From 8d470f5ea388741a7e28dcda78b3b7721622980c Mon Sep 17 00:00:00 2001
From: Olivier Cots <olivier.cots@enseeiht.fr>
Date: Sun, 17 Apr 2022 21:51:17 +0200
Subject: [PATCH] update tp ode
---
tp/ode-etu.ipynb | 2314 +++++++++++++++++++++++-----------------------
1 file changed, 1163 insertions(+), 1151 deletions(-)
diff --git a/tp/ode-etu.ipynb b/tp/ode-etu.ipynb
index 2ea0860..65508a8 100644
--- a/tp/ode-etu.ipynb
+++ b/tp/ode-etu.ipynb
@@ -12,16 +12,16 @@
"metadata": {},
"source": [
"## Introduction\n",
- "Pour la documentation voire \n",
- "[Documentation of DifferentialEquations.jl](https://diffeq.sciml.ai/stable/)\n",
"\n",
+ "Pour la documentation voir\n",
+ "[Documentation of DifferentialEquations.jl](https://diffeq.sciml.ai/stable/)\n",
"\n",
- "Il nous faut tout d'abord importer les bons packages de Julia"
+ "Il nous faut tout d'abord importer les bons packages de Julia."
]
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
@@ -46,18 +46,18 @@
"\\end{array}\\right.\n",
"$$\n",
"\n",
- "- Ecrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t)=f(t,x(t))$\n",
+ "- Ecrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t)=f(t,x(t))$.\n",
"- Vérifier que la solution de $(IVP$ est\n",
"$$\\begin{align*}\n",
"x_1(t)&= (-1/25)(13\\sin t+9\\cos t)\\\\\n",
"x_2(t)&= (-1/25)(3\\sin t+4\\cos t).\n",
"\\end{align*}$$\n",
- "- Coder la fonction $f$"
+ "- Coder la fonction $f$."
]
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 6,
"metadata": {},
"outputs": [
{
@@ -66,336 +66,347 @@
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- " 319.722,1386.4 321.887,1386.4 324.052,1386.4 326.217,1386.4 328.382,1386.4 330.547,1386.4 332.713,1386.4 334.878,1386.4 337.043,1386.4 339.208,1386.4 \n",
- " 341.373,1386.4 343.538,1386.4 345.703,1386.4 347.868,1386.4 350.033,1386.4 352.199,1386.4 354.364,1386.4 356.529,1386.4 358.694,1386.4 360.859,1386.4 \n",
- " 363.024,1386.4 365.189,1386.4 367.354,1386.4 369.519,1386.4 371.684,1386.4 373.85,1386.4 376.015,1386.4 378.18,1386.4 380.345,1386.4 382.51,1386.4 \n",
- " 384.675,1386.4 386.84,1386.4 389.005,1386.4 391.17,1386.4 393.336,1386.4 395.501,1386.4 397.666,1386.4 399.831,1386.4 401.996,1386.4 404.161,1386.4 \n",
- " 406.326,1386.4 408.491,1386.4 410.656,1386.4 412.822,1386.4 414.987,1386.4 417.152,1386.4 419.317,1386.4 421.482,1386.4 423.647,1386.4 425.812,1386.4 \n",
- " 427.977,1386.4 430.142,1386.4 432.307,1386.4 434.473,1386.4 436.638,1386.4 438.803,1386.4 440.968,1386.4 443.133,1386.4 445.298,1386.4 447.463,1386.4 \n",
- " 449.628,1386.4 451.793,1386.4 453.959,1386.4 456.124,1386.4 458.289,1386.4 460.454,1386.4 462.619,1386.4 464.784,1386.4 466.949,1386.4 469.114,1386.4 \n",
- " 471.279,1386.4 473.444,1386.4 475.61,1386.4 477.775,1386.4 479.94,1386.4 482.105,1386.4 484.27,1386.4 486.435,1386.4 488.6,1386.4 490.765,1386.4 \n",
- " 492.93,1386.4 495.096,1386.4 497.261,1386.4 499.426,1386.4 501.591,1386.4 503.756,1386.4 505.921,1386.4 508.086,1386.4 510.251,1386.4 512.416,1386.4 \n",
- " 514.581,1386.4 516.747,1386.4 518.912,1386.4 521.077,1386.4 523.242,1386.4 525.407,1386.4 527.572,1386.4 529.737,1386.4 531.902,1386.4 534.067,1386.4 \n",
- " 536.233,1386.4 538.398,1386.4 540.563,1386.4 542.728,1386.4 544.893,1386.4 547.058,1386.4 549.223,1386.4 551.388,1386.4 553.553,1386.4 555.718,1386.4 \n",
- " 557.884,1386.4 560.049,1386.4 562.214,1386.4 564.379,1386.4 566.544,1386.4 568.709,1386.4 570.874,1386.4 573.039,1386.4 575.204,1386.4 577.37,1386.4 \n",
- " 579.535,1386.4 581.7,1386.4 583.865,1386.4 586.03,1386.4 588.195,1386.4 590.36,1386.4 592.525,1386.4 594.69,1386.4 596.855,1386.4 599.021,1386.4 \n",
- " 601.186,1386.4 603.351,1386.4 605.516,1386.4 607.681,1386.4 609.846,1386.4 612.011,1386.4 614.176,1386.4 616.341,1386.4 618.507,1386.4 620.672,1386.4 \n",
- " 622.837,1386.4 625.002,1386.4 627.167,1386.4 629.332,1386.4 631.497,1386.4 633.662,1386.4 635.827,1386.4 637.992,1386.4 640.158,1386.4 642.323,1386.4 \n",
- " 644.488,1386.4 646.653,1386.4 648.818,1386.4 650.983,1386.4 653.148,1386.4 655.313,1386.4 657.478,1386.4 659.644,1386.4 661.809,1386.4 663.974,1386.4 \n",
- " 666.139,1386.4 668.304,1386.4 670.469,1386.4 672.634,1386.4 674.799,1386.4 676.964,1386.4 679.129,1386.4 681.295,1386.4 683.46,1386.4 685.625,1386.4 \n",
- " 687.79,1386.4 689.955,1386.4 692.12,1386.4 694.285,1386.4 696.45,1386.4 698.615,1386.4 700.781,1386.4 702.946,1386.4 705.111,1386.4 707.276,1386.4 \n",
- " 709.441,1386.4 711.606,1386.4 713.771,1386.4 715.936,1386.4 718.101,1386.4 720.266,1386.4 722.432,1386.4 724.597,1386.4 726.762,1386.4 728.927,1386.4 \n",
- " 731.092,1386.4 733.257,1386.4 735.422,1386.4 737.587,1386.4 739.752,1386.4 741.918,1386.4 744.083,1386.4 746.248,1386.4 748.413,1386.4 750.578,1386.4 \n",
- " 752.743,1386.4 754.908,1386.4 757.073,1386.4 759.238,1386.4 761.403,1386.4 763.569,1386.4 765.734,1386.4 767.899,1386.4 770.064,1386.4 772.229,1386.4 \n",
- " 774.394,1386.4 776.559,1386.4 778.724,1386.4 780.889,1386.4 783.055,1386.4 785.22,1386.4 787.385,1386.4 789.55,1386.4 791.715,1386.4 793.88,1386.4 \n",
- " 796.045,1386.4 798.21,1386.4 800.375,1386.4 802.54,1386.4 804.706,1386.4 806.871,1386.4 809.036,1386.4 811.201,1386.4 813.366,1386.4 815.531,1386.4 \n",
- " 817.696,1386.4 819.861,1386.4 822.026,1386.4 824.192,1386.4 826.357,1386.4 828.522,1386.4 830.687,1386.4 832.852,1386.4 835.017,1386.4 837.182,1386.4 \n",
- " 839.347,1386.4 841.512,1386.4 843.677,1386.4 845.843,1386.4 848.008,1386.4 850.173,1386.4 852.338,1386.4 854.503,1386.4 856.668,1386.4 858.833,1386.4 \n",
- " 860.998,1386.4 863.163,1386.4 865.329,1386.4 867.494,1386.4 869.659,1386.4 871.824,1386.4 873.989,1386.4 876.154,1386.4 878.319,1386.4 880.484,1386.4 \n",
- " 882.649,1386.4 884.814,1386.4 886.98,1386.4 889.145,1386.4 891.31,1386.4 893.475,1386.4 895.64,1386.4 897.805,1386.4 899.97,1386.4 902.135,1386.4 \n",
- " 904.3,1386.4 906.466,1386.4 908.631,1386.4 910.796,1386.4 912.961,1386.4 915.126,1386.4 917.291,1386.4 919.456,1386.4 921.621,1386.4 923.786,1386.4 \n",
- " 925.951,1386.4 928.117,1386.4 930.282,1386.4 932.447,1386.4 934.612,1386.4 936.777,1386.4 938.942,1386.4 941.107,1386.4 943.272,1386.4 945.437,1386.4 \n",
- " 947.603,1386.4 949.768,1386.4 951.933,1386.4 954.098,1386.4 956.263,1386.4 958.428,1386.4 960.593,1386.4 962.758,1386.4 964.923,1386.4 967.088,1386.4 \n",
- " 969.254,1386.4 971.419,1386.4 973.584,1386.4 975.749,1386.4 977.914,1386.4 980.079,1386.4 982.244,1386.4 984.409,1386.4 986.574,1386.4 988.74,1386.4 \n",
- " 990.905,1386.4 993.07,1386.4 995.235,1386.4 997.4,1386.4 999.565,1386.4 1001.73,1386.4 1003.9,1386.4 1006.06,1386.4 1008.23,1386.4 1010.39,1386.4 \n",
- " 1012.56,1386.4 1014.72,1386.4 1016.89,1386.4 1019.05,1386.4 1021.22,1386.4 1023.38,1386.4 1025.55,1386.4 1027.71,1386.4 1029.88,1386.4 1032.04,1386.4 \n",
- " 1034.21,1386.4 1036.37,1386.4 1038.54,1386.4 1040.7,1386.4 1042.87,1386.4 1045.03,1386.4 1047.2,1386.4 1049.36,1386.4 1051.53,1386.4 1053.69,1386.4 \n",
