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# The Lorenz Equations

Have you ever wondered why weather forecasters have such a hard time with predictions more than a few days into the future? They have sophisticated mathematical models, giant computers, data from weather stations, satellites and balloons, yet cant reliably say whether its going to be rainy or sunny next weekend. In 1963, mathematician and meteorologist Edward Lorenz took up this problem, focusing specically on a model for convection within the atmosphere. When air at lower levels of the atmosphere is warmer than air at higher levels, the warmer air tends to rise up through the atmosphere, replaced by the falling cooler (denser) air. See, for example, Figure 3.1.

Figure 3.1: Convection. Warm air rises as cool air falls. The model Lorenz derived is a three-dimensional ordinary dierential equation dx = (x y) dt dy = rx y xz dt dz = bz + xy. dt (3.1) (3.2) (3.3)

In this model x represents the speed of the circulating air in a convective cell. y models the dierence in temperature between ascending and descending currents and z models the distortion of the vertical temperature prole. 13

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When x and y have the same sign, this indicates that warm air is rising and cool air is falling. The constants and r are physical constants, the Prandtl and the Rayleigh numbers, respectively. Reasonable values for these constants are = 10 and r = 8/3. The constant b measures the temperature dierence between the upper atmosphere and the ground.

Visualizing solutions.
After setting initial values for the three components x, y and z, we can picture the solution as a parametric curve winding its way through threedimensional space. It can also be useful to sketch the graphs of the individual components of the solution, x(t), y(t) or z(t) versus t. Equilibrium solutions. The most important solutions in a nonlinear system are the equilibrium solutions. These are constant, steady solutions that do not change with time. In this case, they can be found by solving the algebraic system (x y) = 0 rx y xz = 0 bz + xy = 0. From the rst equation, we see that any equilibrium solution must have y = x. Eliminating y from the second and third equations, we nd x(r 1 z) = 0 bz + x2 = 0. (3.4) (3.5)

In order for Equation (3.4) to be satised, we must have either x = 0 or z = r 1. If x = 0, then from Equation (3.5), we must also have z = 0. So one equilibrium solution is (0, 0, 0). In this case there is no convection, just still air. If z = r 1, then from Equation (3.5), we have x = b(r 1). So, if r > 1, we also have the equilibria ( b(r 1), b(r 1), r 1). These twin states represent steady convection, with warm air rising and cool air descending. Numerical approximation. Outside of this handful of equilibrium solutions, it is often very dicult, if not impossible, to nd explicit solutions of

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nonlinear systems of dierential equations. When Lorenz rst studied his system of equations, he turned to one of the worlds rst o-the-shelf computers, the Royal McBee LGP-30, to numerically approximate solutions. Using punch cards and paper tape, the 740 pound, \$47,000 machine raced through about one iteration per second, not including output time.

Figure 3.2: This ad for the \$47,000, 740 pound LGP-30 appeared in 1957. In the remainder of this chapter, we investigate the solutions of the Lorenz equations using Eulers method on our \$200, 2lb netbooks. Using a simple spreadsheet, our computers slice through 2000 iterations and provide simultaneous beautiful graphical output in well under a second.

Chaos!!
We will investigate this system for small values of the parameter r in Exercises 1 and 2. For small values of r, in particular, r < 1, we will see that most solutions tend asymptotically to the equilibrium point (0, 0, 0), representing a still, motionless state, lacking any convective motion. For intermediate values of r, most solutions tend toward one of the nontrivial equilibria; physically, this means the system approaches a steady convective

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ow. In this section, we consider a larger value of r, r = 28. We would like to determine what happens to the solution in the long run for given initial conditions. In Figure 3.3, we approximate the solution starting at (2, 2, 2). Although this solution starts relatively close to (0, 0, 0), it immediately moves away, never to return. However, it never settles in on one of the other two equilibria either, and it never shows any discernable pattern, switching between positive and negative values in an erratic manner. Although the Lorenz system is deterministic (and does have a single unique solution for a given initial condition), the behavior looks almost random.

Figure 3.3: Two views of a solution of the Lorenz equations. In the lefthand panel, a plot of x vs t is shown for the Lorenz equations with r = 28 and initial point (2, 2, 2). In the right-hand panel, we see the corresponding solution curve (x(t), y(t), z(t)). Note that, although the solution appears to cross itself, this is merely a relic of printing a three-dimensional curve on a at page. The solution curve looks even more interesting as it travels through xyzspace. Although the solution does not approach any equilibrium point in particular, it does seem to wind its way around the two nontrivial equilibria. From nearby one of these equilibria, the solution spirals out slowly. When the radius gets too large, the solution gets ejected and thrown into a neighborhood of the other equilibrium, where it begins another outward spiral. The resulting picture looks something like the wings of a buttery.

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Perhaps even more disquieting is the behavior of solutions with nearby initial conditions. One might expect solutions that start nearby each other, evolving according to the same rules, to stay close together. In Figure 3.4, we show solutions with initial conditions (2, 2, 2) and (2.01, 2, 2). Although the two solutions stay close together for a little while, around t = 9 the solutions diverge. After this time, the solutions appear to have no relation to one another.

Figure 3.4: Solutions of the Lorenz equations with nearby initial conditions. The solution with initial point (2, 2, 2) is shown in dark blue, while the solution starting at (2.01, 2, 2) is shown in light blue. This sensitivity of the long-term behavior to very small changes in the initial conditions is what can make long-term predictions so dicult. In real world systems, it is often impossible to know initial conditions to great levels of precision, and any imprecision can lead to drastically dierent solutions! Moreover, this sensitivity has implications for numerical computations as well. If you use a dierent numerical method, or the same method with a slightly dierent step-size (see Exercise 4), you will introduce small dierences in the computed solution and these small dierences will eventually lead to large deviations. Nonetheless, despite the fact that similar initial conditions can lead to very dierent solutions especially when viewed in an x vs. t graph the solution curve in three-dimensions generally has a very consistent shape. (See Exercise 3.) Essentially all initial conditions generate a similar buttery, as the solution curves get closer and closer to some strange attractor in three-dimensional space.

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3.1

Exercises

A spreadsheet implementing Eulers method for the Lorenz equations is available at www.mathematicallyinclined.org For these exercises, adapt the given spreadsheet to explore the solutions of the Lorenz equations. 1. Choose r = 1/2. Try at least ve dierent initial conditions, and discuss what happens to your solutions as t gets large. For one of your solutions, reproduce the x vs. t graph. 2. Choose r = 10. Trying at least ve dierent initial conditions, discuss what happens to your solutions as t gets large. For one of your solutions, reproduce the x vs. t graph. 3. Choose r = 28. Trying at least ve dierent initial conditions, discuss what happens to your solutions as t gets large. In this case, the x vs. t graphs may vary substantially, but does the buttery in xyz-space look qualitatively dierent for any of your initial conditions? 4. The gures in the text were put together using Eulers method for systems with step size h = 0.01. Keeping all of the system parameters the same ( = 10, b = 8/3, r = 26 and (x0 , y0 , z0 ) = (2, 2, 2)) but increasing the Euler step-size to h = 0.011, how long does it take for the new approximate solution to diverge from the one in the text?
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