##// END OF EJS Templates
Fix typo Lorentz -> Lorenz
Thomas Kluyver -
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@@ -1,13440 +1,13452 b''
1 {
1 {
2 "cells": [
2 "cells": [
3 {
3 {
4 "cell_type": "markdown",
4 "cell_type": "markdown",
5 "metadata": {},
5 "metadata": {},
6 "source": [
6 "source": [
7 "# Exploring the Lorenz System of Differential Equations"
7 "# Exploring the Lorenz System of Differential Equations"
8 ]
8 ]
9 },
9 },
10 {
10 {
11 "cell_type": "markdown",
11 "cell_type": "markdown",
12 "metadata": {},
12 "metadata": {},
13 "source": [
13 "source": [
14 "In this Notebook we explore the Lorenz system of differential equations:\n",
14 "In this Notebook we explore the Lorenz system of differential equations:\n",
15 "\n",
15 "\n",
16 "$$\n",
16 "$$\n",
17 "\\begin{aligned}\n",
17 "\\begin{aligned}\n",
18 "\\dot{x} & = \\sigma(y-x) \\\\\n",
18 "\\dot{x} & = \\sigma(y-x) \\\\\n",
19 "\\dot{y} & = \\rho x - y - xz \\\\\n",
19 "\\dot{y} & = \\rho x - y - xz \\\\\n",
20 "\\dot{z} & = -\\beta z + xy\n",
20 "\\dot{z} & = -\\beta z + xy\n",
21 "\\end{aligned}\n",
21 "\\end{aligned}\n",
22 "$$\n",
22 "$$\n",
23 "\n",
23 "\n",
24 "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."
24 "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."
25 ]
25 ]
26 },
26 },
27 {
27 {
28 "cell_type": "markdown",
28 "cell_type": "markdown",
29 "metadata": {},
29 "metadata": {},
30 "source": [
30 "source": [
31 "## Imports"
31 "## Imports"
32 ]
32 ]
33 },
33 },
34 {
34 {
35 "cell_type": "markdown",
35 "cell_type": "markdown",
36 "metadata": {},
36 "metadata": {},
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37 "source": [
38 "First, we import the needed things from IPython, NumPy, Matplotlib and SciPy."
38 "First, we import the needed things from IPython, NumPy, Matplotlib and SciPy."
39 ]
39 ]
40 },
40 },
41 {
41 {
42 "cell_type": "code",
42 "cell_type": "code",
43 "execution_count": 1,
43 "execution_count": 1,
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44 "metadata": {
45 "collapsed": false
45 "collapsed": false
46 },
46 },
47 "outputs": [],
47 "outputs": [],
48 "source": [
48 "source": [
49 "%matplotlib inline"
49 "%matplotlib inline"
50 ]
50 ]
51 },
51 },
52 {
52 {
53 "cell_type": "code",
53 "cell_type": "code",
54 "execution_count": 2,
54 "execution_count": 2,
55 "metadata": {
55 "metadata": {
56 "collapsed": false
56 "collapsed": false
57 },
57 },
58 "outputs": [],
58 "outputs": [],
59 "source": [
59 "source": [
60 "from IPython.html.widgets import interact, interactive\n",
60 "from IPython.html.widgets import interact, interactive\n",
61 "from IPython.display import clear_output, display, HTML"
61 "from IPython.display import clear_output, display, HTML"
62 ]
62 ]
63 },
63 },
64 {
64 {
65 "cell_type": "code",
65 "cell_type": "code",
66 "execution_count": 3,
66 "execution_count": 3,
67 "metadata": {
67 "metadata": {
68 "collapsed": false
68 "collapsed": false
69 },
69 },
70 "outputs": [],
70 "outputs": [],
71 "source": [
71 "source": [
72 "import numpy as np\n",
72 "import numpy as np\n",
73 "from scipy import integrate\n",
73 "from scipy import integrate\n",
74 "\n",
74 "\n",
75 "from matplotlib import pyplot as plt\n",
75 "from matplotlib import pyplot as plt\n",
76 "from mpl_toolkits.mplot3d import Axes3D\n",
76 "from mpl_toolkits.mplot3d import Axes3D\n",
77 "from matplotlib.colors import cnames\n",
77 "from matplotlib.colors import cnames\n",
78 "from matplotlib import animation"
78 "from matplotlib import animation"
79 ]
79 ]
80 },
80 },
81 {
81 {
82 "cell_type": "markdown",
82 "cell_type": "markdown",
83 "metadata": {},
83 "metadata": {},
84 "source": [
84 "source": [
85 "## Computing the trajectories and plotting the result"
85 "## Computing the trajectories and plotting the result"
86 ]
86 ]
87 },
87 },
88 {
88 {
89 "cell_type": "markdown",
89 "cell_type": "markdown",
90 "metadata": {},
90 "metadata": {},
91 "source": [
91 "source": [
92 "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`)."