- " 1055.86,1386.4 1058.02,1386.4 1060.19,1386.4 1062.35,1386.4 1064.52,1386.4 1066.68,1386.4 1068.85,1386.4 1071.01,1386.4 1073.18,1386.4 1075.34,1386.4 \n",
- " 1077.51,1386.4 1079.67,1386.4 1081.84,1386.4 1084,1386.4 1086.17,1386.4 1088.33,1386.4 1090.5,1386.4 1092.66,1386.4 1094.83,1386.4 1096.99,1386.4 \n",
- " 1099.16,1386.4 1101.33,1386.4 1103.49,1386.4 1105.66,1386.4 1107.82,1386.4 1109.99,1386.4 1112.15,1386.4 1114.32,1386.4 1116.48,1386.4 1118.65,1386.4 \n",
- " 1120.81,1386.4 1122.98,1386.4 1125.14,1386.4 1127.31,1386.4 1129.47,1386.4 1131.64,1386.4 1133.8,1386.4 1135.97,1386.4 1138.13,1386.4 1140.3,1386.4 \n",
- " 1142.46,1386.4 1144.63,1386.4 1146.79,1386.4 1148.96,1386.4 1151.12,1386.4 1153.29,1386.4 1155.45,1386.4 1157.62,1386.4 1159.78,1386.4 1161.95,1386.4 \n",
- " 1164.11,1386.4 1166.28,1386.4 1168.44,1386.4 1170.61,1386.4 1172.77,1386.4 1174.94,1386.4 1177.1,1386.4 1179.27,1386.4 1181.43,1386.4 1183.6,1386.4 \n",
- " 1185.76,1386.4 1187.93,1386.4 1190.09,1386.4 1192.26,1386.4 1194.42,1386.4 1196.59,1386.4 1198.75,1386.4 1200.92,1386.4 1203.08,1386.4 1205.25,1386.4 \n",
- " 1207.42,1386.4 1209.58,1386.4 1211.75,1386.4 1213.91,1386.4 1216.08,1386.4 1218.24,1386.4 1220.41,1386.4 1222.57,1386.4 1224.74,1386.4 1226.9,1386.4 \n",
- " 1229.07,1386.4 1231.23,1386.4 1233.4,1386.4 1235.56,1386.4 1237.73,1386.4 1239.89,1386.4 1242.06,1386.4 1244.22,1386.4 1246.39,1386.4 1248.55,1386.4 \n",
- " 1250.72,1386.4 1252.88,1386.4 1255.05,1386.4 1257.21,1386.4 1259.38,1386.4 1261.54,1386.4 1263.71,1386.4 1265.87,1386.4 1268.04,1386.4 1270.2,1386.4 \n",
- " 1272.37,1386.4 1274.53,1386.4 1276.7,1386.4 1278.86,1386.4 1281.03,1386.4 1283.19,1386.4 1285.36,1386.4 1287.52,1386.4 1289.69,1386.4 1291.85,1386.4 \n",
- " 1294.02,1386.4 1296.18,1386.4 1298.35,1386.4 1300.51,1386.4 1302.68,1386.4 1304.84,1386.4 1307.01,1386.4 1309.18,1386.4 1311.34,1386.4 1313.51,1386.4 \n",
- " 1315.67,1386.4 1317.84,1386.4 1320,1386.4 1322.17,1386.4 1324.33,1386.4 1326.5,1386.4 1328.66,1386.4 1330.83,1386.4 1332.99,1386.4 1335.16,1386.4 \n",
- " 1337.32,1386.4 1339.49,1386.4 1341.65,1386.4 1343.82,1386.4 1345.98,1386.4 1348.15,1386.4 1350.31,1386.4 1352.48,1386.4 1354.64,1386.4 1356.81,1386.4 \n",
- " 1358.97,1386.4 1361.14,1386.4 1363.3,1386.4 1365.47,1386.4 1367.63,1386.4 1369.8,1386.4 1371.96,1386.4 1374.13,1386.4 1376.29,1386.4 1378.46,1386.4 \n",
- " 1380.62,1386.4 1382.79,1386.4 1384.95,1386.4 1387.12,1386.4 1389.28,1386.4 1391.45,1386.4 1393.61,1386.4 1395.78,1386.4 1397.94,1386.4 1400.11,1386.4 \n",
- " 1402.27,1386.4 1404.44,1386.4 1406.6,1386.4 1408.77,1386.4 1410.94,1386.4 1413.1,1386.4 1415.27,1386.4 1417.43,1386.4 1419.6,1386.4 1421.76,1386.4 \n",
- " 1423.93,1386.4 1426.09,1386.4 1428.26,1386.4 1430.42,1386.4 1432.59,1386.4 1434.75,1386.4 1436.92,1386.4 1439.08,1386.4 1441.25,1386.4 1443.41,1386.4 \n",
- " 1445.58,1386.4 1447.74,1386.4 1449.91,1386.4 1452.07,1386.4 1454.24,1386.4 1456.4,1386.4 1458.57,1386.4 1460.73,1386.4 1462.9,1386.4 1465.06,1386.4 \n",
- " 1467.23,1386.4 1469.39,1386.4 1471.56,1386.4 1473.72,1386.4 1475.89,1386.4 1478.05,1386.4 1480.22,1386.4 1482.38,1386.4 1484.55,1386.4 1486.71,1386.4 \n",
- " 1488.88,1386.4 1491.04,1386.4 1493.21,1386.4 1495.37,1386.4 1497.54,1386.4 1499.7,1386.4 1501.87,1386.4 1504.03,1386.4 1506.2,1386.4 1508.36,1386.4 \n",
- " 1510.53,1386.4 1512.7,1386.4 1514.86,1386.4 1517.03,1386.4 1519.19,1386.4 1521.36,1386.4 1523.52,1386.4 1525.69,1386.4 1527.85,1386.4 1530.02,1386.4 \n",
- " 1532.18,1386.4 1534.35,1386.4 1536.51,1386.4 1538.68,1386.4 1540.84,1386.4 1543.01,1386.4 1545.17,1386.4 1547.34,1386.4 1549.5,1386.4 1551.67,1386.4 \n",
- " 1553.83,1386.4 1556,1386.4 1558.16,1386.4 1560.33,1386.4 1562.49,1386.4 1564.66,1386.4 1566.82,1386.4 1568.99,1386.4 1571.15,1386.4 1573.32,1386.4 \n",
- " 1575.48,1386.4 1577.65,1386.4 1579.81,1386.4 1581.98,1386.4 1584.14,1386.4 1586.31,1386.4 1588.47,1386.4 1590.64,1386.4 1592.8,1386.4 1594.97,1386.4 \n",
- " 1597.13,1386.4 1599.3,1386.4 1601.46,1386.4 1603.63,1386.4 1605.79,1386.4 1607.96,1386.4 1610.12,1386.4 1612.29,1386.4 1614.45,1386.4 1616.62,1386.4 \n",
- " 1618.79,1386.4 1620.95,1386.4 1623.12,1386.4 1625.28,1386.4 1627.45,1386.4 1629.61,1386.4 1631.78,1386.4 1633.94,1386.4 1636.11,1386.4 1638.27,1386.4 \n",
- " 1640.44,1386.4 1642.6,1386.4 1644.77,1386.4 1646.93,1386.4 1649.1,1386.4 1651.26,1386.4 1653.43,1386.4 1655.59,1386.4 1657.76,1386.4 1659.92,1386.4 \n",
- " 1662.09,1386.4 1664.25,1386.4 1666.42,1386.4 1668.58,1386.4 1670.75,1386.4 1672.91,1386.4 1675.08,1386.4 1677.24,1386.4 1679.41,1386.4 1681.57,1386.4 \n",
- " 1683.74,1386.4 1685.9,1386.4 1688.07,1386.4 1690.23,1386.4 1692.4,1386.4 1694.56,1386.4 1696.73,1386.4 1698.89,1386.4 1701.06,1386.4 1703.22,1386.4 \n",
- " 1705.39,1386.4 1707.55,1386.4 1709.72,1386.4 1711.88,1386.4 1714.05,1386.4 1716.21,1386.4 1718.38,1386.4 1720.55,1386.4 1722.71,1386.4 1724.88,1386.4 \n",
- " 1727.04,1386.4 1729.21,1386.4 1731.37,1386.4 1733.54,1386.4 1735.7,1386.4 1737.87,1386.4 1740.03,1386.4 1742.2,1386.4 1744.36,1386.4 1746.53,1386.4 \n",
- " 1748.69,1386.4 1750.86,1386.4 1753.02,1386.4 1755.19,1386.4 1757.35,1386.4 1759.52,1386.4 1761.68,1386.4 1763.85,1386.4 1766.01,1386.4 1768.18,1386.4 \n",
- " 1770.34,1386.4 1772.51,1386.4 1774.67,1386.4 1776.84,1386.4 1779,1386.4 1781.17,1386.4 1783.33,1386.4 1785.5,1386.4 1787.66,1386.4 1789.83,1386.4 \n",
- " 1791.99,1386.4 1794.16,1386.4 1796.32,1386.4 1798.49,1386.4 1800.65,1386.4 1802.82,1386.4 1804.98,1386.4 1807.15,1386.4 1809.31,1386.4 1811.48,1386.4 \n",
- " 1813.64,1386.4 1815.81,1386.4 1817.97,1386.4 1820.14,1386.4 1822.31,1386.4 1824.47,1386.4 1826.64,1386.4 1828.8,1386.4 1830.97,1386.4 1833.13,1386.4 \n",
- " 1835.3,1386.4 1837.46,1386.4 1839.63,1386.4 1841.79,1386.4 1843.96,1386.4 1846.12,1386.4 1848.29,1386.4 1850.45,1386.4 1852.62,1386.4 1854.78,1386.4 \n",
- " 1856.95,1386.4 1859.11,1386.4 1861.28,1386.4 1863.44,1386.4 1865.61,1386.4 1867.77,1386.4 1869.94,1386.4 1872.1,1386.4 1874.27,1386.4 1876.43,1386.4 \n",
- " 1878.6,1386.4 1880.76,1386.4 1882.93,1386.4 1885.09,1386.4 1887.26,1386.4 1889.42,1386.4 1891.59,1386.4 1893.75,1386.4 1895.92,1386.4 1898.08,1386.4 \n",
- " 1900.25,1386.4 1902.41,1386.4 1904.58,1386.4 1906.74,1386.4 1908.91,1386.4 1911.07,1386.4 1913.24,1386.4 1915.4,1386.4 1917.57,1386.4 1919.73,1386.4 \n",
- " 1921.9,1386.4 1924.07,1386.4 1926.23,1386.4 1928.4,1386.4 1930.56,1386.4 1932.73,1386.4 1934.89,1386.4 1937.06,1386.4 1939.22,1386.4 1941.39,1386.4 \n",
- " 1943.55,1386.4 1945.72,1386.4 1947.88,1386.4 1950.05,1386.4 1952.21,1386.4 1954.38,1386.4 1956.54,1386.4 1958.71,1386.4 1960.87,1386.4 1963.04,1386.4 \n",
- " 1965.2,1386.4 1967.37,1386.4 1969.53,1386.4 1971.7,1386.4 1973.86,1386.4 1976.03,1386.4 1978.19,1386.4 1980.36,1386.4 1982.52,1386.4 1984.69,1386.4 \n",
- " 1986.85,1386.4 1989.02,1386.4 1991.18,1386.4 1993.35,1386.4 1995.51,1386.4 1997.68,1386.4 1999.84,1386.4 2002.01,1386.4 2004.17,1386.4 2006.34,1386.4 \n",
