Really Interactive Blog Posts

This notebook first appeared as a blog post on Tim Head's blog.

License: MIT

(C) 2016, Tim Head. Feel free to use, distribute, and modify with the above attribution.

Update (13 Jan 2016): Added links to the DIY section.

A few days ago I started making my blog posts interactive. It was cool, but required you to surf to a different page for the interactive experience, while the original post was still non-interactive.

Alex pointed out that really you wanted it all in one page. Basically he was:

My blog setup follows Jake Vanderplas' pretty closely. So I created a new liquid_tags plugin that has its own template for nbconvert which generates HTML that thebe understands.

No downloading, no installing, no browsing to a separate page! Just interactive blog posts! (Scroll down to see it in action if you do not care how it was done.)

Do it yourself

If you have a pelican site take a look at my fork of the liquid_tags plugin. In addition I made a small gist that shows how to convert notebook to interactive HTML with plain nbconvert. The most important part is using the following template with nbconvert:

{%- extends 'basic.tpl' -%}

{% block codecell %}
<pre data-executable>
{{ cell.source }}
</pre>
{% endblock codecell %}

{% block markdowncell scoped %}
<div class="cellOOO border-box-sizing text_cell rendered">
{{ self.empty_in_prompt() }}
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
{{ cell.source  | markdown2html | strip_files_prefix }}
</div>
</div>
</div>
{%- endblock markdowncell %}

It embeds code cells in simple <pre> tags and modifies non-code cells so that they do not match the selectors used inside the notebook machinery. That is it. Then stick a bit of CSS and JS in the <head> of your web page and you are good to go (use the source of this page for inspiration).

Credits

Compared to my previous post this setup now only relies on thebe, tmpnb, and the kind people at rackspace who sponsor the computing power for tmpnb.

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

Experiment with using %matplotlib notebook for interactive matplotlib figures. Thanks to Thomas Caswell for that tip! If you use this you will have to modify the interact() call below a bit, but I'll leave that as an exercise for the reader.

