The Lorenz System

LORENZ_ODE is a MATLAB program which approximates solutions to the Lorenz system, 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:

FLAME_ODE, a MATLAB library which considers an ordinary differential equation (ODE) which models the growth of a ball of flame in a combustion process.

GNUPLOT, MATLAB programs which illustrate the use of the gnuplot graphics program.

GRAPHICS_EXAMPLES_GNUPLOT, gnuplot scripts which illustrate how various kinds of data can be displayed and analyzed graphically using the interactive executable graphics program GNUPLOT.

LORENZ_CLUSTER, a MATLAB library which takes a set of N points on a trajectory of solutions to the Lorenz equations, and applies the K-means algorithm to organize the data into K clusters.

PENDULUM_ODE, a MATLAB library which looks at some simple topics involving the linear and nonlinear ordinary differential equations (ODEs) that represent the behavior of a pendulum of length L under a gravitational force of strength G.

PREDATOR_PREY_ODE, a MATLAB program which solves a pair of predator prey ordinary differential equations (ODE's) using MATLAB's ode23() solver.

SPRING_ODE2, a MATLAB program which shows how gnuplot graphics can be used to illustrate a solution of the ordinary differential equation (ODE) that describes the motion of a weight attached to a spring.


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

Source Code:

Examples and Tests:

Last revised on 16 February 2019.