- " 2008.5,1386.4 2010.67,1386.4 2012.83,1386.4 2015,1386.4 2017.16,1386.4 2019.33,1386.4 2021.49,1386.4 2023.66,1386.4 2025.82,1386.4 2027.99,1386.4 \n",
- " 2030.16,1386.4 2032.32,1386.4 2034.49,1386.4 2036.65,1386.4 2038.82,1386.4 2040.98,1386.4 2043.15,1386.4 2045.31,1386.4 2047.48,1386.4 2049.64,1386.4 \n",
- " 2051.81,1386.4 2053.97,1386.4 2056.14,1386.4 2058.3,1386.4 2060.47,1386.4 2062.63,1386.4 2064.8,1386.4 2066.96,1386.4 2069.13,1386.4 2071.29,1386.4 \n",
- " 2073.46,1386.4 2075.62,1386.4 2077.79,1386.4 2079.95,1386.4 2082.12,1386.4 2084.28,1386.4 2086.45,1386.4 2088.61,1386.4 2090.78,1386.4 2092.94,1386.4 \n",
- " 2095.11,1386.4 2097.27,1386.4 2099.44,1386.4 2101.6,1386.4 2103.77,1386.4 2105.93,1386.4 2108.1,1386.4 2110.26,1386.4 2112.43,1386.4 2114.59,1386.4 \n",
- " 2116.76,1386.4 2118.92,1386.4 2121.09,1386.4 2123.25,1386.4 2125.42,1386.4 2127.58,1386.4 2129.75,1386.4 2131.92,1386.4 2134.08,1386.4 2136.25,1386.4 \n",
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- " 1575.48,160.256 1577.65,160.256 1579.81,160.256 1581.98,160.256 1584.14,160.256 1586.31,160.256 1588.47,160.256 1590.64,160.256 1592.8,160.256 1594.97,160.256 \n",
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- " 1770.34,160.256 1772.51,160.256 1774.67,160.256 1776.84,160.256 1779,160.256 1781.17,160.256 1783.33,160.256 1785.5,160.256 1787.66,160.256 1789.83,160.256 \n",
- " 1791.99,160.256 1794.16,160.256 1796.32,160.256 1798.49,160.256 1800.65,160.256 1802.82,160.256 1804.98,160.256 1807.15,160.256 1809.31,160.256 1811.48,160.256 \n",
- " 1813.64,160.256 1815.81,160.256 1817.97,160.256 1820.14,160.256 1822.31,160.256 1824.47,160.256 1826.64,160.256 1828.8,160.256 1830.97,160.256 1833.13,160.256 \n",
- " 1835.3,160.256 1837.46,160.256 1839.63,160.256 1841.79,160.256 1843.96,160.256 1846.12,160.256 1848.29,160.256 1850.45,160.256 1852.62,160.256 1854.78,160.256 \n",
- " 1856.95,160.256 1859.11,160.256 1861.28,160.256 1863.44,160.256 1865.61,160.256 1867.77,160.256 1869.94,160.256 1872.1,160.256 1874.27,160.256 1876.43,160.256 \n",
- " 1878.6,160.256 1880.76,160.256 1882.93,160.256 1885.09,160.256 1887.26,160.256 1889.42,160.256 1891.59,160.256 1893.75,160.256 1895.92,160.256 1898.08,160.256 \n",
- " 1900.25,160.256 1902.41,160.256 1904.58,160.256 1906.74,160.256 1908.91,160.256 1911.07,160.256 1913.24,160.256 1915.4,160.256 1917.57,160.256 1919.73,160.256 \n",
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- "execution_count": 10,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
+ "# Remarque : \n",
+ "# un problème IVP est une équation différentielle ordinaire \n",
+ "# (EDO en français, ODE en anglais)\n",
+ "# avec en plus une condition initiale.\n",
+ "\n",
"function exemple1(x,p,t)\n",
- "# second member of the IVP\n",
- "# x : state\n",
- "# p : paramter vector\n",
- "# t : time, variable not use here\n",
+ "# Second membre de l'IVP\n",
+ "# p : vecteur de paramètres\n",
+ "# t : variable de temps\n",
+ " xpoint = similar(x)\n",
+ " \n",
+ " # A COMPLETER/MODIFIER\n",
" xpoint = zeros(size(x))\n",
- "\n",
- "# TO COMPLETE\n",
+ " \n",
" return xpoint\n",
"end\n",
- "p = []\n",
- "t0 = 0.; tf = 10\n",
- "tspan = (t0,tf)\n",
- "x0 = [-9/25 -4/25]\n",
- "prob = ODEProblem(exemple1,x0,tspan,p) # défini le problème en Julia\n",
- "sol = solve(prob)\n",
- "p1 = plot(sol, label = [\"x_1(t)\" \"x_2(t)\"], title = \"default options\", lw = 0.5) # lw = linewidth"
+ "\n",
+ "p = [] # pas de paramètres\n",
+ "\n",
+ "t0 = 0.\n",
+ "tf = 10\n",
+ "tspan = (t0, tf) # intervalle d'intégration\n",
+ "\n",
+ "x0 = [-9/25; -4/25] # condition initiale\n",
+ "\n",
+ "prob = ODEProblem(exemple1, x0, tspan, p) # définition du problème IVP\n",
+ "sol = solve(prob) # résolution du problème IVP\n",
+ "p1 = plot(sol, label = [\"x_1(t)\" \"x_2(t)\"], title = \"default options\", lw = 0.5) # affichage de la solution"
]
},
{
@@ -403,8 +414,9 @@
"metadata": {},
"source": [
"## Le contrôle du pas\n",
- "Les bons intégrateurs numériques ont une mise à jour automatique du pas. Pour les méthodes de Runge-Kutta par exemple, sur un pas $[t_0, t_{1}]$ on calcule par 2 schémas différents deux solutions approchées à l'instant $t_{1}~: x_{1}$ et $\\hat{x}_{1}$ dont l'erreur local est respectivement en $O(h^p)$ et $O(h^{p+1})$ (les err\n",
- "eurs globales sont elles en $O(h^{p-1})$ et $O(h^p)$). Ainsi on peut estimer l'erreur locale du schéma le moins précis à l'aide de la différence $x_{1}-\\hat{x}_{1}$. On peut alors estimer le pas de façon à avoir la norme de cette différence inférieur à une tolérance donnée $Tol$ ou encore à avoir \n",
+ "Les bons intégrateurs numériques ont une mise à jour automatique du pas. Pour les \n",
+ "[méthodes de Runge-Kutta](https://fr.wikipedia.org/wiki/Méthodes_de_Runge-Kutta) \n",
+ "par exemple, sur un pas $[t_0, t_{1}]$ on calcule par 2 schémas différents deux solutions approchées à l'instant $t_{1}$ : $x_{1}$ et $\\hat{x}_{1}$ dont l'erreur local est respectivement en $O(h^p)$ et $O(h^{p+1})$ (les erreurs globales sont elles en $O(h^{p-1})$ et $O(h^p)$). Ainsi on peut estimer l'erreur locale du schéma le moins précis à l'aide de la différence $x_{1}-\\hat{x}_{1}$. On peut alors estimer le pas de façon à avoir la norme de cette différence inférieure à une tolérance donnée $Tol$ ou encore à avoir \n",
"$$\\frac{\\|x_{1}-\\hat{x}_{1}\\|}{Tol}<1.$$\n",
"\n",
"En pratique on considère des tolérences absolue et relative composante par composante et on définit : $sc_i = Atol_i + \\max(|x_{0i}|,|x_{1i}|)Rtol_i$. On calcule alors le pas de façon à avoir \n",
@@ -439,7 +451,7 @@
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{
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- " 385.116,1034.42 387.257,1034.42 389.399,1034.42 391.54,1034.42 393.681,1034.42 395.822,1034.42 397.964,1034.42 400.105,1034.42 402.246,1034.42 404.387,1034.42 \n",
- " 406.529,1034.42 408.67,1034.42 410.811,1034.42 412.952,1034.42 415.094,1034.42 417.235,1034.42 419.376,1034.42 421.517,1034.42 423.659,1034.42 425.8,1034.42 \n",
- " 427.941,1034.42 430.082,1034.42 432.224,1034.42 434.365,1034.42 436.506,1034.42 438.647,1034.42 440.789,1034.42 442.93,1034.42 445.071,1034.42 447.212,1034.42 \n",
- " 449.354,1034.42 451.495,1034.42 453.636,1034.42 455.777,1034.42 457.919,1034.42 460.06,1034.42 462.201,1034.42 464.342,1034.42 466.484,1034.42 468.625,1034.42 \n",
- " 470.766,1034.42 472.908,1034.42 475.049,1034.42 477.19,1034.42 479.331,1034.42 481.473,1034.42 483.614,1034.42 485.755,1034.42 487.896,1034.42 490.038,1034.42 \n",
- " 492.179,1034.42 494.32,1034.42 496.461,1034.42 498.603,1034.42 500.744,1034.42 502.885,1034.42 505.026,1034.42 507.168,1034.42 509.309,1034.42 511.45,1034.42 \n",
- " 513.591,1034.42 515.733,1034.42 517.874,1034.42 520.015,1034.42 522.156,1034.42 524.298,1034.42 526.439,1034.42 528.58,1034.42 530.721,1034.42 532.863,1034.42 \n",