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,
         0.02002002,  0.02402402,  0.02802803,  0.03203203,  0.03603604,
         0.04004004,  0.04404404,  0.04804805,  0.05205205,  0.05605606,
         0.06006006,  0.06406406,  0.06806807,  0.07207207,  0.07607608,
         0.08008008,  0.08408408,  0.08808809,  0.09209209,  0.0960961 ,
         0.1001001 ,  0.1041041 ,  0.10810811,  0.11211211,  0.11611612,
         0.12012012,  0.12412412,  0.12812813,  0.13213213,  0.13613614,
         0.14014014,  0.14414414,  0.14814815,  0.15215215,  0.15615616,
         0.16016016,  0.16416416,  0.16816817,  0.17217217,  0.17617618,
         0.18018018,  0.18418418,  0.18818819,  0.19219219,  0.1961962 ,
         0.2002002 ,  0.2042042 ,  0.20820821,  0.21221221,  0.21621622,
         0.22022022,  0.22422422,  0.22822823,  0.23223223,  0.23623624,
         0.24024024,  0.24424424,  0.24824825,  0.25225225,  0.25625626,
         0.26026026,  0.26426426,  0.26826827,  0.27227227,  0.27627628,
         0.28028028,  0.28428428,  0.28828829,  0.29229229,  0.2962963 ,
         0.3003003 ,  0.3043043 ,  0.30830831,  0.31231231,  0.31631632,
         0.32032032,  0.32432432,  0.32832833,  0.33233233,  0.33633634,
         0.34034034,  0.34434434,  0.34834835,  0.35235235,  0.35635636,
         0.36036036,  0.36436436,  0.36836837,  0.37237237,  0.37637638,
         0.38038038,  0.38438438,  0.38838839,  0.39239239,  0.3963964 ,
         0.4004004 ,  0.4044044 ,  0.40840841,  0.41241241,  0.41641642,
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         0.48048048,  0.48448448,  0.48848849,  0.49249249,  0.4964965 ,
         0.5005005 ,  0.5045045 ,  0.50850851,  0.51251251,  0.51651652,
         0.52052052,  0.52452452,  0.52852853,  0.53253253,  0.53653654,
         0.54054054,  0.54454454,  0.54854855,  0.55255255,  0.55655656,
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         0.76076076,  0.76476476,  0.76876877,  0.77277277,  0.77677678,
         0.78078078,  0.78478478,  0.78878879,  0.79279279,  0.7967968 ,
         0.8008008 ,  0.8048048 ,  0.80880881,  0.81281281,  0.81681682,
         0.82082082,  0.82482482,  0.82882883,  0.83283283,  0.83683684,
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         0.88088088,  0.88488488,  0.88888889,  0.89289289,  0.8968969 ,
         0.9009009 ,  0.9049049 ,  0.90890891,  0.91291291,  0.91691692,
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         0.98098098,  0.98498498,  0.98898899,  0.99299299,  0.996997  ,
         1.001001  ,  1.00500501,  1.00900901,  1.01301301,  1.01701702,
         1.02102102,  1.02502503,  1.02902903,  1.03303303,  1.03703704,
         1.04104104,  1.04504505,  1.04904905,  1.05305305,  1.05705706,
         1.06106106,  1.06506507,  1.06906907,  1.07307307,  1.07707708,
         1.08108108,  1.08508509,  1.08908909,  1.09309309,  1.0970971 ,
         1.1011011 ,  1.10510511,  1.10910911,  1.11311311,  1.11711712,
         1.12112112,  1.12512513,  1.12912913,  1.13313313,  1.13713714,
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         1.18118118,  1.18518519,  1.18918919,  1.19319319,  1.1971972 ,
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         1.32132132,  1.32532533,  1.32932933,  1.33333333,  1.33733734,
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         1.38138138,  1.38538539,  1.38938939,  1.39339339,  1.3973974 ,
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         1.42142142,  1.42542543,  1.42942943,  1.43343343,  1.43743744,
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         1.46146146,  1.46546547,  1.46946947,  1.47347347,  1.47747748,
         1.48148148,  1.48548549,  1.48948949,  1.49349349,  1.4974975 ,
         1.5015015 ,  1.50550551,  1.50950951,  1.51351351,  1.51751752,
         1.52152152,  1.52552553,  1.52952953,  1.53353353,  1.53753754,
         1.54154154,  1.54554555,  1.54954955,  1.55355355,  1.55755756,
         1.56156156,  1.56556557,  1.56956957,  1.57357357,  1.57757758,
         1.58158158,  1.58558559,  1.58958959,  1.59359359,  1.5975976 ,
         1.6016016 ,  1.60560561,  1.60960961,  1.61361361,  1.61761762,
         1.62162162,  1.62562563,  1.62962963,  1.63363363,  1.63763764,
         1.64164164,  1.64564565,  1.64964965,  1.65365365,  1.65765766,
         1.66166166,  1.66566567,  1.66966967,  1.67367367,  1.67767768,
         1.68168168,  1.68568569,  1.68968969,  1.69369369,  1.6976977 ,
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         1.72172172,  1.72572573,  1.72972973,  1.73373373,  1.73773774,
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         2.04204204,  2.04604605,  2.05005005,  2.05405405,  2.05805806,
         2.06206206,  2.06606607,  2.07007007,  2.07407407,  2.07807808,
         2.08208208,  2.08608609,  2.09009009,  2.09409409,  2.0980981 ,
         2.1021021 ,  2.10610611,  2.11011011,  2.11411411,  2.11811812,
         2.12212212,  2.12612613,  2.13013013,  2.13413413,  2.13813814,
         2.14214214,  2.14614615,  2.15015015,  2.15415415,  2.15815816,
         2.16216216,  2.16616617,  2.17017017,  2.17417417,  2.17817818,
         2.18218218,  2.18618619,  2.19019019,  2.19419419,  2.1981982 ,
         2.2022022 ,  2.20620621,  2.21021021,  2.21421421,  2.21821822,
         2.22222222,  2.22622623,  2.23023023,  2.23423423,  2.23823824,
         2.24224224,  2.24624625,  2.25025025,  2.25425425,  2.25825826,
         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,
         2.32232232,  2.32632633,  2.33033033,  2.33433433,  2.33833834,
         2.34234234,  2.34634635,  2.35035035,  2.35435435,  2.35835836,
         2.36236236,  2.36636637,  2.37037037,  2.37437437,  2.37837838,
         2.38238238,  2.38638639,  2.39039039,  2.39439439,  2.3983984 ,
         2.4024024 ,  2.40640641,  2.41041041,  2.41441441,  2.41841842,
         2.42242242,  2.42642643,  2.43043043,  2.43443443,  2.43843844,
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         2.48248248,  2.48648649,  2.49049049,  2.49449449,  2.4984985 ,
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         2.56256256,  2.56656657,  2.57057057,  2.57457457,  2.57857858,
         2.58258258,  2.58658659,  2.59059059,  2.59459459,  2.5985986 ,
         2.6026026 ,  2.60660661,  2.61061061,  2.61461461,  2.61861862,
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         2.74274274,  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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         2.88288288,  2.88688689,  2.89089089,  2.89489489,  2.8988989 ,
         2.9029029 ,  2.90690691,  2.91091091,  2.91491491,  2.91891892,
         2.92292292,  2.92692693,  2.93093093,  2.93493493,  2.93893894,
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         2.98298298,  2.98698699,  2.99099099,  2.99499499,  2.998999  ,
         3.003003  ,  3.00700701,  3.01101101,  3.01501502,  3.01901902,
         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,
         3.14314314,  3.14714715,  3.15115115,  3.15515516,  3.15915916,
         3.16316316,  3.16716717,  3.17117117,  3.17517518,  3.17917918,
         3.18318318,  3.18718719,  3.19119119,  3.1951952 ,  3.1991992 ,
         3.2032032 ,  3.20720721,  3.21121121,  3.21521522,  3.21921922,
         3.22322322,  3.22722723,  3.23123123,  3.23523524,  3.23923924,
         3.24324324,  3.24724725,  3.25125125,  3.25525526,  3.25925926,
         3.26326326,  3.26726727,  3.27127127,  3.27527528,  3.27927928,
         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>