the two rolling states spirals away from it, about one turn every $0.62$ time units, with its distance growing by about $6$ percent per turn, so solutions do not settle there.

Phase space

Definition The phase space, or state space, of a system is the set of all its possible states, with each state drawn as one point. For the Lorenz system a state is a triple $(x, y, z)$ , so the phase space is three-dimensional space with axes $x$ , $y$ and $z$ . The equations give every point a velocity, made of the three rates $dx/dt$ , $dy/dt$ and $dz/dt$ , and as time passes the state moves along a curve called a trajectory. Because the rates depend only on the current state and change smoothly, exactly one trajectory passes through each point. Two different trajectories therefore never meet, and a trajectory can come back to a point it has already visited only if it is periodic. Apparent crossings in a picture come from drawing a three-dimensional curve on a flat page. A point of phase space is a state of the whole model, not a position in the fluid, so the two-lobed curve is not the path of any parcel of air. For the logistic map the phase space is the interval $[0, 1]$ , and time moves in steps.

In plain words Phase space shows the whole state of the system at one moment as a single point. Watching that point move is the same as watching all three quantities change together.

Example At the state $(1, 1, 1)$ the three rates are

\begin{aligned} \frac{dx}{dt} &= 10\,(1 - 1) = 0, \\[0.4em] \frac{dy}{dt} &= 1 \cdot 27 - 1 = 26, \\[0.4em] \frac{dz}{dt} &= 1 - \tfrac{8}{3} = -\tfrac{5}{3}. \end{aligned}

So at that moment the point moves almost exactly in the direction of increasing $y$ , with $x$ momentarily not changing and $z$ slowly decreasing. One instant later the rates are different, because the state is different.

Sensitive dependence on initial conditions

Definition A system has sensitive dependence on initial conditions on a set $A$ of states when there is a distance $d > 0$ with the following property. For every state $p$ in $A$ and every $\varepsilon > 0$ , however small, there is a state $q$ in $A$ within distance $\varepsilon$ of $p$ such that, at some later time, the trajectories from $p$ and $q$ are more than $d$ apart. It does not say that every nearby pair separates, or that the separation comes soon. For the Lorenz system with the standard parameters, the distance between two close trajectories on the attractor grows on average like $\varepsilon\,e^{\lambda t}$ , where $\varepsilon$ is the starting distance and $\lambda \approx 0.9$ is the largest Lyapunov exponent, the average rate of exponential growth of small gaps. This lasts until the distance is comparable to the size of the attractor, beyond which it cannot grow. The growth is uneven, and for short stretches the distance can even shrink.

In plain words Two starting states that are almost the same can lead, given enough time, to futures that are far apart. Knowing the equations exactly does not help if the start is known only approximately.

Example Follow the trajectories from $(1, 1, 1)$ and from $(1.00000001, 1, 1)$ , which start $10^{-8}$ apart. They were computed with two independent high-accuracy methods, one of them carrying $30$ significant digits, and both give the distances below.

Time $t$	Distance between the two states
$0$	$1.0 \times 10^{-8}$
$10$	$1.8 \times 10^{-8}$
$15$	$2.4 \times 10^{-7}$
$20$	$9.3 \times 10^{-6}$
$25$	$2.0 \times 10^{-3}$
$28$	$0.021$

For the first ten time units the distance hardly grows, and then it grows fast. It first exceeds $1$ at about $t = 29.1$ , and at about $t = 31.7$ the two states are on different lobes for the first time. Soon afterward the distance between them is as large as the distance between two unrelated states on the attractor, at times more than $40$ .

Butterfly effect

Definition The butterfly effect is the popular name for sensitive dependence on initial conditions, used especially about weather forecasting. It comes from the title of a talk Lorenz gave in 1972, which asked whether the flap of a butterfly’s wings in Brazil could set off a tornado in Texas. Its precise content is that in a system with sensitive dependence an error in the measured starting state, however small, can grow until the forecast state and the true state are as far apart as two unrelated states of the system. Because the error grows roughly exponentially, making the starting error ten times smaller extends a useful forecast only by a fixed amount of time, about $\ln(10)/\lambda$ , rather than making it ten times longer. The butterfly effect does not say that small causes regularly produce large effects, that a particular butterfly causes a particular storm, or that nothing about the system can be predicted. The region of states the system keeps visiting stays the same.