- " 535.004,1034.42 537.145,1034.42 539.286,1034.42 541.428,1034.42 543.569,1034.42 545.71,1034.42 547.851,1034.42 549.993,1034.42 552.134,1034.42 554.275,1034.42 \n",
- " 556.416,1034.42 558.558,1034.42 560.699,1034.42 562.84,1034.42 564.981,1034.42 567.123,1034.42 569.264,1034.42 571.405,1034.42 573.547,1034.42 575.688,1034.42 \n",
- " 577.829,1034.42 579.97,1034.42 582.112,1034.42 584.253,1034.42 586.394,1034.42 588.535,1034.42 590.677,1034.42 592.818,1034.42 594.959,1034.42 597.1,1034.42 \n",
- " 599.242,1034.42 601.383,1034.42 603.524,1034.42 605.665,1034.42 607.807,1034.42 609.948,1034.42 612.089,1034.42 614.23,1034.42 616.372,1034.42 618.513,1034.42 \n",
- " 620.654,1034.42 622.795,1034.42 624.937,1034.42 627.078,1034.42 629.219,1034.42 631.36,1034.42 633.502,1034.42 635.643,1034.42 637.784,1034.42 639.925,1034.42 \n",
- " 642.067,1034.42 644.208,1034.42 646.349,1034.42 648.49,1034.42 650.632,1034.42 652.773,1034.42 654.914,1034.42 657.055,1034.42 659.197,1034.42 661.338,1034.42 \n",
- " 663.479,1034.42 665.62,1034.42 667.762,1034.42 669.903,1034.42 672.044,1034.42 674.186,1034.42 676.327,1034.42 678.468,1034.42 680.609,1034.42 682.751,1034.42 \n",
- " 684.892,1034.42 687.033,1034.42 689.174,1034.42 691.316,1034.42 693.457,1034.42 695.598,1034.42 697.739,1034.42 699.881,1034.42 702.022,1034.42 704.163,1034.42 \n",
- " 706.304,1034.42 708.446,1034.42 710.587,1034.42 712.728,1034.42 714.869,1034.42 717.011,1034.42 719.152,1034.42 721.293,1034.42 723.434,1034.42 725.576,1034.42 \n",
- " 727.717,1034.42 729.858,1034.42 731.999,1034.42 734.141,1034.42 736.282,1034.42 738.423,1034.42 740.564,1034.42 742.706,1034.42 744.847,1034.42 746.988,1034.42 \n",
- " 749.129,1034.42 751.271,1034.42 753.412,1034.42 755.553,1034.42 757.694,1034.42 759.836,1034.42 761.977,1034.42 764.118,1034.42 766.259,1034.42 768.401,1034.42 \n",
- " 770.542,1034.42 772.683,1034.42 774.824,1034.42 776.966,1034.42 779.107,1034.42 781.248,1034.42 783.39,1034.42 785.531,1034.42 787.672,1034.42 789.813,1034.42 \n",
- " 791.955,1034.42 794.096,1034.42 796.237,1034.42 798.378,1034.42 800.52,1034.42 802.661,1034.42 804.802,1034.42 806.943,1034.42 809.085,1034.42 811.226,1034.42 \n",
- " 813.367,1034.42 815.508,1034.42 817.65,1034.42 819.791,1034.42 821.932,1034.42 824.073,1034.42 826.215,1034.42 828.356,1034.42 830.497,1034.42 832.638,1034.42 \n",
- " 834.78,1034.42 836.921,1034.42 839.062,1034.42 841.203,1034.42 843.345,1034.42 845.486,1034.42 847.627,1034.42 849.768,1034.42 851.91,1034.42 854.051,1034.42 \n",
- " 856.192,1034.42 858.333,1034.42 860.475,1034.42 862.616,1034.42 864.757,1034.42 866.898,1034.42 869.04,1034.42 871.181,1034.42 873.322,1034.42 875.463,1034.42 \n",
- " 877.605,1034.42 879.746,1034.42 881.887,1034.42 884.029,1034.42 886.17,1034.42 888.311,1034.42 890.452,1034.42 892.594,1034.42 894.735,1034.42 896.876,1034.42 \n",
- " 899.017,1034.42 901.159,1034.42 903.3,1034.42 905.441,1034.42 907.582,1034.42 909.724,1034.42 911.865,1034.42 914.006,1034.42 916.147,1034.42 918.289,1034.42 \n",
- " 920.43,1034.42 922.571,1034.42 924.712,1034.42 926.854,1034.42 928.995,1034.42 931.136,1034.42 933.277,1034.42 935.419,1034.42 937.56,1034.42 939.701,1034.42 \n",
- " 941.842,1034.42 943.984,1034.42 946.125,1034.42 948.266,1034.42 950.407,1034.42 952.549,1034.42 954.69,1034.42 956.831,1034.42 958.972,1034.42 961.114,1034.42 \n",
- " 963.255,1034.42 965.396,1034.42 967.537,1034.42 969.679,1034.42 971.82,1034.42 973.961,1034.42 976.102,1034.42 978.244,1034.42 980.385,1034.42 982.526,1034.42 \n",
- " 984.667,1034.42 986.809,1034.42 988.95,1034.42 991.091,1034.42 993.233,1034.42 995.374,1034.42 997.515,1034.42 999.656,1034.42 1001.8,1034.42 1003.94,1034.42 \n",
- " 1006.08,1034.42 1008.22,1034.42 1010.36,1034.42 1012.5,1034.42 1014.65,1034.42 1016.79,1034.42 1018.93,1034.42 1021.07,1034.42 1023.21,1034.42 1025.35,1034.42 \n",
- " 1027.49,1034.42 1029.63,1034.42 1031.78,1034.42 1033.92,1034.42 1036.06,1034.42 1038.2,1034.42 1040.34,1034.42 1042.48,1034.42 1044.62,1034.42 1046.76,1034.42 \n",
- " 1048.91,1034.42 1051.05,1034.42 1053.19,1034.42 1055.33,1034.42 1057.47,1034.42 1059.61,1034.42 1061.75,1034.42 1063.89,1034.42 1066.04,1034.42 1068.18,1034.42 \n",
- " 1070.32,1034.42 1072.46,1034.42 1074.6,1034.42 1076.74,1034.42 1078.88,1034.42 1081.02,1034.42 1083.17,1034.42 1085.31,1034.42 1087.45,1034.42 1089.59,1034.42 \n",
- " 1091.73,1034.42 1093.87,1034.42 1096.01,1034.42 1098.15,1034.42 1100.3,1034.42 1102.44,1034.42 1104.58,1034.42 1106.72,1034.42 1108.86,1034.42 1111,1034.42 \n",
- " 1113.14,1034.42 1115.28,1034.42 1117.43,1034.42 1119.57,1034.42 1121.71,1034.42 1123.85,1034.42 1125.99,1034.42 1128.13,1034.42 1130.27,1034.42 1132.41,1034.42 \n",
- " 1134.56,1034.42 1136.7,1034.42 1138.84,1034.42 1140.98,1034.42 1143.12,1034.42 1145.26,1034.42 1147.4,1034.42 1149.54,1034.42 1151.69,1034.42 1153.83,1034.42 \n",
- " 1155.97,1034.42 1158.11,1034.42 1160.25,1034.42 1162.39,1034.42 1164.53,1034.42 1166.67,1034.42 1168.82,1034.42 1170.96,1034.42 1173.1,1034.42 1175.24,1034.42 \n",
- " 1177.38,1034.42 1179.52,1034.42 1181.66,1034.42 1183.8,1034.42 1185.95,1034.42 1188.09,1034.42 1190.23,1034.42 1192.37,1034.42 1194.51,1034.42 1196.65,1034.42 \n",
- " 1198.79,1034.42 1200.93,1034.42 1203.08,1034.42 1205.22,1034.42 1207.36,1034.42 1209.5,1034.42 1211.64,1034.42 1213.78,1034.42 1215.92,1034.42 1218.06,1034.42 \n",
- " 1220.21,1034.42 1222.35,1034.42 1224.49,1034.42 1226.63,1034.42 1228.77,1034.42 1230.91,1034.42 1233.05,1034.42 1235.19,1034.42 1237.34,1034.42 1239.48,1034.42 \n",
- " 1241.62,1034.42 1243.76,1034.42 1245.9,1034.42 1248.04,1034.42 1250.18,1034.42 1252.32,1034.42 1254.47,1034.42 1256.61,1034.42 1258.75,1034.42 1260.89,1034.42 \n",
- " 1263.03,1034.42 1265.17,1034.42 1267.31,1034.42 1269.45,1034.42 1271.6,1034.42 1273.74,1034.42 1275.88,1034.42 1278.02,1034.42 1280.16,1034.42 1282.3,1034.42 \n",
- " 1284.44,1034.42 1286.58,1034.42 1288.73,1034.42 1290.87,1034.42 1293.01,1034.42 1295.15,1034.42 1297.29,1034.42 1299.43,1034.42 1301.57,1034.42 1303.71,1034.42 \n",
- " 1305.86,1034.42 1308,1034.42 1310.14,1034.42 1312.28,1034.42 1314.42,1034.42 1316.56,1034.42 1318.7,1034.42 1320.84,1034.42 1322.99,1034.42 1325.13,1034.42 \n",
- " 1327.27,1034.42 1329.41,1034.42 1331.55,1034.42 1333.69,1034.42 1335.83,1034.42 1337.97,1034.42 1340.12,1034.42 1342.26,1034.42 1344.4,1034.42 1346.54,1034.42 \n",
- " 1348.68,1034.42 1350.82,1034.42 1352.96,1034.42 1355.1,1034.42 1357.25,1034.42 1359.39,1034.42 1361.53,1034.42 1363.67,1034.42 1365.81,1034.42 1367.95,1034.42 \n",
- " 1370.09,1034.42 1372.23,1034.42 1374.38,1034.42 1376.52,1034.42 1378.66,1034.42 1380.8,1034.42 1382.94,1034.42 1385.08,1034.42 1387.22,1034.42 1389.36,1034.42 \n",
- " 1391.51,1034.42 1393.65,1034.42 1395.79,1034.42 1397.93,1034.42 1400.07,1034.42 1402.21,1034.42 1404.35,1034.42 1406.49,1034.42 1408.64,1034.42 1410.78,1034.42 \n",
