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919 | 919 | "cell_type": "markdown",
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920 | 920 | "metadata": {},
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921 | 921 | "source": [
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922 |
| - "This system of ODEs is the Lorenz system:\n", |
| 922 | + "The [Lorenz system](https://en.wikipedia.org/wiki/Lorenz_system) is a very simple\n", |
| 923 | + "model of convection in our atmosphere, but demonstrates the idea of chaos well.\n", |
| 924 | + "\n", |
| 925 | + "This system of ODEs for the Lorenz system is:\n", |
923 | 926 | "\n",
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924 | 927 | "$$\\frac{dx}{dt} = \\sigma (y - x)$$\n",
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925 | 928 | "$$\\frac{dy}{dt} = rx - y - xz$$\n",
|
|
929 | 932 | "\n",
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930 | 933 | "$${\\bf f}({\\bf x}) = \n",
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931 | 934 | "\\left (\n",
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932 |
| - "\\sigma (y -x), \n", |
933 |
| - "rx - y -xz, \n", |
| 935 | + "\\begin{array}{c}\n", |
| 936 | + "\\sigma (y -x) \\\\\n", |
| 937 | + "rx - y -xz \\\\\n", |
934 | 938 | "xy - bz\n",
|
935 |
| - "\\right )^\\intercal\n", |
| 939 | + "\\end{array}\n", |
| 940 | + "\\right )\n", |
936 | 941 | "= 0$$\n"
|
937 | 942 | ]
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938 | 943 | },
|
|
1019 | 1024 | "fig.set_size_inches(8.0,6.0)"
|
1020 | 1025 | ]
|
1021 | 1026 | },
|
| 1027 | + { |
| 1028 | + "cell_type": "markdown", |
| 1029 | + "metadata": {}, |
| 1030 | + "source": [ |
| 1031 | + "```{admonition} try it\n", |
| 1032 | + "Rerun the integration, but change the initial conditions by 1 part in $10^6$ for one of the components.\n", |
| 1033 | + "The make a plot of $x$ vs. $t$ comparing the solutions. You'll see that the 2 solutions track well\n", |
| 1034 | + "for some time but then greatly diverged. This is the sensitivity to initial conditions that is the\n", |
| 1035 | + "hallmark of chaos.\n", |
| 1036 | + "```" |
| 1037 | + ] |
| 1038 | + }, |
1022 | 1039 | {
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1023 | 1040 | "cell_type": "markdown",
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1024 | 1041 | "metadata": {},
|
|
1246 | 1263 | "\n",
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1247 | 1264 | "plt.legend(loc=\"best\", frameon=False)"
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1248 | 1265 | ]
|
| 1266 | + }, |
| 1267 | + { |
| 1268 | + "cell_type": "markdown", |
| 1269 | + "metadata": {}, |
| 1270 | + "source": [ |
| 1271 | + "```{admonition} try it\n", |
| 1272 | + "Redo this integration, but now use the `RK45` solver instead of `BDF`. Does it work?\n", |
| 1273 | + "\n", |
| 1274 | + "You may need to use the `kernel` menu in Jupyter to interrupt the kernel if you get impatient.\n", |
| 1275 | + "```" |
| 1276 | + ] |
| 1277 | + }, |
| 1278 | + { |
| 1279 | + "cell_type": "code", |
| 1280 | + "execution_count": null, |
| 1281 | + "metadata": {}, |
| 1282 | + "outputs": [], |
| 1283 | + "source": [] |
1249 | 1284 | }
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1250 | 1285 | ],
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1251 | 1286 | "metadata": {
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