""" .. _chaosbutterfly: 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 .. math:: \\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´, .. math:: \\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, .. math:: \\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. # # .. admonition:: Think yourself! # # Does anything strike you as familiar in this plot? # Compare it to the time series for the :ref:`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) ############################################################################## # The same tent-like shape is seen here as we saw :ref:`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 :ref:`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 `_ #