- " 1412.92,1034.42 1415.06,1034.42 1417.2,1034.42 1419.34,1034.42 1421.48,1034.42 1423.62,1034.42 1425.77,1034.42 1427.91,1034.42 1430.05,1034.42 1432.19,1034.42 \n",
- " 1434.33,1034.42 1436.47,1034.42 1438.61,1034.42 1440.75,1034.42 1442.9,1034.42 1445.04,1034.42 1447.18,1034.42 1449.32,1034.42 1451.46,1034.42 1453.6,1034.42 \n",
- " 1455.74,1034.42 1457.88,1034.42 1460.03,1034.42 1462.17,1034.42 1464.31,1034.42 1466.45,1034.42 1468.59,1034.42 1470.73,1034.42 1472.87,1034.42 1475.01,1034.42 \n",
- " 1477.16,1034.42 1479.3,1034.42 1481.44,1034.42 1483.58,1034.42 1485.72,1034.42 1487.86,1034.42 1490,1034.42 1492.14,1034.42 1494.29,1034.42 1496.43,1034.42 \n",
- " 1498.57,1034.42 1500.71,1034.42 1502.85,1034.42 1504.99,1034.42 1507.13,1034.42 1509.27,1034.42 1511.42,1034.42 1513.56,1034.42 1515.7,1034.42 1517.84,1034.42 \n",
- " 1519.98,1034.42 1522.12,1034.42 1524.26,1034.42 1526.41,1034.42 1528.55,1034.42 1530.69,1034.42 1532.83,1034.42 1534.97,1034.42 1537.11,1034.42 1539.25,1034.42 \n",
- " 1541.39,1034.42 1543.54,1034.42 1545.68,1034.42 1547.82,1034.42 1549.96,1034.42 1552.1,1034.42 1554.24,1034.42 1556.38,1034.42 1558.52,1034.42 1560.67,1034.42 \n",
- " 1562.81,1034.42 1564.95,1034.42 1567.09,1034.42 1569.23,1034.42 1571.37,1034.42 1573.51,1034.42 1575.65,1034.42 1577.8,1034.42 1579.94,1034.42 1582.08,1034.42 \n",
- " 1584.22,1034.42 1586.36,1034.42 1588.5,1034.42 1590.64,1034.42 1592.78,1034.42 1594.93,1034.42 1597.07,1034.42 1599.21,1034.42 1601.35,1034.42 1603.49,1034.42 \n",
- " 1605.63,1034.42 1607.77,1034.42 1609.91,1034.42 1612.06,1034.42 1614.2,1034.42 1616.34,1034.42 1618.48,1034.42 1620.62,1034.42 1622.76,1034.42 1624.9,1034.42 \n",
- " 1627.04,1034.42 1629.19,1034.42 1631.33,1034.42 1633.47,1034.42 1635.61,1034.42 1637.75,1034.42 1639.89,1034.42 1642.03,1034.42 1644.17,1034.42 1646.32,1034.42 \n",
- " 1648.46,1034.42 1650.6,1034.42 1652.74,1034.42 1654.88,1034.42 1657.02,1034.42 1659.16,1034.42 1661.3,1034.42 1663.45,1034.42 1665.59,1034.42 1667.73,1034.42 \n",
- " 1669.87,1034.42 1672.01,1034.42 1674.15,1034.42 1676.29,1034.42 1678.43,1034.42 1680.58,1034.42 1682.72,1034.42 1684.86,1034.42 1687,1034.42 1689.14,1034.42 \n",
- " 1691.28,1034.42 1693.42,1034.42 1695.56,1034.42 1697.71,1034.42 1699.85,1034.42 1701.99,1034.42 1704.13,1034.42 1706.27,1034.42 1708.41,1034.42 1710.55,1034.42 \n",
- " 1712.69,1034.42 1714.84,1034.42 1716.98,1034.42 1719.12,1034.42 1721.26,1034.42 1723.4,1034.42 1725.54,1034.42 1727.68,1034.42 1729.82,1034.42 1731.97,1034.42 \n",
- " 1734.11,1034.42 1736.25,1034.42 1738.39,1034.42 1740.53,1034.42 1742.67,1034.42 1744.81,1034.42 1746.95,1034.42 1749.1,1034.42 1751.24,1034.42 1753.38,1034.42 \n",
- " 1755.52,1034.42 1757.66,1034.42 1759.8,1034.42 1761.94,1034.42 1764.08,1034.42 1766.23,1034.42 1768.37,1034.42 1770.51,1034.42 1772.65,1034.42 1774.79,1034.42 \n",
- " 1776.93,1034.42 1779.07,1034.42 1781.21,1034.42 1783.36,1034.42 1785.5,1034.42 1787.64,1034.42 1789.78,1034.42 1791.92,1034.42 1794.06,1034.42 1796.2,1034.42 \n",
- " 1798.34,1034.42 1800.49,1034.42 1802.63,1034.42 1804.77,1034.42 1806.91,1034.42 1809.05,1034.42 1811.19,1034.42 1813.33,1034.42 1815.47,1034.42 1817.62,1034.42 \n",
- " 1819.76,1034.42 1821.9,1034.42 1824.04,1034.42 1826.18,1034.42 1828.32,1034.42 1830.46,1034.42 1832.6,1034.42 1834.75,1034.42 1836.89,1034.42 1839.03,1034.42 \n",
- " 1841.17,1034.42 1843.31,1034.42 1845.45,1034.42 1847.59,1034.42 1849.73,1034.42 1851.88,1034.42 1854.02,1034.42 1856.16,1034.42 1858.3,1034.42 1860.44,1034.42 \n",
- " 1862.58,1034.42 1864.72,1034.42 1866.86,1034.42 1869.01,1034.42 1871.15,1034.42 1873.29,1034.42 1875.43,1034.42 1877.57,1034.42 1879.71,1034.42 1881.85,1034.42 \n",
- " 1883.99,1034.42 1886.14,1034.42 1888.28,1034.42 1890.42,1034.42 1892.56,1034.42 1894.7,1034.42 1896.84,1034.42 1898.98,1034.42 1901.12,1034.42 1903.27,1034.42 \n",
- " 1905.41,1034.42 1907.55,1034.42 1909.69,1034.42 1911.83,1034.42 1913.97,1034.42 1916.11,1034.42 1918.25,1034.42 1920.4,1034.42 1922.54,1034.42 1924.68,1034.42 \n",
- " 1926.82,1034.42 1928.96,1034.42 1931.1,1034.42 1933.24,1034.42 1935.38,1034.42 1937.53,1034.42 1939.67,1034.42 1941.81,1034.42 1943.95,1034.42 1946.09,1034.42 \n",
- " 1948.23,1034.42 1950.37,1034.42 1952.51,1034.42 1954.66,1034.42 1956.8,1034.42 1958.94,1034.42 1961.08,1034.42 1963.22,1034.42 1965.36,1034.42 1967.5,1034.42 \n",
- " 1969.64,1034.42 1971.79,1034.42 1973.93,1034.42 1976.07,1034.42 1978.21,1034.42 1980.35,1034.42 1982.49,1034.42 1984.63,1034.42 1986.77,1034.42 1988.92,1034.42 \n",
- " 1991.06,1034.42 1993.2,1034.42 1995.34,1034.42 1997.48,1034.42 1999.62,1034.42 2001.76,1034.42 2003.9,1034.42 2006.05,1034.42 2008.19,1034.42 2010.33,1034.42 \n",
- " 2012.47,1034.42 2014.61,1034.42 2016.75,1034.42 2018.89,1034.42 2021.03,1034.42 2023.18,1034.42 2025.32,1034.42 2027.46,1034.42 2029.6,1034.42 2031.74,1034.42 \n",
- " 2033.88,1034.42 2036.02,1034.42 2038.17,1034.42 2040.31,1034.42 2042.45,1034.42 2044.59,1034.42 2046.73,1034.42 2048.87,1034.42 2051.01,1034.42 2053.15,1034.42 \n",
- " 2055.3,1034.42 2057.44,1034.42 2059.58,1034.42 2061.72,1034.42 2063.86,1034.42 2066,1034.42 2068.14,1034.42 2070.28,1034.42 2072.43,1034.42 2074.57,1034.42 \n",
- " 2076.71,1034.42 2078.85,1034.42 2080.99,1034.42 2083.13,1034.42 2085.27,1034.42 2087.41,1034.42 2089.56,1034.42 2091.7,1034.42 2093.84,1034.42 2095.98,1034.42 \n",
- " 2098.12,1034.42 2100.26,1034.42 2102.4,1034.42 2104.54,1034.42 2106.69,1034.42 2108.83,1034.42 2110.97,1034.42 2113.11,1034.42 2115.25,1034.42 2117.39,1034.42 \n",
- " 2119.53,1034.42 2121.67,1034.42 2123.82,1034.42 2125.96,1034.42 2128.1,1034.42 2130.24,1034.42 2132.38,1034.42 2134.52,1034.42 2136.66,1034.42 2138.8,1034.42 \n",
- " 2140.95,1034.42 2143.09,1034.42 2145.23,1034.42 2147.37,1034.42 2149.51,1034.42 2151.65,1034.42 2153.79,1034.42 2155.93,1034.42 2158.08,1034.42 2160.22,1034.42 \n",
- " 2162.36,1034.42 2164.5,1034.42 2166.64,1034.42 2168.78,1034.42 2170.92,1034.42 2173.06,1034.42 2175.21,1034.42 2177.35,1034.42 2179.49,1034.42 2181.63,1034.42 \n",
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- " 1477.16,135.325 1479.3,135.325 1481.44,135.325 1483.58,135.325 1485.72,135.325 1487.86,135.325 1490,135.325 1492.14,135.325 1494.29,135.325 1496.43,135.325 \n",
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- " 1798.34,135.325 1800.49,135.325 1802.63,135.325 1804.77,135.325 1806.91,135.325 1809.05,135.325 1811.19,135.325 1813.33,135.325 1815.47,135.325 1817.62,135.325 \n",
- " 1819.76,135.325 1821.9,135.325 1824.04,135.325 1826.18,135.325 1828.32,135.325 1830.46,135.325 1832.6,135.325 1834.75,135.325 1836.89,135.325 1839.03,135.325 \n",
- " 1841.17,135.325 1843.31,135.325 1845.45,135.325 1847.59,135.325 1849.73,135.325 1851.88,135.325 1854.02,135.325 1856.16,135.325 1858.3,135.325 1860.44,135.325 \n",
- " 1862.58,135.325 1864.72,135.325 1866.86,135.325 1869.01,135.325 1871.15,135.325 1873.29,135.325 1875.43,135.325 1877.57,135.325 1879.71,135.325 1881.85,135.325 \n",