92 "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`)."
93 ]
93 ]
94 },
94 },
95 {
95 {
96 "cell_type": "code",
96 "cell_type": "code",
97 "execution_count": 4,
97 "execution_count": 4,
98 "metadata": {
98 "metadata": {
99 "collapsed": false
99 "collapsed": false
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101 "outputs": [],
102 "source": [
102 "source": [
103 "def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):\n",
103 "def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):\n",
104 "\n",
104 "\n",
105 " fig = plt.figure()\n",
105 " fig = plt.figure()\n",
106 " ax = fig.add_axes([0, 0, 1, 1], projection='3d')\n",
106 " ax = fig.add_axes([0, 0, 1, 1], projection='3d')\n",
107 " ax.axis('off')\n",
107 " ax.axis('off')\n",
108 "\n",
108 "\n",
109 " # prepare the axes limits\n",
109 " # prepare the axes limits\n",
110 " ax.set_xlim((-25, 25))\n",
110 " ax.set_xlim((-25, 25))\n",
111 " ax.set_ylim((-35, 35))\n",
111 " ax.set_ylim((-35, 35))\n",
112 " ax.set_zlim((5, 55))\n",
112 " ax.set_zlim((5, 55))\n",
113 " \n",
113 " \n",
114 " def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):\n",
114 " def lorenz_deriv(x_y_z, t0, sigma=sigma, beta=beta, rho=rho):\n",
115 " \"\"\"Compute the time-derivative of a Lorentz system.\"\"\"\n",
115 " \"\"\"Compute the time-derivative of a Lorenz system.\"\"\"\n",
116 " x, y, z = x_y_z\n",
116 " x, y, z = x_y_z\n",
117 " return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]\n",
117 " return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]\n",
118 "\n",
118 "\n",
119 " # Choose random starting points, uniformly distributed from -15 to 15\n",
119 " # Choose random starting points, uniformly distributed from -15 to 15\n",
120 " np.random.seed(1)\n",
120 " np.random.seed(1)\n",
121 " x0 = -15 + 30 * np.random.random((N, 3))\n",
121 " x0 = -15 + 30 * np.random.random((N, 3))\n",
122 "\n",
122 "\n",
123 " # Solve for the trajectories\n",
123 " # Solve for the trajectories\n",
124 " t = np.linspace(0, max_time, int(250*max_time))\n",
124 " t = np.linspace(0, max_time, int(250*max_time))\n",
125 " x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)\n",
125 " x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)\n",
126 " for x0i in x0])\n",
126 " for x0i in x0])\n",
127 " \n",
127 " \n",
128 " # choose a different color for each trajectory\n",
128 " # choose a different color for each trajectory\n",
129 " colors = plt.cm.jet(np.linspace(0, 1, N))\n",
129 " colors = plt.cm.jet(np.linspace(0, 1, N))\n",
130 "\n",
130 "\n",
131 " for i in range(N):\n",
131 " for i in range(N):\n",
132 " x, y, z = x_t[i,:,:].T\n",
132 " x, y, z = x_t[i,:,:].T\n",
133 " lines = ax.plot(x, y, z, '-', c=colors[i])\n",
133 " lines = ax.plot(x, y, z, '-', c=colors[i])\n",
134 " plt.setp(lines, linewidth=2)\n",
134 " plt.setp(lines, linewidth=2)\n",
135 "\n",
135 "\n",
136 " ax.view_init(30, angle)\n",
136 " ax.view_init(30, angle)\n",
137 " plt.show()\n",
137 " plt.show()\n",
138 "\n",
138 "\n",
139 " return t, x_t"
139 " return t, x_t"
140 ]
140 ]
141 },
141 },
142 {
142 {
143 "cell_type": "markdown",
