LORENZ_ODE
The Lorenz System

LORENZ_ODE is a C++ 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
      

        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
      

Licensing:

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

Languages:

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:

GNUPLOT, C++ 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.

SPRING_ODE2, a C++ 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.

Reference:

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

Source Code:

• lorenz_ode.cpp, the source code.

Examples and Tests:

• lorenz_ode_output.txt, the output file.
• lorenz_ode_data.txt, the graphics data file.
• lorenz_ode_commands.txt, the graphics command file.
• xyz_3d.png, a plot, created by gnuplot, of (X(t),Y(t),Z(t)).
• xyz_time.png, a plot, created by gnuplot, of (t,X(t)), (t,Y(t), and (t,Z(t)).

List of Routines:

• LORENZ_RHS evaluates the right hand side of the Lorenz ODE.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• RK4VEC takes one Runge-Kutta step for a vector ODE.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.