- " 1883.99,135.325 1886.14,135.325 1888.28,135.325 1890.42,135.325 1892.56,135.325 1894.7,135.325 1896.84,135.325 1898.98,135.325 1901.12,135.325 1903.27,135.325 \n",
- " 1905.41,135.325 1907.55,135.325 1909.69,135.325 1911.83,135.325 1913.97,135.325 1916.11,135.325 1918.25,135.325 1920.4,135.325 1922.54,135.325 1924.68,135.325 \n",
- " 1926.82,135.325 1928.96,135.325 1931.1,135.325 1933.24,135.325 1935.38,135.325 1937.53,135.325 1939.67,135.325 1941.81,135.325 1943.95,135.325 1946.09,135.325 \n",
- " 1948.23,135.325 1950.37,135.325 1952.51,135.325 1954.66,135.325 1956.8,135.325 1958.94,135.325 1961.08,135.325 1963.22,135.325 1965.36,135.325 1967.5,135.325 \n",
- " 1969.64,135.325 1971.79,135.325 1973.93,135.325 1976.07,135.325 1978.21,135.325 1980.35,135.325 1982.49,135.325 1984.63,135.325 1986.77,135.325 1988.92,135.325 \n",
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},
- "execution_count": 11,
+ "execution_count": 7,
"metadata": {},
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}
@@ -1048,7 +1060,7 @@
"sol = solve(prob, reltol = 1.e-3, abstol = 1.e-6)\n",
"p1 = plot(sol, label = [\"x1(t)\" \"x2(t)\"], title = \"reltol=1e-3, abstol=1e-6\", lw = 0.5) # lw = linewidth\n",
"#\n",
- "# TO COMPLETE\n",
+ "# A COMPLETER/MODIFIER\n",
"p2 = plot()\n",
"p3 = plot()\n",
"\n",
@@ -1077,26 +1089,26 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"function pendule(x,p,t)\n",
- "# second member of the IVP\n",
- "# x : state\n",
- "# p : paramter vector\n",
- "# t : time, variable not use here\n",
- " xpoint = zeros(size(x))\n",
- "\n",
- "# TO COMPLETE\n",
+ "# Second membre de l'IVP\n",
+ "# p : vecteur de paramètres\n",
+ "# t : variable de temps. Ici le temps n'intervient pas explicitement.\n",
+ " xpoint = similar(x)\n",
" \n",
+ " # A COMPLETER/MODIFIER\n",
+ " xpoint = zeros(size(x))\n",
+ " \n",
" return xpoint \n",
"end;"
]
},
{
"cell_type": "code",
- "execution_count": 13,
+ "execution_count": 9,
"metadata": {},
"outputs": [
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- " 1352.1,86.1857 1354.28,86.1857 1356.46,86.1857 1358.64,86.1857 1360.82,86.1857 1363,86.1857 1365.18,86.1857 1367.36,86.1857 1369.54,86.1857 1371.72,86.1857 \n",
- " 1373.9,86.1857 1376.08,86.1857 1378.26,86.1857 1380.44,86.1857 1382.62,86.1857 1384.8,86.1857 1386.98,86.1857 1389.16,86.1857 1391.34,86.1857 1393.52,86.1857 \n",
- " 1395.7,86.1857 1397.88,86.1857 1400.06,86.1857 1402.24,86.1857 1404.42,86.1857 1406.6,86.1857 1408.78,86.1857 1410.96,86.1857 1413.14,86.1857 1415.32,86.1857 \n",
- " 1417.5,86.1857 1419.68,86.1857 1421.86,86.1857 1424.04,86.1857 1426.22,86.1857 1428.4,86.1857 1430.58,86.1857 1432.76,86.1857 1434.94,86.1857 1437.12,86.1857 \n",
- " 1439.3,86.1857 1441.48,86.1857 1443.67,86.1857 1445.85,86.1857 1448.03,86.1857 1450.21,86.1857 1452.39,86.1857 1454.57,86.1857 1456.75,86.1857 1458.93,86.1857 \n",
- " 1461.11,86.1857 1463.29,86.1857 1465.47,86.1857 1467.65,86.1857 1469.83,86.1857 1472.01,86.1857 1474.19,86.1857 1476.37,86.1857 1478.55,86.1857 1480.73,86.1857 \n",
- " 1482.91,86.1857 1485.09,86.1857 1487.27,86.1857 1489.45,86.1857 1491.63,86.1857 1493.81,86.1857 1495.99,86.1857 1498.17,86.1857 1500.35,86.1857 1502.53,86.1857 \n",
- " 1504.71,86.1857 1506.89,86.1857 1509.07,86.1857 1511.25,86.1857 1513.43,86.1857 1515.61,86.1857 1517.79,86.1857 1519.97,86.1857 1522.15,86.1857 1524.33,86.1857 \n",
- " 1526.51,86.1857 1528.69,86.1857 1530.87,86.1857 1533.05,86.1857 1535.23,86.1857 1537.41,86.1857 1539.59,86.1857 1541.77,86.1857 1543.95,86.1857 1546.13,86.1857 \n",
- " 1548.31,86.1857 1550.49,86.1857 1552.67,86.1857 1554.85,86.1857 1557.03,86.1857 1559.21,86.1857 1561.39,86.1857 1563.57,86.1857 1565.75,86.1857 1567.93,86.1857 \n",
- " 1570.11,86.1857 1572.29,86.1857 1574.47,86.1857 1576.65,86.1857 1578.83,86.1857 1581.01,86.1857 1583.19,86.1857 1585.37,86.1857 1587.55,86.1857 1589.73,86.1857 \n",
- " 1591.91,86.1857 1594.09,86.1857 1596.27,86.1857 1598.45,86.1857 1600.63,86.1857 1602.81,86.1857 1604.99,86.1857 1607.17,86.1857 1609.35,86.1857 1611.53,86.1857 \n",
- " 1613.71,86.1857 1615.89,86.1857 1618.07,86.1857 1620.25,86.1857 1622.43,86.1857 1624.61,86.1857 1626.79,86.1857 1628.97,86.1857 1631.15,86.1857 1633.33,86.1857 \n",
- " 1635.51,86.1857 1637.69,86.1857 1639.87,86.1857 1642.05,86.1857 1644.23,86.1857 1646.41,86.1857 1648.59,86.1857 1650.77,86.1857 1652.95,86.1857 1655.13,86.1857 \n",
- " 1657.31,86.1857 1659.49,86.1857 1661.67,86.1857 1663.85,86.1857 1666.03,86.1857 1668.21,86.1857 1670.39,86.1857 1672.57,86.1857 1674.75,86.1857 1676.93,86.1857 \n",
- " 1679.11,86.1857 1681.29,86.1857 1683.47,86.1857 1685.65,86.1857 1687.83,86.1857 1690.01,86.1857 1692.19,86.1857 1694.37,86.1857 1696.55,86.1857 1698.73,86.1857 \n",
- " 1700.91,86.1857 1703.09,86.1857 1705.27,86.1857 1707.45,86.1857 1709.63,86.1857 1711.81,86.1857 1713.99,86.1857 1716.17,86.1857 1718.35,86.1857 1720.53,86.1857 \n",
- " 1722.71,86.1857 1724.89,86.1857 1727.07,86.1857 1729.25,86.1857 1731.43,86.1857 1733.61,86.1857 1735.79,86.1857 1737.98,86.1857 1740.16,86.1857 1742.34,86.1857 \n",
- " 1744.52,86.1857 1746.7,86.1857 1748.88,86.1857 1751.06,86.1857 1753.24,86.1857 1755.42,86.1857 1757.6,86.1857 1759.78,86.1857 1761.96,86.1857 1764.14,86.1857 \n",
- " 1766.32,86.1857 1768.5,86.1857 1770.68,86.1857 1772.86,86.1857 1775.04,86.1857 1777.22,86.1857 1779.4,86.1857 1781.58,86.1857 1783.76,86.1857 1785.94,86.1857 \n",
- " 1788.12,86.1857 1790.3,86.1857 1792.48,86.1857 1794.66,86.1857 1796.84,86.1857 1799.02,86.1857 1801.2,86.1857 1803.38,86.1857 1805.56,86.1857 1807.74,86.1857 \n",
- " 1809.92,86.1857 1812.1,86.1857 1814.28,86.1857 1816.46,86.1857 1818.64,86.1857 1820.82,86.1857 1823,86.1857 1825.18,86.1857 1827.36,86.1857 1829.54,86.1857 \n",
- " 1831.72,86.1857 1833.9,86.1857 1836.08,86.1857 1838.26,86.1857 1840.44,86.1857 1842.62,86.1857 1844.8,86.1857 1846.98,86.1857 1849.16,86.1857 1851.34,86.1857 \n",
- " 1853.52,86.1857 1855.7,86.1857 1857.88,86.1857 1860.06,86.1857 1862.24,86.1857 1864.42,86.1857 1866.6,86.1857 1868.78,86.1857 1870.96,86.1857 1873.14,86.1857 \n",
- " 1875.32,86.1857 1877.5,86.1857 1879.68,86.1857 1881.86,86.1857 1884.04,86.1857 1886.22,86.1857 1888.4,86.1857 1890.58,86.1857 1892.76,86.1857 1894.94,86.1857 \n",
- " 1897.12,86.1857 1899.3,86.1857 1901.48,86.1857 1903.66,86.1857 1905.84,86.1857 1908.02,86.1857 1910.2,86.1857 1912.38,86.1857 1914.56,86.1857 1916.74,86.1857 \n",
- " 1918.92,86.1857 1921.1,86.1857 1923.28,86.1857 1925.46,86.1857 1927.64,86.1857 1929.82,86.1857 1932,86.1857 1934.18,86.1857 1936.36,86.1857 1938.54,86.1857 \n",
- " 1940.72,86.1857 1942.9,86.1857 1945.08,86.1857 1947.26,86.1857 1949.44,86.1857 1951.62,86.1857 1953.8,86.1857 1955.98,86.1857 1958.16,86.1857 1960.34,86.1857 \n",
- " 1962.52,86.1857 1964.7,86.1857 1966.88,86.1857 1969.06,86.1857 1971.24,86.1857 1973.42,86.1857 1975.6,86.1857 1977.78,86.1857 1979.96,86.1857 1982.14,86.1857 \n",
