Interactive Blog Posts

This post is out of date. Interactivity on a different page was cool, but Alex was "meh, same page!", so now there are really interactive posts! Interactivity on the same page as the post.

From today on all my blog posts based on jupyter notebooks will be interactive!!

The combined magical powers of tmpnb, gistexec and thebe mean you can run and modify my blog posts as you read them. No downloading, no installing, just one click and you can change parameters or plot things in your favourite colour scheme.

Try it out

Try out the interactive version of this post. I will wait. As the beginning of the post is not particularly interesting to run, below the well used jupyter notebook on exploring the Lorenz system of differential equations.

This is a very dull post if you can't change it, but as it is interactive you can slide the ipywidgets based sliders to change things!

I am preparing a write up of how to make your own pelican site have an interactive section.

Right now you can peak at the source of notebookexec for the dirty details.

Credit where credit is due

Really all the hard work had already been done by Kyle Kelley who wrote large parts of both tmpnb and gistexec.

Below, the work of the jupyter development team, licensed under the 3 clause BSD license.

Get in touch on twitter @betatim.

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.jet(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 [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.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))
display(w)

(array([ 0.        ,  0.004004  ,  0.00800801,  0.01201201,  0.01601602,
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         0.08008008,  0.08408408,  0.08808809,  0.09209209,  0.0960961 ,
         0.1001001 ,  0.1041041 ,  0.10810811,  0.11211211,  0.11611612,
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         2.18218218,  2.18618619,  2.19019019,  2.19419419,  2.1981982 ,
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         2.26226226,  2.26626627,  2.27027027,  2.27427427,  2.27827828,
         2.28228228,  2.28628629,  2.29029029,  2.29429429,  2.2982983 ,
         2.3023023 ,  2.30630631,  2.31031031,  2.31431431,  2.31831832,
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         2.58258258,  2.58658659,  2.59059059,  2.59459459,  2.5985986 ,
         2.6026026 ,  2.60660661,  2.61061061,  2.61461461,  2.61861862,
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         2.68268268,  2.68668669,  2.69069069,  2.69469469,  2.6986987 ,
         2.7027027 ,  2.70670671,  2.71071071,  2.71471471,  2.71871872,
         2.72272272,  2.72672673,  2.73073073,  2.73473473,  2.73873874,
         2.74274274,  2.74674675,  2.75075075,  2.75475475,  2.75875876,
         2.76276276,  2.76676677,  2.77077077,  2.77477477,  2.77877878,
         2.78278278,  2.78678679,  2.79079079,  2.79479479,  2.7987988 ,
         2.8028028 ,  2.80680681,  2.81081081,  2.81481481,  2.81881882,
         2.82282282,  2.82682683,  2.83083083,  2.83483483,  2.83883884,
         2.84284284,  2.84684685,  2.85085085,  2.85485485,  2.85885886,
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         2.9029029 ,  2.90690691,  2.91091091,  2.91491491,  2.91891892,
         2.92292292,  2.92692693,  2.93093093,  2.93493493,  2.93893894,
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         3.02302302,  3.02702703,  3.03103103,  3.03503504,  3.03903904,
         3.04304304,  3.04704705,  3.05105105,  3.05505506,  3.05905906,
         3.06306306,  3.06706707,  3.07107107,  3.07507508,  3.07907908,
         3.08308308,  3.08708709,  3.09109109,  3.0950951 ,  3.0990991 ,
         3.1031031 ,  3.10710711,  3.11111111,  3.11511512,  3.11911912,
         3.12312312,  3.12712713,  3.13113113,  3.13513514,  3.13913914,
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         3.16316316,  3.16716717,  3.17117117,  3.17517518,  3.17917918,
         3.18318318,  3.18718719,  3.19119119,  3.1951952 ,  3.1991992 ,
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         3.22322322,  3.22722723,  3.23123123,  3.23523524,  3.23923924,
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         3.28328328,  3.28728729,  3.29129129,  3.2952953 ,  3.2992993 ,
         3.3033033 ,  3.30730731,  3.31131131,  3.31531532,  3.31931932,
         3.32332332,  3.32732733,  3.33133133,  3.33533534,  3.33933934,
         3.34334334,  3.34734735,  3.35135135,  3.35535536,  3.35935936,
         3.36336336,  3.36736737,  3.37137137,  3.37537538,  3.37937938,
         3.38338338,  3.38738739,  3.39139139,  3.3953954 ,  3.3993994 ,
         3.4034034 ,  3.40740741,  3.41141141,  3.41541542,  3.41941942,
         3.42342342,  3.42742743,  3.43143143,  3.43543544,  3.43943944,
         3.44344344,  3.44744745,  3.45145145,  3.45545546,  3.45945946,
         3.46346346,  3.46746747,  3.47147147,  3.47547548,  3.47947948,
         3.48348348,  3.48748749,  3.49149149,  3.4954955 ,  3.4994995 ,
         3.5035035 ,  3.50750751,  3.51151151,  3.51551552,  3.51951952,
         3.52352352,  3.52752753,  3.53153153,  3.53553554,  3.53953954,
         3.54354354,  3.54754755,  3.55155155,  3.55555556,  3.55955956,
         3.56356356,  3.56756757,  3.57157157,  3.57557558,  3.57957958,
         3.58358358,  3.58758759,  3.59159159,  3.5955956 ,  3.5995996 ,
         3.6036036 ,  3.60760761,  3.61161161,  3.61561562,  3.61961962,
         3.62362362,  3.62762763,  3.63163163,  3.63563564,  3.63963964,
         3.64364364,  3.64764765,  3.65165165,  3.65565566,  3.65965966,
         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([[[ -2.48933986e+00,   6.60973480e+00,  -1.49965688e+01],
         [ -2.14077645e+00,   6.18646806e+00,  -1.48962127e+01],
         [ -1.82130748e+00,   5.82299967e+00,  -1.47853208e+01],
         ..., 
         [  6.87667416e+00,   1.07734499e+01,   1.78286316e+01],
         [  7.03431517e+00,   1.10115546e+01,   1.79410297e+01],
         [  7.19515729e+00,   1.12517722e+01,   1.80659207e+01]],
 
