lorenz_ode, a Python code which approximates solutions to the Lorenz system of ordinary differential equations (ODE), which exhibit sensitive dependence on the initial conditions.
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. 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 MIT license
lorenz_ode is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.
python_ode, Python codes which sets up various systems of ordinary differential equations (ODE).