- " 1984.32,86.1857 1986.5,86.1857 1988.68,86.1857 1990.86,86.1857 1993.04,86.1857 1995.22,86.1857 1997.4,86.1857 1999.58,86.1857 2001.76,86.1857 2003.94,86.1857 \n",
- " 2006.12,86.1857 2008.3,86.1857 2010.48,86.1857 2012.66,86.1857 2014.84,86.1857 2017.02,86.1857 2019.2,86.1857 2021.38,86.1857 2023.56,86.1857 2025.74,86.1857 \n",
- " 2027.92,86.1857 2030.1,86.1857 2032.29,86.1857 2034.47,86.1857 2036.65,86.1857 2038.83,86.1857 2041.01,86.1857 2043.19,86.1857 2045.37,86.1857 2047.55,86.1857 \n",
- " 2049.73,86.1857 2051.91,86.1857 2054.09,86.1857 2056.27,86.1857 2058.45,86.1857 2060.63,86.1857 2062.81,86.1857 2064.99,86.1857 2067.17,86.1857 2069.35,86.1857 \n",
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- " 2093.33,86.1857 2095.51,86.1857 2097.69,86.1857 2099.87,86.1857 2102.05,86.1857 2104.23,86.1857 2106.41,86.1857 2108.59,86.1857 2110.77,86.1857 2112.95,86.1857 \n",
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- " 2202.33,86.1857 2204.51,86.1857 2206.69,86.1857 2208.87,86.1857 2211.05,86.1857 2213.23,86.1857 2215.41,86.1857 2217.59,86.1857 2219.77,86.1857 2221.95,86.1857 \n",
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- " 1264.9,1384.24 1267.08,1384.24 1269.26,1384.24 1271.44,1384.24 1273.62,1384.24 1275.8,1384.24 1277.98,1384.24 1280.16,1384.24 1282.34,1384.24 1284.52,1384.24 \n",
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- " 1308.5,1384.24 1310.68,1384.24 1312.86,1384.24 1315.04,1384.24 1317.22,1384.24 1319.4,1384.24 1321.58,1384.24 1323.76,1384.24 1325.94,1384.24 1328.12,1384.24 \n",
- " 1330.3,1384.24 1332.48,1384.24 1334.66,1384.24 1336.84,1384.24 1339.02,1384.24 1341.2,1384.24 1343.38,1384.24 1345.56,1384.24 1347.74,1384.24 1349.92,1384.24 \n",
- " 1352.1,1384.24 1354.28,1384.24 1356.46,1384.24 1358.64,1384.24 1360.82,1384.24 1363,1384.24 1365.18,1384.24 1367.36,1384.24 1369.54,1384.24 1371.72,1384.24 \n",
- " 1373.9,1384.24 1376.08,1384.24 1378.26,1384.24 1380.44,1384.24 1382.62,1384.24 1384.8,1384.24 1386.98,1384.24 1389.16,1384.24 1391.34,1384.24 1393.52,1384.24 \n",
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- " 1548.31,1384.24 1550.49,1384.24 1552.67,1384.24 1554.85,1384.24 1557.03,1384.24 1559.21,1384.24 1561.39,1384.24 1563.57,1384.24 1565.75,1384.24 1567.93,1384.24 \n",
- " 1570.11,1384.24 1572.29,1384.24 1574.47,1384.24 1576.65,1384.24 1578.83,1384.24 1581.01,1384.24 1583.19,1384.24 1585.37,1384.24 1587.55,1384.24 1589.73,1384.24 \n",
- " 1591.91,1384.24 1594.09,1384.24 1596.27,1384.24 1598.45,1384.24 1600.63,1384.24 1602.81,1384.24 1604.99,1384.24 1607.17,1384.24 1609.35,1384.24 1611.53,1384.24 \n",
- " 1613.71,1384.24 1615.89,1384.24 1618.07,1384.24 1620.25,1384.24 1622.43,1384.24 1624.61,1384.24 1626.79,1384.24 1628.97,1384.24 1631.15,1384.24 1633.33,1384.24 \n",
- " 1635.51,1384.24 1637.69,1384.24 1639.87,1384.24 1642.05,1384.24 1644.23,1384.24 1646.41,1384.24 1648.59,1384.24 1650.77,1384.24 1652.95,1384.24 1655.13,1384.24 \n",
- " 1657.31,1384.24 1659.49,1384.24 1661.67,1384.24 1663.85,1384.24 1666.03,1384.24 1668.21,1384.24 1670.39,1384.24 1672.57,1384.24 1674.75,1384.24 1676.93,1384.24 \n",
- " 1679.11,1384.24 1681.29,1384.24 1683.47,1384.24 1685.65,1384.24 1687.83,1384.24 1690.01,1384.24 1692.19,1384.24 1694.37,1384.24 1696.55,1384.24 1698.73,1384.24 \n",
- " 1700.91,1384.24 1703.09,1384.24 1705.27,1384.24 1707.45,1384.24 1709.63,1384.24 1711.81,1384.24 1713.99,1384.24 1716.17,1384.24 1718.35,1384.24 1720.53,1384.24 \n",
- " 1722.71,1384.24 1724.89,1384.24 1727.07,1384.24 1729.25,1384.24 1731.43,1384.24 1733.61,1384.24 1735.79,1384.24 1737.98,1384.24 1740.16,1384.24 1742.34,1384.24 \n",
- " 1744.52,1384.24 1746.7,1384.24 1748.88,1384.24 1751.06,1384.24 1753.24,1384.24 1755.42,1384.24 1757.6,1384.24 1759.78,1384.24 1761.96,1384.24 1764.14,1384.24 \n",
- " 1766.32,1384.24 1768.5,1384.24 1770.68,1384.24 1772.86,1384.24 1775.04,1384.24 1777.22,1384.24 1779.4,1384.24 1781.58,1384.24 1783.76,1384.24 1785.94,1384.24 \n",
- " 1788.12,1384.24 1790.3,1384.24 1792.48,1384.24 1794.66,1384.24 1796.84,1384.24 1799.02,1384.24 1801.2,1384.24 1803.38,1384.24 1805.56,1384.24 1807.74,1384.24 \n",
- " 1809.92,1384.24 1812.1,1384.24 1814.28,1384.24 1816.46,1384.24 1818.64,1384.24 1820.82,1384.24 1823,1384.24 1825.18,1384.24 1827.36,1384.24 1829.54,1384.24 \n",
- " 1831.72,1384.24 1833.9,1384.24 1836.08,1384.24 1838.26,1384.24 1840.44,1384.24 1842.62,1384.24 1844.8,1384.24 1846.98,1384.24 1849.16,1384.24 1851.34,1384.24 \n",
- " 1853.52,1384.24 1855.7,1384.24 1857.88,1384.24 1860.06,1384.24 1862.24,1384.24 1864.42,1384.24 1866.6,1384.24 1868.78,1384.24 1870.96,1384.24 1873.14,1384.24 \n",
- " 1875.32,1384.24 1877.5,1384.24 1879.68,1384.24 1881.86,1384.24 1884.04,1384.24 1886.22,1384.24 1888.4,1384.24 1890.58,1384.24 1892.76,1384.24 1894.94,1384.24 \n",
- " 1897.12,1384.24 1899.3,1384.24 1901.48,1384.24 1903.66,1384.24 1905.84,1384.24 1908.02,1384.24 1910.2,1384.24 1912.38,1384.24 1914.56,1384.24 1916.74,1384.24 \n",
- " 1918.92,1384.24 1921.1,1384.24 1923.28,1384.24 1925.46,1384.24 1927.64,1384.24 1929.82,1384.24 1932,1384.24 1934.18,1384.24 1936.36,1384.24 1938.54,1384.24 \n",
- " 1940.72,1384.24 1942.9,1384.24 1945.08,1384.24 1947.26,1384.24 1949.44,1384.24 1951.62,1384.24 1953.8,1384.24 1955.98,1384.24 1958.16,1384.24 1960.34,1384.24 \n",
- " 1962.52,1384.24 1964.7,1384.24 1966.88,1384.24 1969.06,1384.24 1971.24,1384.24 1973.42,1384.24 1975.6,1384.24 1977.78,1384.24 1979.96,1384.24 1982.14,1384.24 \n",
- " 1984.32,1384.24 1986.5,1384.24 1988.68,1384.24 1990.86,1384.24 1993.04,1384.24 1995.22,1384.24 1997.4,1384.24 1999.58,1384.24 2001.76,1384.24 2003.94,1384.24 \n",
- " 2006.12,1384.24 2008.3,1384.24 2010.48,1384.24 2012.66,1384.24 2014.84,1384.24 2017.02,1384.24 2019.2,1384.24 2021.38,1384.24 2023.56,1384.24 2025.74,1384.24 \n",
- " 2027.92,1384.24 2030.1,1384.24 2032.29,1384.24 2034.47,1384.24 2036.65,1384.24 2038.83,1384.24 2041.01,1384.24 2043.19,1384.24 2045.37,1384.24 2047.55,1384.24 \n",
- " 2049.73,1384.24 2051.91,1384.24 2054.09,1384.24 2056.27,1384.24 2058.45,1384.24 2060.63,1384.24 2062.81,1384.24 2064.99,1384.24 2067.17,1384.24 2069.35,1384.24 \n",
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- " 2093.33,1384.24 2095.51,1384.24 2097.69,1384.24 2099.87,1384.24 2102.05,1384.24 2104.23,1384.24 2106.41,1384.24 2108.59,1384.24 2110.77,1384.24 2112.95,1384.24 \n",
- " 2115.13,1384.24 2117.31,1384.24 2119.49,1384.24 2121.67,1384.24 2123.85,1384.24 2126.03,1384.24 2128.21,1384.24 2130.39,1384.24 2132.57,1384.24 2134.75,1384.24 \n",
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]
},
- "execution_count": 13,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -1420,16 +1432,20 @@
"source": [
"# Main\n",
"#\n",
- "g = 9.81; l = 10; k = 0.;\n",
- "p = [g,l,k] # constantes\n",
+ "g = 9.81; l = 10; k = 0.; m = 1.;\n",
+ "p = [g, l, k, m] # paramètres constants\n",
+ "\n",
"theta0 = pi/3\n",
+ "x0 = [theta0, 0.] # état initial\n",
+ "\n",
"t0 = 0.\n",
