# lorenz_ode

lorenz_ode, a FORTRAN90 code which approximates solutions to the Lorenz ordinary differential equations (ODEs), 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
```

### Languages:

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.

### Related Data and Programs:

gnuplot_test, FORTRAN90 codes which illustrate the use of the gnuplot graphics program.

predator_prey_ode, a FORTRAN90 code which solves a pair of predator prey ordinary differential equations (ODE).

spring_ode, a FORTRAN90 code 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.

stiff_ode, a FORTRAN90 code which considers an ordinary differential equation (ODE) which is an example of a stiff ODE.

### Reference:

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

### Source Code:

Last revised on 28 July 2020.