Lorenz Attractor

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");

MIT LICENSE

Copyright (C) 2014 Paulo Marques (pjp.marques@gmail.com)

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.