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146 "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. "
146 "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. "
147 ]
147 ]
148 },
148 },
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151 "execution_count": 5,
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5384 ],
5384 ],
5385 "text/plain": [
5385 "text/plain": [
5386 "<matplotlib.figure.Figure at 0x10c01fc90>"
5386 "<matplotlib.figure.Figure at 0x10c01fc90>"
5387 ]
5387 ]
5388 },
5388 },
5389 "metadata": {
5389 "metadata": {
5390 "image/png": {
5390 "image/png": {
5391 "height": 459,
5391 "height": 459,
5392 "width": 734
5392 "width": 734
5393 }
5393 }
5394 },
5394 },
5395 "output_type": "display_data"
5395 "output_type": "display_data"
5396 }
5396 }
5397 ],
5397 ],
5398 "source": [
5398 "source": [
5399 "t, x_t = solve_lorenz(angle=0, N=10)"
5399 "t, x_t = solve_lorenz(angle=0, N=10)"
5400 ]
5400 ]
5401 },
5401 },
5402 {
5402 {
5403 "cell_type": "markdown",
5403 "cell_type": "markdown",
5404 "metadata": {},
5404 "metadata": {},
5405 "source": [
5405 "source": [
5406 "Using IPython's `interactive` function, we can explore how the trajectories behave as we change the various parameters."
5406 "Using IPython's `interactive` function, we can explore how the trajectories behave as we change the various parameters."
5407 ]
5407 ]
5408 },
5408 },
5409 {
5409 {
5410 "cell_type": "code",
5410 "cell_type": "code",
5411 "execution_count": 6,
5411 "execution_count": 6,
5412 "metadata": {
5412 "metadata": {
5413 "collapsed": false
5413 "collapsed": false
5414 },
5414 },
5415 "outputs": [
5415 "outputs": [
5416 {
5416 {
5417 "data": {
5417 "data": {
5418 "image/png": [
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12790 "text/plain": [
12790 "text/plain": [
12791 "<matplotlib.figure.Figure at 0x10c08c690>"
12791 "<matplotlib.figure.Figure at 0x10c08c690>"
12792 ]
12792 ]
12793 },
12793 },
12794 "metadata": {
12794 "metadata": {
12795 "image/png": {
12795 "image/png": {
12796 "height": 459,
12796 "height": 459,
12797 "width": 734
12797 "width": 734
12798 }
12798 }
12799 },
12799 },
12800 "output_type": "display_data"
12800 "output_type": "display_data"
12801 }
12801 }
12802 ],
12802 ],
12803 "source": [
12803 "source": [
12804 "w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))\n",
12804 "w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))\n",
12805 "display(w)"
12805 "display(w)"
12806 ]
12806 ]
12807 },
12807 },
12808 {
12808 {
12809 "cell_type": "markdown",
12809 "cell_type": "markdown",
12810 "metadata": {},
12810 "metadata": {},
12811 "source": [
12811 "source": [
12812 "The object returned by `interactive` is a `Widget` object and it has attributes that contain the current result and arguments:"
12812 "The object returned by `interactive` is a `Widget` object and it has attributes that contain the current result and arguments:"
12813 ]
12813 ]
12814 },
12814 },
12815 {
12815 {
12816 "cell_type": "code",
12816 "cell_type": "code",
12817 "execution_count": 7,
12817 "execution_count": 7,
12818 "metadata": {
