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 (y2 − y1),
y2′ = r y1 − y2 −
y1 y3,
y3′ = y1 y2 − b
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
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
References:
Developers: Evan VanderZee and Michael Heath