lorenz_ode_sensitivity_test, a Python code which demonstrates sensitivity to initial conditions in the Lorenz system of ordinary differential equations (ODE), using an approach suggested by John D Cook.
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 zwhere 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. In this program, we consider two initial conditions at t=0:
x1 = 1 y1 = 1 z1 = 1and
x1 = 1 y1 = 1 z1 = 1.00001and allow the system to evolve over 0 ≤ t ≤ 40. Plotting the values of x1-x2 over time shows the rather large and chaotic variation in the solutions.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
lorenz_ode_sensitivity_test is available in a MATLAB version and a Python version.
Original Python code by John D Cook. Modifications by John Burkardt.
lorenz_ode, a Python code which approximates solutions to the Lorenz system of ordinary differential equations (ODE), creating graphics output using matplotlib.