Exploring the Lorenz System of Differential Equations

In this Notebook we explore the Lorenz system of differential equations:

$$ \begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned} $$

This is one of the classic systems in non-linear differential equations. It exhibits a range of different behaviors as the parameters ($\sigma$, $\beta$, $\rho$) are varied.

Imports

First, we import the needed things from IPython, NumPy, Matplotlib and SciPy.



In [1]:

    
%matplotlib inline



In [2]:

    
from ipywidgets import interact, interactive
from IPython.display import clear_output, display, HTML



In [3]:

    
import numpy as np
from scipy import integrate

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.colors import cnames
from matplotlib import animation

Computing the trajectories and plotting the result

We define a function that can integrate the differential equations numerically and then plot the solutions. This function has arguments that control the parameters of the differential equation ($\sigma$, $\beta$, $\rho$), the numerical integration (N, max_time) and the visualization (angle).



In [4]:

    
def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):

    fig = plt.figure()
    ax = fig.add_axes([0, 0, 1, 1], projection='3d')
    ax.axis('off')

    # prepare the axes limits
    ax.set_xlim((-25, 25))
    ax.set_ylim((-35, 35))
    ax.set_zlim((5, 55))
    
    def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):
        """Compute the time-derivative of a Lorenz system."""
        x, y, z = x_y_z
        return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]

    # Choose random starting points, uniformly distributed from -15 to 15
    np.random.seed(1)
    x0 = -15 + 30 * np.random.random((N, 3))

    # Solve for the trajectories
    t = np.linspace(0, max_time, int(250*max_time))
    x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)
                      for x0i in x0])
    
    # choose a different color for each trajectory
    colors = plt.cm.viridis(np.linspace(0, 1, N))

    for i in range(N):
        x, y, z = x_t[i,:,:].T
        lines = ax.plot(x, y, z, '-', c=colors[i])
        plt.setp(lines, linewidth=2)

    ax.view_init(30, angle)
    plt.show()

    return t, x_t

Let's call the function once to view the solutions. For this set of parameters, we see the trajectories swirling around two points, called attractors.



In [7]:

    
import seaborn









    



---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-7-085c0287ecb5> in <module>()
----> 1 import seaborn

ImportError: No module named 'seaborn'



In [5]:

    
t, x_t = solve_lorenz(angle=0, N=10)

Using IPython's interactive function, we can explore how the trajectories behave as we change the various parameters.



In [6]:

    
w = interactive(solve_lorenz, angle=(0.,360.), max_time=(0.1, 4.0), 
                N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))
display(w)









    












    





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         3.66366366,  3.66766767,  3.67167167,  3.67567568,  3.67967968,
         3.68368368,  3.68768769,  3.69169169,  3.6956957 ,  3.6996997 ,
         3.7037037 ,  3.70770771,  3.71171171,  3.71571572,  3.71971972,
         3.72372372,  3.72772773,  3.73173173,  3.73573574,  3.73973974,
         3.74374374,  3.74774775,  3.75175175,  3.75575576,  3.75975976,
         3.76376376,  3.76776777,  3.77177177,  3.77577578,  3.77977978,
         3.78378378,  3.78778779,  3.79179179,  3.7957958 ,  3.7997998 ,
         3.8038038 ,  3.80780781,  3.81181181,  3.81581582,  3.81981982,
         3.82382382,  3.82782783,  3.83183183,  3.83583584,  3.83983984,
         3.84384384,  3.84784785,  3.85185185,  3.85585586,  3.85985986,
         3.86386386,  3.86786787,  3.87187187,  3.87587588,  3.87987988,
         3.88388388,  3.88788789,  3.89189189,  3.8958959 ,  3.8998999 ,
         3.9039039 ,  3.90790791,  3.91191191,  3.91591592,  3.91991992,
         3.92392392,  3.92792793,  3.93193193,  3.93593594,  3.93993994,
         3.94394394,  3.94794795,  3.95195195,  3.95595596,  3.95995996,
         3.96396396,  3.96796797,  3.97197197,  3.97597598,  3.97997998,
         3.98398398,  3.98798799,  3.99199199,  3.995996  ,  4.        ]),
 array([[[ -2.48933986,   6.6097348 , -14.99656876],
         [ -1.27447027,   6.27778097, -14.88707257],
         [ -0.25722317,   6.12979892, -14.74948804],
         ..., 
         [  5.32036927,   6.22605165,  17.23097523],
         [  5.45539837,   6.39261398,  17.18564024],
         [  5.59508392,   6.56430812,  17.14795467]],
 
        [[ -5.93002282, -10.59732328, -12.22984216],
         [ -6.62900726, -11.51134832, -11.82513862],
         [ -7.36381423, -12.51878894, -11.36636957],
         ..., 
         [ -2.85220055,  -3.17884274,  19.26071859],
         [ -2.90136148,  -3.24571225,  19.09550647],
         [ -2.95314777,  -3.31562455,  18.93347025]],
 
        [[ -9.41219366,  -4.63317819,  -3.09697577],
         [ -8.83423854,  -5.67287479,  -2.87758692],
         [ -8.47187066,  -6.64619187,  -2.63545814],
         ..., 
         [  9.9353102 ,   9.63098629,  27.04185483],
         [  9.88843899,   9.55329252,  27.13693426],
         [  9.83714455,   9.4724735 ,  27.22602599]],
 
        ..., 
        [[ 13.94520141,   4.90324493,   3.65087161],
         [ 12.79576674,   6.07398768,   3.90437431],
         [ 11.95486056,   7.13841575,   4.1885436 ],
         ..., 
         [ -9.68022335, -10.27613923,  23.34360663],
         [ -9.76591775, -10.33918286,  23.4979851 ],
         [ -9.84814472, -10.39674056,  23.65654846]],
 
        [[-11.55762081,  13.48467776,  -1.502636  ],
         [ -8.22824618,  12.33592752,  -1.99268875],
         [ -5.48564653,  11.51766268,  -2.29537794],
         ..., 
         [-10.16394423, -10.12097423,  26.17169753],
         [-10.15492846, -10.07488319,  26.3058404 ],
         [-10.14049547, -10.02362007,  26.43613619]],
 
        [[  2.35168843,  -2.75589592,  -7.88919059],
         [  1.67624957,  -2.47319098,  -7.82728918],
         [  1.12533125,  -2.27483902,  -7.75834743],
         ..., 
         [  8.39101357,   9.03991244,  22.18472012],
         [  8.48609466,   9.1346013 ,  22.25747636],
         [  8.58102139,   9.22780333,  22.33610257]]]))

The object returned by interactive is a Widget object and it has attributes that contain the current result and arguments:



In [7]:

    
t, x_t = w.result



In [8]:

    
w.kwargs









    Out[8]:





{'N': 10,
 'angle': 0.0,
 'beta': 2.6666666666666665,
 'max_time': 4.0,
 'rho': 28.0,
 'sigma': 10.0}

After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in $x$, $y$ and $z$.



In [9]:

    
xyz_avg = x_t.mean(axis=1)



In [10]:

    
xyz_avg.shape









    Out[10]:





(10, 3)

Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors.



In [11]:

    
plt.hist(xyz_avg[:,0])
plt.title('Average $x(t)$');



In [12]:

    
plt.hist(xyz_avg[:,1])
plt.title('Average $y(t)$');



In [ ]: