{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intégration numérique avec Julia, une introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import Pkg; Pkg.add(\"DifferentialEquations\")\n",
    "# Pkg.add(\"Plots\")\n",
    "using DifferentialEquations\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exemple 1\n",
    "\n",
    "Nous allons ici résoudre le problème à valeur initiale\n",
    "$$(IVP)\\left\\{\\begin{array}{l}\n",
    "\\dot{x}_1(t)=x_1(t)+x_2(t)+\\sin t\\\\\n",
    "\\dot{x}_2(t)=-x_1(t)+3x_2(t)\\\\\n",
    "x_1(0)=-9/25\\\\\n",
    "x_2(0)=-4/25,\n",
    "\\end{array}\\right.\n",
    "$$\n",
    "\n",
    "- Ecrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t)=f(t,x(t))$.\n",
    "- 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 $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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       "  394.561,1386.4 396.692,1386.4 398.823,1386.4 400.953,1386.4 403.084,1386.4 405.215,1386.4 407.346,1386.4 409.477,1386.4 411.607,1386.4 413.738,1386.4 \n",
       "  415.869,1386.4 418,1386.4 420.13,1386.4 422.261,1386.4 424.392,1386.4 426.523,1386.4 428.654,1386.4 430.784,1386.4 432.915,1386.4 435.046,1386.4 \n",
       "  437.177,1386.4 439.308,1386.4 441.438,1386.4 443.569,1386.4 445.7,1386.4 447.831,1386.4 449.961,1386.4 452.092,1386.4 454.223,1386.4 456.354,1386.4 \n",
       "  458.485,1386.4 460.615,1386.4 462.746,1386.4 464.877,1386.4 467.008,1386.4 469.139,1386.4 471.269,1386.4 473.4,1386.4 475.531,1386.4 477.662,1386.4 \n",
       "  479.793,1386.4 481.923,1386.4 484.054,1386.4 486.185,1386.4 488.316,1386.4 490.446,1386.4 492.577,1386.4 494.708,1386.4 496.839,1386.4 498.97,1386.4 \n",
       "  501.1,1386.4 503.231,1386.4 505.362,1386.4 507.493,1386.4 509.624,1386.4 511.754,1386.4 513.885,1386.4 516.016,1386.4 518.147,1386.4 520.278,1386.4 \n",
       "  522.408,1386.4 524.539,1386.4 526.67,1386.4 528.801,1386.4 530.931,1386.4 533.062,1386.4 535.193,1386.4 537.324,1386.4 539.455,1386.4 541.585,1386.4 \n",
       "  543.716,1386.4 545.847,1386.4 547.978,1386.4 550.109,1386.4 552.239,1386.4 554.37,1386.4 556.501,1386.4 558.632,1386.4 560.763,1386.4 562.893,1386.4 \n",
       "  565.024,1386.4 567.155,1386.4 569.286,1386.4 571.416,1386.4 573.547,1386.4 575.678,1386.4 577.809,1386.4 579.94,1386.4 582.07,1386.4 584.201,1386.4 \n",
       "  586.332,1386.4 588.463,1386.4 590.594,1386.4 592.724,1386.4 594.855,1386.4 596.986,1386.4 599.117,1386.4 601.247,1386.4 603.378,1386.4 605.509,1386.4 \n",
       "  607.64,1386.4 609.771,1386.4 611.901,1386.4 614.032,1386.4 616.163,1386.4 618.294,1386.4 620.425,1386.4 622.555,1386.4 624.686,1386.4 626.817,1386.4 \n",
       "  628.948,1386.4 631.079,1386.4 633.209,1386.4 635.34,1386.4 637.471,1386.4 639.602,1386.4 641.732,1386.4 643.863,1386.4 645.994,1386.4 648.125,1386.4 \n",
       "  650.256,1386.4 652.386,1386.4 654.517,1386.4 656.648,1386.4 658.779,1386.4 660.91,1386.4 663.04,1386.4 665.171,1386.4 667.302,1386.4 669.433,1386.4 \n",
       "  671.564,1386.4 673.694,1386.4 675.825,1386.4 677.956,1386.4 680.087,1386.4 682.217,1386.4 684.348,1386.4 686.479,1386.4 688.61,1386.4 690.741,1386.4 \n",
       "  692.871,1386.4 695.002,1386.4 697.133,1386.4 699.264,1386.4 701.395,1386.4 703.525,1386.4 705.656,1386.4 707.787,1386.4 709.918,1386.4 712.049,1386.4 \n",
       "  714.179,1386.4 716.31,1386.4 718.441,1386.4 720.572,1386.4 722.702,1386.4 724.833,1386.4 726.964,1386.4 729.095,1386.4 731.226,1386.4 733.356,1386.4 \n",
       "  735.487,1386.4 737.618,1386.4 739.749,1386.4 741.88,1386.4 744.01,1386.4 746.141,1386.4 748.272,1386.4 750.403,1386.4 752.534,1386.4 754.664,1386.4 \n",
       "  756.795,1386.4 758.926,1386.4 761.057,1386.4 763.187,1386.4 765.318,1386.4 767.449,1386.4 769.58,1386.4 771.711,1386.4 773.841,1386.4 775.972,1386.4 \n",
       "  778.103,1386.4 780.234,1386.4 782.365,1386.4 784.495,1386.4 786.626,1386.4 788.757,1386.4 790.888,1386.4 793.018,1386.4 795.149,1386.4 797.28,1386.4 \n",