In plain words A tiny uncertainty about the present can become a complete uncertainty about the detailed state later. Measuring better helps, but each extra digit of accuracy buys only a fixed amount of extra forecast time, while the overall range of possible states remains predictable.

Example For the standard Lorenz system $\lambda \approx 0.9$ , so each extra factor of $10$ in starting accuracy buys about

\frac{\ln 10}{0.9} \approx 2.6

time units of forecast. The two trajectories of the last card show what stays predictable. After about $t = 32$ their detailed routes have nothing in common, and an observer who knew the start only to eight decimal places could not say which lobe the state would be on. Yet both trajectories stay in the same region, with $x$ between about $-20$ and $20$ and $z$ between about $0$ and $50$ , and both keep circling the same two steady states.

Check yourself

Change the heating to $\rho = 10$ and keep $\sigma = 10$ and $\beta = 8/3$ . Find all the steady states. Why does the formula for the two rolling states need $\rho > 1$ ?

Show the answer

A steady state needs all three rates to be zero. The first equation gives $y = x$ . The second then reads $x(\rho - z) - x = x(\rho - 1 - z) = 0$ , so either $x = 0$ or $z = \rho - 1 = 9$ . The third reads $x^2 - \tfrac83 z = 0$ . With $x = 0$ it gives $z = 0$ , the origin. With $z = 9$ it gives $x^2 = 24$ , so

\begin{aligned} x = y &= \pm\sqrt{24} = \pm 2\sqrt{6}, \\[0.4em] z &= 9, \end{aligned}

which is about $(\pm 4.899, \pm 4.899, 9)$ . The equation $x^2 = \beta(\rho - 1)$ has real solutions other than $0$ only when $\rho > 1$ . For $\rho \le 1$ the origin is the only steady state, because the heating is too weak to keep the fluid rolling.

For the Lorenz system with $\sigma > \beta + 1$ and $\rho > 1$ , the two rolling steady states are stable exactly when

\rho < \frac{\sigma(\sigma + \beta + 3)}{\sigma - \beta - 1}.

Compute this threshold for $\sigma = 10$ and $\beta = 8/3$ . What does it say about $\rho = 10$ and about $\rho = 28$ ?

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First check the condition. Here $\sigma = 10$ is larger than $\beta + 1 = 11/3$ , and the threshold is

\begin{aligned} &\frac{10\,(10 + \tfrac83 + 3)}{10 - \tfrac83 - 1} \\[0.4em] &\quad = \frac{10 \cdot \tfrac{47}{3}}{\tfrac{19}{3}} = \frac{470}{19} \\[0.4em] &\quad \approx 24.74. \end{aligned}

The value $\rho = 10$ lies below the threshold, so the two rolling states are stable, and solutions computed from many starting states settle at one of them, with no lasting chaos. The value $\rho = 28$ lies above it, so both rolling states are unstable, which is why the solutions keep moving around the attractor instead of settling. Stability of a steady state is a statement about states near it only, so on its own it does not decide what happens to solutions that start far away.

The two trajectories from $(1, 1, 1)$ and $(1.00001, 1, 1)$ , which start $10^{-5}$ apart, are first more than $1$ apart at about $t = 21.6$ . Use $\lambda \approx 0.9$ to predict when the pair from the sensitive dependence card, which starts $10^{-8}$ apart, first gets more than $1$ apart. Compare with the computed value.

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On average the distance is multiplied by $e^{0.9}$ in each time unit, so making the starting distance $1000$ times smaller should delay the moment the distance reaches $1$ by about

\frac{\ln 1000}{0.9} \approx 7.7

time units. The prediction is about $21.6 + 7.7 = 29.3$ , and the computed value is about $29.1$ . A start known a thousand times more accurately buys less than eight extra time units, because growth by a factor of $1000$ takes only about $7.7$ time units at this rate.