Solve

\begin{cases} \dot{x} &= \sigma \cdot (y - x) \\ \dot{y} &= x \cdot (r - z) - y \\ \dot{z} &= x y - b z \\ \end{cases}

for 0 < t < 40 with initial conditions

\begin{cases} x(0) = -11 \\ y(0) = -16 \\ z(0) = 22.5 \\ \end{cases}

and \sigma=10, r=28 and b=8/3, which are the classical parameters that generate the butterfly as presented by Edward Lorenz back in his seminal 1963 paper Deterministic non-periodic flow. This example’s input file ressembles the parameters, inital conditions and differential equations of the problem as naturally as possible with an ASCII file.

```
PHASE_SPACE x y z # Lorenz attractor’s phase space is x-y-z
end_time = 40 # we go from t=0 to 40 non-dimensional units
= 10 # the original parameters from the 1963 paper
sigma = 28
r = 8/3
b
= -11 # initial conditions
x_0 = -16
y_0 = 22.5
z_0
# the dynamical system's equations written as naturally as possible
= sigma*(y - x)
x_dot = x*(r - z) - y
y_dot = x*y - b*z
z_dot
PRINT t x y z # four-column plain-ASCII output
```

```
$ feenox lorenz.fee > lorenz.dat
$ gnuplot lorenz.gp
$ python3 lorenz.py
$ sh lorenz2x3d.sh < lorenz.dat > lorenz.html
```