- "tf = 3*pi*sqrt(l/g)*(1 + theta0^2/16 + theta0^4/3072) # 2*approximation of the period\n",
- "tspan = (t0,tf) # instant initiel et terminal\n",
- "x0 = [theta0,0] # état initial\n",
- "prob = ODEProblem(pendule,x0,tspan,p) # défini le problème en Julia\n",
- "sol = solve(prob) # réalise l'intégration numérique\n",
- "plot(sol, label = [\"x1(t)\" \"x2(t)\"], lw = 0.5) "
+ "tf = 3*pi*sqrt(l/g)*(1 + theta0^2/16 + theta0^4/3072) # 2*approximation de la période\n",
+ "tspan = (t0, tf) # instant initial et terminal\n",
+ "\n",
+ "prob = ODEProblem(pendule,x0,tspan,p) # défini le problème en Julia\n",
+ "sol = solve(prob) # réalise l'intégration numérique\n",
+ "\n",
+ "plot(sol, label = [\"x1(t)\" \"x2(t)\"], lw = 0.5) # affichage de la solution"
]
},
{
@@ -5157,53 +5173,55 @@
"source": [
"# Main\n",
"#\n",
- "g = 9.81; l = 10; k = 0.;\n",
- "p = [g l k]\n",
+ "g = 9.81; l = 1.5; k = 0.; m = 1.;\n",
+ "p = [g, l, k, m] # paramètres constants\n",
+ "\n",
"plot()\n",
"for theta0 in 0:(2*pi)/10:2*pi\n",
" theta0_princ = theta0\n",
" tf = 3*pi*sqrt(l/g)*(1 + theta0_princ^2/16 + theta0_princ^4/3072) # 2*approximation of the period\n",
" tspan = (0.0,tf)\n",
" x0 = [theta0 0]\n",
- " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " prob = ODEProblem(pendule, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
" sol = solve(prob)\n",
- " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5, color=\"blue\") # lw = linewidth\n",
"end\n",
+ "\n",
"theta0 = pi-10*eps()\n",
"x0 = [theta0 0]\n",
- "tf = 70 # problem for tf=50 1/4 of the period!\n",
+ "tf = 50 # problem for tf=50 1/4 of the period!\n",
"tspan = (0.0,tf)\n",
- "prob = ODEProblem(pendule,x0,tspan,p)\n",
+ "prob = ODEProblem(pendule, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
"sol = solve(prob)\n",
- "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5, color=\"green\") # lw = linewidth\n",
"\n",
"theta0 = pi+10*eps()\n",
"x0 = [theta0 0]\n",
- "tf = 70\n",
+ "tf = 50\n",
"tspan = (0.0,tf)\n",
- "prob = ODEProblem(pendule,x0,tspan,p)\n",
+ "prob = ODEProblem(pendule, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
"sol = solve(prob)\n",
- "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ "plot!(sol,vars=(1,2), xlims = (-2*pi,4*pi), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5, color=\"green\") # lw = linewidth\n",
"\n",
"# circulation case\n",
- "for thetapoint0 in 0:0.2:2 \n",
+ "for thetapoint0 in 0:1.:4 \n",
" tf = 10\n",
" tspan = (0.,tf)\n",
" x0 = [-pi thetapoint0] # thetapoint0 > 0 so theta increases from -pi to ...\n",
- " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " prob = ODEProblem(pendule, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
" sol = solve(prob)\n",
- " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5, color=\"red\") # lw = linewidth\n",
"end\n",
- "for thetapoint0 in -2:0.2:0\n",
+ "for thetapoint0 in -4:1.:0\n",
" tf = 10\n",
" tspan = (0.,tf)\n",
" x0 = [3*pi thetapoint0] # thetapoint0 < 0 so theta decreases from 3pi to ...\n",
- " prob = ODEProblem(pendule,x0,tspan,p)\n",
+ " prob = ODEProblem(pendule, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
" sol = solve(prob)\n",
- " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5) # lw = linewidth\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x1\", ylabel = \"x2\", legend = false, lw = 0.5, color=\"purple\") # lw = linewidth\n",
"end\n",
"plot!([-pi 0 pi 2*pi 3*pi], [0 0 0 0 0], seriestype=:scatter)\n",
- "plot!(xlims = (-pi,3*pi))\n"
+ "plot!(xlims = (-pi,3*pi), size=(900, 600))\n"
]
},
{
@@ -5258,9 +5276,10 @@
"# xpoint : vector of velocity\n",
"# same as x\n",
"\n",
+ " xpoint = similar(x)\n",
+ " \n",
+ " # A COMPLETER/MODIFIER\n",
" xpoint = zeros(size(x))\n",
- "\n",
- "# TO COMPLETE\n",
" \n",
" return xpoint\n",
"end;"
@@ -6406,31 +6425,38 @@
"source": [
"# Main\n",
"#\n",
- "tf = 6.6632868593231301896996820305\n",
- "tspan = (0.0, tf)\n",
- "x0 = [2.00861986087484313650940188,0]\n",
- "mu = 1\n",
- "p = [mu]\n",
"t0 = 0.;\n",
- "tspan = (t0,tf)\n",
- "println(\"x0 = $x0, p = $p, tspan = $tspan\")\n",
- "prob = ODEProblem(vdp,x0,tspan,p)\n",
- "sol = solve(prob)\n",
- "plot(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false)\n",
- "for x01 in -2:0.4:0 #4.5:4.5\n",
+ "tf = 6.6632868593231301896996820305\n",
+ "tspan = (t0, tf)\n",
+ "mu = 1; p = [mu]\n",
+ "\n",
+ "plot()\n",
+ "\n",
+ "# Trajecctoires intérieures\n",
+ "for x01 in -2:0.4:0\n",
" x0 = [x01,0]\n",
- " prob = ODEProblem(vdp,x0,tspan,p)\n",
- " sol = solve(prob)#,force_dtmin=true)# \n",
- " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false) \n",
+ " prob = ODEProblem(vdp, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
+ " sol = solve(prob)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false, color = \"blue\") \n",
"end\n",
- "for x02 in 2:1:4\n",
+ "\n",
+ "# Trajecctoires extérieures\n",
+ "for x02 in 2.5:0.5:4\n",
" x0 = [0.,x02]\n",
- " prob = ODEProblem(vdp,x0,tspan,p)\n",
- " sol = solve(prob)#solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)\n",
- " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false)\n",
+ " prob = ODEProblem(vdp, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
+ " sol = solve(prob)\n",
+ " plot!(sol,vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false, color = \"green\")\n",
"end\n",
+ "\n",
+ "# Trajecctoires périodique\n",
+ "x0 = [2.00861986087484313650940188,0]\n",
+ "prob = ODEProblem(vdp, x0, tspan, p, reltol = 1.e-8, abstol = 1.e-8)\n",
+ "sol = solve(prob)\n",
+ "plot!(sol, vars=(1,2), xlabel = \"x_1\", ylabel = \"x_2\", legend = false, color = \"red\", lw = 2.)\n",
+ "\n",
+ "#\n",
"plot!([0], [0], seriestype=:scatter) # point visualisation\n",
- "plot!(xlims = (-2.5,5))"
+ "plot!(xlims = (-2.5,5), size=(900, 600))"
]
},
{
@@ -6439,13 +6465,13 @@
"source": [
"## Flots en dimension 2\n",
"\n",
- "### Flot des extrémales pour le problème de conrôle optimal \n",
+ "### Flot des extrémales pour le problème de contrôle optimal \n",
"\n",
"On considère le problème de contrôle optimal suivant:\n",
"\n",
"$$ \\frac{1}{2}\\int_0^{1} u^2(t)\\,\\mathrm{d}t \\to \\min, $$\n",
"\n",
- "où\n",
+ "sous les contraintes\n",
"\n",
"$$ \\dot{x}(t) = -x(t) + u(t)$$\n",
"\n",
@@ -7724,7 +7750,7 @@
"\n",
"$$ \\frac{1}{2}\\int_0^{1} u^2(t)\\,\\mathrm{d}t \\to \\min, $$\n",
"\n",
- "où\n",
+ "sous les contraintes\n",
"\n",
"$$ \\dot{x}(t) = -x(t) + \\alpha x^2(t) + u(t)$$\n",
"\n",
@@ -8906,33 +8932,19 @@
"plot!(plt1,[0.,0.],[0.,4], linestyle = :dot, color = :black)\n",
"plot(plt1)"
]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
}
],
"metadata": {
"kernelspec": {
- "display_name": "Julia 1.4.1",
+ "display_name": "Julia 1.6.0",
"language": "julia",
- "name": "julia-1.4"
+ "name": "julia-1.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
- "version": "1.4.1"
+ "version": "1.6.0"
}
},
"nbformat": 4,
--
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