The butterfly effect

The three differential equations proposed by Edward Lorenz in his 1963 paper can be thought of (roughly)in terms of weather on a tropical island. We can think of the variables as strength of breeze (:math:´X´), which increases with :math:´Y´ but decreases as a function of itself

\[\frac{dX}{dt} = s(Y-X)\]

Temperature difference between currents (:math:´Y´)) which increases with :math:´X´ but decreases as a function of itself and of :math:´Z´,

\[\frac{dY}{dt} = rX - Y - XZ\]

And land/air temperature distortion (:math:´Z´) which increases with :math:´X´ but decreases as a function of itself,

\[\frac{dZ}{dt} = XY - bZ\]

There are three parameter values, which are set to :math:´s=10´, :math:´r=28´ and :math:´b=8/3´ to create the butterfly (th Lorenz attractor).

Simulating chaos

Let’s first define a function which gives the derivatives at each point

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
from pylab import rcParams
matplotlib.font_manager.FontProperties(family='Helvetica',size=11)

from scipy import integrate
from mpl_toolkits import mplot3d

def lorenz(XYZ,t=0):


    dXdt = s*(XYZ[1] - XYZ[0])                  # X
    dYdt = r*XYZ[0] - XYZ[1] - XYZ[0]*XYZ[2]    # Y
    dZdt = XYZ[0]*XYZ[1] - b*XYZ[2]             # Z

    return np.array([dXdt, dYdt, dZdt])


# Parameter values
s=10
r=28
b=8/3

Now we solve the equations numerically. By plotting them in three dimensions we can see how the weather never repeats.

endtime=300
dt = 0.01

t = np.linspace(0,endtime, int(endtime/dt))               # time
XYZ0 = np.array([0.3, 1.2, 5.05])          # initial conditions
XYZ = integrate.odeint(lorenz, XYZ0, t)

start=int(200/dt)
finish=int(start+40/dt)

rcParams['figure.figsize'] = 14/2.54, 14/2.54
ax=plt.subplot(projection='3d')
ax.set_xticks(np.arange(-20,21,step=10))
ax.set_yticks(np.arange(-20,21,step=10))
ax.set_zticks(np.arange(0,40,step=10))
ax.yaxis.labelpad=10
ax.plot3D(XYZ[start:finish,0], XYZ[start:finish,1], XYZ[start:finish,2], lw=1,color='k')
ax.set_xlabel("Strength of breeze ($X$)")
ax.set_ylabel("Temperature difference \n between currents ($Y$)")
ax.set_zlabel("Land/air temperature \n distortion ($Z$) ")

# Format the axes
ax.xaxis.pane.fill = False
ax.yaxis.pane.fill = False
ax.zaxis.pane.fill = False
ax.xaxis.pane.set_edgecolor('w')
ax.yaxis.pane.set_edgecolor('w')
ax.zaxis.pane.set_edgecolor('w')
ax.set_xlim(-21,21)
ax.set_zlim(0,40)
ax.set_ylim(-21,21)


plt.show()

This is the butterfly of chaos in all its glory.

How variables change over time

In order to better understand chaos, Fetter and Lorenz measured the height of consecutive peaks in the temperature distortion Z. Before we do this, let’s start by plotting the three variables over time.

rcParams['figure.figsize'] = 14/2.54, 7/2.54

finish=int(start+20/dt)

fig,ax=plt.subplots(num=1)
ax.plot(XYZ[start:finish,0], color='black')
ax.plot(XYZ[start:finish,1], color='green')
ax.plot(XYZ[start:finish,2], color='red')
plt.show()

To identify the peaks we note each time Z reaches a maximum. This is done using a function argrelextrema from the Scipy package.

from scipy.signal import argrelextrema

# for local maxima
zm= XYZ[:,2]
maxz=argrelextrema(zm, np.greater)
zm=zm[maxz]

zm=zm[20:]
fig,ax=plt.subplots(num=1)
ax.plot(np.arange(len(zm)),zm, color='black')
ax.set_ylabel('Maximum value of $Z$')
ax.set_xlabel('Iteration round the butterfly')
ax.set_ylim((27,52))
ax.set_xlim((0,50))
ax.set_xticks(range(0,60,10))
ax.set_yticks(range(30,60,10))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)

plt.show()

To be clear, this is not a plot of Z itself, but rather a plot of the maximum values it takes on each loop round the butterfly shape.

Think yourself!

Does anything strike you as familiar in this plot? Compare it to the time series for the doubling rule. The rise and fall of the maxima is not enirely disimilar to the map we saw there. Run the code above for different intitial conditions and look at how the sequence of maxima changes.

The chaos in Lorenz equation can be teased out further by plotting consecutive maxima of Z to create a map of maxima from one time to the next. So, each time Z reaches a maximum the size of that value is noted and then consecutive values of Z are plotted against each other.

rcParams['figure.figsize'] = 10/2.54, 10/2.54
fig,ax=plt.subplots(num=1)
ax.plot(zm[:-1],zm[1:],linestyle='none',marker='.',color='k')
ax.set_ylabel('Next Maximum of $Z$')
ax.set_xlabel('Previous Maximum of $Z$')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.set_xticks(np.arange(20,51,step=10))
ax.set_yticks(np.arange(20,51,step=10))
ax.set_xlim(27,52)
ax.set_ylim(27,52)

(27.0, 52.0)

The same tent-like shape is seen here as we saw earlier. For Lorenz article, Ellen Fetter made similar plots for the height of consecutive peaks. This tells us that in the model, when temperature distortions are small, the temperature distortion typically doubled, while large distortions are followed by very small distortions.

Lorenz noted the similarity to the tent map and started to sketch out an argument as to why this meant that any two close-by points will soon move apart. In doing so, he provided the first argument as to why the weather is chaotic. It wasn’t a rigorous proof at that stage, but it was the starting point of an explanation.

Learn more

The original paper by Lorenz:

Edward N. Lorenz, Deterministic nonperiodic flow, Journal of Atmospheric Sciences 20, no. 2 (1963): 130‒41

A more mathematical description of the Lorenz equations and their relationship to the doubling map (which we looked at here):

Étienne Ghys, The Lorenz attractor, a paradigm for chaos, Chaos (2013): 1‒54, p. 20

A beautiful and thorough analysis of Lorenz equations:

Colin Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Vol. 41, Springer Science and Business Media, 2012