12818 "metadata": {
12819 "collapsed": false
12819 "collapsed": false
12820 },
12820 },
12821 "outputs": [],
12821 "outputs": [],
12822 "source": [
12822 "source": [
12823 "t, x_t = w.result"
12823 "t, x_t = w.result"
12824 ]
12824 ]
12825 },
12825 },
12826 {
12826 {
12827 "cell_type": "code",
12827 "cell_type": "code",
12828 "execution_count": 8,
12828 "execution_count": 8,
12829 "metadata": {
12829 "metadata": {
12830 "collapsed": false
12830 "collapsed": false
12831 },
12831 },
12832 "outputs": [
12832 "outputs": [
12833 {
12833 {
12834 "data": {
12834 "data": {
12835 "text/plain": [
12835 "text/plain": [
12836 "{u'N': 25,\n",
12836 "{u'N': 25,\n",
12837 " u'angle': 180.0,\n",
12837 " u'angle': 180.0,\n",
12838 " u'beta': 2.6666666666666665,\n",
12838 " u'beta': 2.6666666666666665,\n",
12839 " u'max_time': 4.0,\n",
12839 " u'max_time': 4.0,\n",
12840 " u'rho': 25.0,\n",
12840 " u'rho': 25.0,\n",
12841 " u'sigma': 25.0}"
12841 " u'sigma': 25.0}"
12842 ]
12842 ]
12843 },
12843 },
12844 "execution_count": 8,
12844 "execution_count": 8,
12845 "metadata": {},
12845 "metadata": {},
12846 "output_type": "execute_result"
12846 "output_type": "execute_result"
12847 }
12847 }
12848 ],
12848 ],
12849 "source": [
12849 "source": [
12850 "w.kwargs"
12850 "w.kwargs"
12851 ]
12851 ]
12852 },
12852 },
12853 {
12853 {
12854 "cell_type": "markdown",
12854 "cell_type": "markdown",
12855 "metadata": {},
12855 "metadata": {},
12856 "source": [
12856 "source": [
12857 "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 $z$."
12857 "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 $z$."
12858 ]
12858 ]
12859 },
12859 },
12860 {
12860 {
12861 "cell_type": "code",
12861 "cell_type": "code",
12862 "execution_count": 9,
12862 "execution_count": 9,
12863 "metadata": {
12863 "metadata": {
12864 "collapsed": false
12864 "collapsed": false
12865 },
12865 },
12866 "outputs": [],
12866 "outputs": [],
12867 "source": [
12867 "source": [
12868 "xyz_avg = x_t.mean(axis=1)"
12868 "xyz_avg = x_t.mean(axis=1)"
12869 ]
12869 ]
12870 },
12870 },
12871 {
12871 {
12872 "cell_type": "code",
12872 "cell_type": "code",
12873 "execution_count": 10,
12873 "execution_count": 10,
12874 "metadata": {
12874 "metadata": {
12875 "collapsed": false
12875 "collapsed": false
12876 },
12876 },
12877 "outputs": [
12877 "outputs": [
12878 {
12878 {
12879 "data": {
12879 "data": {
12880 "text/plain": [
12880 "text/plain": [
12881 "(25, 3)"
12881 "(25, 3)"
12882 ]
12882 ]
12883 },
12883 },
12884 "execution_count": 10,
12884 "execution_count": 10,
12885 "metadata": {},
12885 "metadata": {},
12886 "output_type": "execute_result"
12886 "output_type": "execute_result"
12887 }
12887 }
12888 ],
12888 ],
12889 "source": [
12889 "source": [
12890 "xyz_avg.shape"
12890 "xyz_avg.shape"
12891 ]
12891 ]
12892 },
12892 },
12893 {
12893 {
12894 "cell_type": "markdown",
12894 "cell_type": "markdown",
12895 "metadata": {},
12895 "metadata": {},
12896 "source": [
12896 "source": [
12897 "Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors."
12897 "Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors."
