#. # Lorenz’ attractor (the one with the butterfly)
#.
#. 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](http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2).
#. 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

sigma = 10            # the original parameters from the 1963 paper
r = 28
b = 8/3

x_0 = -11             # initial conditions
y_0 = -16
z_0 = 22.5

# the dynamical system's equations written as naturally as possible
x_dot = sigma*(y - x)
y_dot = x*(r - z) - y
z_dot = x*y - b*z

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
#> ::: {#fig:ex_lorenz}
#> ![Gnuplot](lorenz-gnuplot.svg){width=48%}
#> ![Matplotlib](lorenz-matplotlib.png){width=48%}
#> 
#> The Lorenz attractor computed with FeenoX plotted with two different tools
#> :::
