# Lorenz System

This module illustrates the numerical solution of the Lorenz system of ordinary differential equations, a crude model for atmospheric circulation. Ordinary differential equations (ODEs) have applications in a wide variety of contexts, including meteorology. One such application is modeling atmospheric circulation. In 1963 meteorologist Edward Lorenz published a model given by the system of ODEs

y1′ = σ (y2y1),
y2′ = r y1y2y1 y3,
y3′ = y1 y2b y3.

This system of ODEs results from a spectral discretization of a partial differential equation describing convective motion in a two-dimensional fluid cell that is warmed from below and cooled from above, crudely modeling atmospheric circulation. The variable y1 represents the rate of convective overturning, y2 the horizontal temperature variation, and y3 the vertical temperature variation. The parameter σ represents the Prandtl number, r the Rayleigh number, and b the geometric proportions of the problem domain. This system of ODEs was one of the first shown to exhibit chaotic behavior, in which the solution oscillates in a seemingly random way, never settling into either stationary nor truly periodic behavior.

The user first selects values for the parameters σ, r, and b, whose default values are those used by Lorenz in his original paper. Next the user selects the initial conditions for each variable and how far forward in time to compute the solution. Finally, the user selects the step size and numerical method to be used in computing an approximate solution for the ODE system.

Clicking Solve then calculates the approximate solution using the chosen numerical method and step size, and displays the results graphically. In the upper left graph, each component of the solution is plotted as a function of time. The other three graphs plot the trajectories (y1(t), y2(t)), (y1(t), y3(t)), and (y2(t), y3(t)), showing three different views of the three-dimensional solution trajectory. Degradation in the quality of the solution using a lower-order method or coarser step size is clearly visible. Indeed, the approximate solution may be so poor that it does not fit within the displayed portion of the graph, in which case using a higher-order method or smaller step size will yield a more accurate solution.

References:

1. Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Computer Problem 9.6 on page 419.
2. Edward N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20:130-141, 1963.

Developers: Evan VanderZee and Michael Heath