12898 ]
12898 ]
12899 },
12899 },
12900 {
12900 {
12901 "cell_type": "code",
12901 "cell_type": "code",
12902 "execution_count": 11,
12902 "execution_count": 11,
12903 "metadata": {
12903 "metadata": {
12904 "collapsed": false
12904 "collapsed": false
12905 },
12905 },
12906 "outputs": [
12906 "outputs": [
12907 {
12907 {
12908 "data": {
12908 "data": {
12909 "text/plain": [
12909 "text/plain": [
12910 "<matplotlib.text.Text at 0x10c0387d0>"
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12913 "execution_count": 11,
12914 "metadata": {},
12914 "metadata": {},
12915 "output_type": "execute_result"
12915 "output_type": "execute_result"
12916 },
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13142 ],
13142 ],
13143 "text/plain": [
13143 "text/plain": [
13144 "<matplotlib.figure.Figure at 0x10c08c610>"
13144 "<matplotlib.figure.Figure at 0x10c08c610>"
13145 ]
13145 ]
13146 },
13146 },
13147 "metadata": {
13147 "metadata": {
13148 "image/png": {
13148 "image/png": {
13149 "height": 391,
13149 "height": 391,
13150 "width": 586
13150 "width": 586
13151 }
13151 }
13152 },
13152 },
13153 "output_type": "display_data"
13153 "output_type": "display_data"
13154 }
13154 }
13155 ],
13155 ],
13156 "source": [
13156 "source": [
13157 "plt.hist(xyz_avg[:,0])\n",
13157 "plt.hist(xyz_avg[:,0])\n",
13158 "plt.title('Average $x(t)$')"
13158 "plt.title('Average $x(t)$')"
13159 ]
13159 ]
13160 },
13160 },
13161 {
13161 {
13162 "cell_type": "code",
13162 "cell_type": "code",
13163 "execution_count": 12,
13163 "execution_count": 12,
13164 "metadata": {
13164 "metadata": {
13165 "collapsed": false
13165 "collapsed": false
13166 },
13166 },
13167 "outputs": [
13167 "outputs": [
13168 {
13168 {
13169 "data": {
13169 "data": {
13170 "text/plain": [
13170 "text/plain": [
13171 "<matplotlib.text.Text at 0x10cadb610>"
13171 "<matplotlib.text.Text at 0x10cadb610>"
13172 ]
13172 ]
13173 },
13173 },
13174 "execution_count": 12,
13174 "execution_count": 12,
13175 "metadata": {},
13175 "metadata": {},
13176 "output_type": "execute_result"
13176 "output_type": "execute_result"
13177 },
13177 },
13178 {
13178 {
13179 "data": {
13179 "data": {
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13412 ],
13412 ],
13413 "text/plain": [
13413 "text/plain": [
13414 "<matplotlib.figure.Figure at 0x10c033e10>"
13414 "<matplotlib.figure.Figure at 0x10c033e10>"
13415 ]
13415 ]
13416 },
13416 },
13417 "metadata": {
13417 "metadata": {
13418 "image/png": {
13418 "image/png": {
13419 "height": 391,
13419 "height": 391,
13420 "width": 586
13420 "width": 586
13421 }
13421 }
13422 },
13422 },
13423 "output_type": "display_data"
13423 "output_type": "display_data"
13424 }
13424 }
13425 ],
13425 ],
13426 "source": [
13426 "source": [
13427 "plt.hist(xyz_avg[:,1])\n",
13427 "plt.hist(xyz_avg[:,1])\n",
13428 "plt.title('Average $y(t)$')"
13428 "plt.title('Average $y(t)$')"
13429 ]
13429 ]
13430 }
13430 }
13431 ],
13431 ],
13432 "metadata": {
13432 "metadata": {
13433 "kernelspec": {
13433 "kernelspec": {
13434 "display_name": "Python 3",
13434 "display_name": "Python 3",
13435 "name": "python3"
13435 "name": "python3"
13436 },
13437 "language_info": {
13438 "codemirror_mode": {
13439 "name": "ipython",
13440 "version": 3
13441 },
13442 "file_extension": ".py",
13443 "mimetype": "text/x-python",
13444 "name": "python",
13445 "nbconvert_exporter": "python",
13446 "pygments_lexer": "ipython3",
13447 "version": "3.4.2"
13436 }
13448 }
13437 },
13449 },
13438 "nbformat": 4,
13450 "nbformat": 4,
13439 "nbformat_minor": 0
13451 "nbformat_minor": 0
13440 }
13452 }
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