        [[ -5.93002282e+00,  -1.05973233e+01,  -1.22298422e+01],
         [ -6.13136261e+00,  -1.15193465e+01,  -1.18344052e+01],
         [ -6.36138540e+00,  -1.24619735e+01,  -1.14104771e+01],
         ..., 
         [ -1.11085203e+01,  -1.62261151e+01,   2.32341935e+01],
         [ -1.13121656e+01,  -1.63640151e+01,   2.37150767e+01],
         [ -1.15128567e+01,  -1.64827701e+01,   2.42098140e+01]],
 
        [[ -9.41219366e+00,  -4.63317819e+00,  -3.09697577e+00],
         [ -9.24717733e+00,  -5.76936651e+00,  -2.87083263e+00],
         [ -9.13260659e+00,  -6.87477643e+00,  -2.60897637e+00],
         ..., 
         [  8.94933599e+00,   1.00204877e+01,   2.61339939e+01],
         [  8.99188828e+00,   1.00458891e+01,   2.62149083e+01],
         [  9.03371359e+00,   1.00685518e+01,   2.62975123e+01]],
 
        ..., 
        [[  1.40478473e+01,  -5.59727466e+00,   5.76967847e+00],
         [  1.33015844e+01,  -4.35137818e+00,   5.43762841e+00],
         [  1.26324566e+01,  -3.15868945e+00,   5.18607122e+00],
         ..., 
         [ -5.62183444e+00,  -9.08227752e+00,   1.57576353e+01],
         [ -5.76241216e+00,  -9.32397964e+00,   1.57989149e+01],
         [ -5.90705639e+00,  -9.57059699e+00,   1.58506576e+01]],
 
        [[  1.12916746e+01,   1.18381999e+01,  -1.24486737e+01],
         [  1.13481065e+01,   1.36039497e+01,  -1.17430178e+01],
         [  1.14710812e+01,   1.53431579e+01,  -1.09606825e+01],
         ..., 
         [ -1.47677316e-02,  -1.23193351e-02,   1.01977111e+01],
         [ -1.46914304e-02,  -1.33208729e-02,   1.00894066e+01],
         [ -1.46573919e-02,  -1.43208005e-02,   9.98225246e+00]],
 
        [[ -1.38283565e+01,  -9.90508741e+00,   1.13442751e+01],
         [ -1.36915606e+01,  -1.07690154e+01,   1.17903119e+01],
         [ -1.35933180e+01,  -1.15964765e+01,   1.22727276e+01],
         ..., 
         [  1.61306741e+01,   1.93584734e+01,   3.43088188e+01],
         [  1.62473985e+01,   1.88446346e+01,   3.51764615e+01],
         [  1.63382575e+01,   1.82747191e+01,   3.60076113e+01]]]))

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 $x(t)$')
Out[11]:
<matplotlib.text.Text at 0x109d07160>
In [12]:
plt.hist(xyz_avg[:,1])
plt.title('Average $y(t)$')
Out[12]:
<matplotlib.text.Text at 0x109df9048>
In [ ]:

This post started life as a jupyter notebook, download it or view it online.

Copyright © 2014-2021 - Tim Head