The Lorenz System

LORENZ_ODE is a Python program which approximates solutions to the Lorenz system of ordinary differential equations (ODE's), creating output files that can be displayed by Gnuplot.

The Lorenz system, originally intended as a simplified model of atmospheric convection, has instead become a standard example of sensitive dependence on initial conditions; that is, tiny differences in the starting condition for the system rapidly become magnified. The system also exhibits what is known as the "Lorenz attractor", that is, the collection of trajectories for different starting points tends to approach a peculiar butterfly-shaped region.

The Lorenz system includes three ordinary differential equations:

        dx/dt = sigma ( y - x )
        dy/dt = x ( rho - z ) - y
        dz/dt = xy - beta z
where the parameters beta, rho and sigma are usually assumed to be positive. The classic case uses the parameter values
        beta = 8 / 3
        rho = 28
        sigma - 10

The initial conditions for this system are not often specified; rather, investigators simply note that the trajectories associated with two very close starting points will eventually separate. However, simply to get started, we can suggest the following starting values at t=0:

        x = 8
        y = 1
        z = 1


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


LORENZ_ODE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

GRAPHICS_EXAMPLES, Python scripts which illustrate how various kinds of data can be displayed and analyzed graphically using the graphics facilities of Python.

RK4, a Python library which applies the fourth order Runge-Kutta (RK) algorithm to estimate the solution of an ordinary differential equation (ODE) at the next time step.

RKF45, a Python library which implements the Runge-Kutta-Fehlberg solver for systems of ordinary differential equations (ODE's).


  1. Edward Lorenz,
    Deterministic Nonperiodic Flow,
    Journal of the Atmospheric Sciences,
    Volume 20, Number 2, 1963, pages 130-141.

Source Code:

You can go up one level to the Python source codes.

Last revised on 24 May 2016.