by Paulo Marques, 2014/01/28
This notebook implements the beautiful Lorenz Attractor in Julia. The Lorenz Attractor is probably the most illustrative example of a system that exhibits chaotic behavior. Slightly changing the initial conditions of the system leads to completely different solutions. The system itself corresponds to the movement of a point particle in a 3D space over time.
The system is formally described by three different differential equations. These equations represent the movement of a point $(x, y, z)$ in space over time. In the following equations, $t$ represents time, $\sigma$, $\rho$, $\beta$ are constants.
$$ \frac{dx}{dt} = \sigma (y - x) $$$$ \frac{dy}{dt} = x (\rho - z) - y $$$$ \frac{dz}{dt} = x y - \beta z $$Let's implement it in Julia.
Let's start by importing some basic libraries.
In [1]:
using PyPlot
using ODE;
We need to define the system of differential equations as an equation of the form: ${\bf r}' = {\bf f}(t, {\bf r})$ where ${\bf r} = (x, y, z)$ and ${\bf f}(t, {\bf r})$ is the mapping function and $t$ is time.
In [2]:
function f(t, r)
# Extract the coordinates from the r vector
(x, y, z) = r
# The Lorenz equations
dx_dt = sigma*(y - x)
dy_dt = x*(rho - z) - y
dz_dt = x*y - bet*z
# Return the derivatives as a vector
[dx_dt; dy_dt; dz_dt]
end;
Let's define the initial conditions of the system ${\bf r}_0 = (x_0, y_0, z_0)$, the constants $\sigma$, $\rho$ and $\beta$ and a time grid.
In [3]:
# Define time vector and interval grid
dt = 0.001
tf = 100.0
t = collect(0:dt:tf)
# Initial position in space
r0 = [0.1; 0.0; 0.0]
# Constants sigma, rho and beta
sigma = 10.0
rho = 28.0
bet = 8.0/3.0;
Now let's solve the differencial equations numericaly and extract the corresponding $(x, y, z)$:
In [4]:
(t, pos) = ode23(f, r0, t)
x = map(v -> v[1], pos)
y = map(v -> v[2], pos)
z = map(v -> v[3], pos);
Let's see how it looks in 3D.
In [5]:
plot3D(x, y, z);
Let's see different cuts around the axes:
In [6]:
fig, ax = subplots(1, 3, sharex=true, sharey=true, figsize=(16,8))
ax[1][:plot](x, y)
ax[1][:set_title]("X-Y cut")
ax[2][:plot](x, z)
ax[2][:set_title]("X-Z cut")
ax[3][:plot](y, z)
ax[3][:set_title]("Y-Z cut");
Copyright (C) 2014 Paulo Marques (pjp.marques@gmail.com)
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