       "  799.411,1386.4 801.542,1386.4 803.672,1386.4 805.803,1386.4 807.934,1386.4 810.065,1386.4 812.196,1386.4 814.326,1386.4 816.457,1386.4 818.588,1386.4 \n",
       "  820.719,1386.4 822.85,1386.4 824.98,1386.4 827.111,1386.4 829.242,1386.4 831.373,1386.4 833.503,1386.4 835.634,1386.4 837.765,1386.4 839.896,1386.4 \n",
       "  842.027,1386.4 844.157,1386.4 846.288,1386.4 848.419,1386.4 850.55,1386.4 852.681,1386.4 854.811,1386.4 856.942,1386.4 859.073,1386.4 861.204,1386.4 \n",
       "  863.335,1386.4 865.465,1386.4 867.596,1386.4 869.727,1386.4 871.858,1386.4 873.988,1386.4 876.119,1386.4 878.25,1386.4 880.381,1386.4 882.512,1386.4 \n",
       "  884.642,1386.4 886.773,1386.4 888.904,1386.4 891.035,1386.4 893.166,1386.4 895.296,1386.4 897.427,1386.4 899.558,1386.4 901.689,1386.4 903.82,1386.4 \n",
       "  905.95,1386.4 908.081,1386.4 910.212,1386.4 912.343,1386.4 914.473,1386.4 916.604,1386.4 918.735,1386.4 920.866,1386.4 922.997,1386.4 925.127,1386.4 \n",
       "  927.258,1386.4 929.389,1386.4 931.52,1386.4 933.651,1386.4 935.781,1386.4 937.912,1386.4 940.043,1386.4 942.174,1386.4 944.304,1386.4 946.435,1386.4 \n",
       "  948.566,1386.4 950.697,1386.4 952.828,1386.4 954.958,1386.4 957.089,1386.4 959.22,1386.4 961.351,1386.4 963.482,1386.4 965.612,1386.4 967.743,1386.4 \n",
       "  969.874,1386.4 972.005,1386.4 974.136,1386.4 976.266,1386.4 978.397,1386.4 980.528,1386.4 982.659,1386.4 984.789,1386.4 986.92,1386.4 989.051,1386.4 \n",
       "  991.182,1386.4 993.313,1386.4 995.443,1386.4 997.574,1386.4 999.705,1386.4 1001.84,1386.4 1003.97,1386.4 1006.1,1386.4 1008.23,1386.4 1010.36,1386.4 \n",
       "  1012.49,1386.4 1014.62,1386.4 1016.75,1386.4 1018.88,1386.4 1021.01,1386.4 1023.14,1386.4 1025.27,1386.4 1027.41,1386.4 1029.54,1386.4 1031.67,1386.4 \n",
       "  1033.8,1386.4 1035.93,1386.4 1038.06,1386.4 1040.19,1386.4 1042.32,1386.4 1044.45,1386.4 1046.58,1386.4 1048.71,1386.4 1050.84,1386.4 1052.97,1386.4 \n",
       "  1055.11,1386.4 1057.24,1386.4 1059.37,1386.4 1061.5,1386.4 1063.63,1386.4 1065.76,1386.4 1067.89,1386.4 1070.02,1386.4 1072.15,1386.4 1074.28,1386.4 \n",
       "  1076.41,1386.4 1078.54,1386.4 1080.67,1386.4 1082.81,1386.4 1084.94,1386.4 1087.07,1386.4 1089.2,1386.4 1091.33,1386.4 1093.46,1386.4 1095.59,1386.4 \n",
       "  1097.72,1386.4 1099.85,1386.4 1101.98,1386.4 1104.11,1386.4 1106.24,1386.4 1108.38,1386.4 1110.51,1386.4 1112.64,1386.4 1114.77,1386.4 1116.9,1386.4 \n",
       "  1119.03,1386.4 1121.16,1386.4 1123.29,1386.4 1125.42,1386.4 1127.55,1386.4 1129.68,1386.4 1131.81,1386.4 1133.94,1386.4 1136.08,1386.4 1138.21,1386.4 \n",
       "  1140.34,1386.4 1142.47,1386.4 1144.6,1386.4 1146.73,1386.4 1148.86,1386.4 1150.99,1386.4 1153.12,1386.4 1155.25,1386.4 1157.38,1386.4 1159.51,1386.4 \n",
       "  1161.64,1386.4 1163.78,1386.4 1165.91,1386.4 1168.04,1386.4 1170.17,1386.4 1172.3,1386.4 1174.43,1386.4 1176.56,1386.4 1178.69,1386.4 1180.82,1386.4 \n",
       "  1182.95,1386.4 1185.08,1386.4 1187.21,1386.4 1189.35,1386.4 1191.48,1386.4 1193.61,1386.4 1195.74,1386.4 1197.87,1386.4 1200,1386.4 1202.13,1386.4 \n",
       "  1204.26,1386.4 1206.39,1386.4 1208.52,1386.4 1210.65,1386.4 1212.78,1386.4 1214.91,1386.4 1217.05,1386.4 1219.18,1386.4 1221.31,1386.4 1223.44,1386.4 \n",
       "  1225.57,1386.4 1227.7,1386.4 1229.83,1386.4 1231.96,1386.4 1234.09,1386.4 1236.22,1386.4 1238.35,1386.4 1240.48,1386.4 1242.61,1386.4 1244.75,1386.4 \n",
       "  1246.88,1386.4 1249.01,1386.4 1251.14,1386.4 1253.27,1386.4 1255.4,1386.4 1257.53,1386.4 1259.66,1386.4 1261.79,1386.4 1263.92,1386.4 1266.05,1386.4 \n",
       "  1268.18,1386.4 1270.32,1386.4 1272.45,1386.4 1274.58,1386.4 1276.71,1386.4 1278.84,1386.4 1280.97,1386.4 1283.1,1386.4 1285.23,1386.4 1287.36,1386.4 \n",
       "  1289.49,1386.4 1291.62,1386.4 1293.75,1386.4 1295.88,1386.4 1298.02,1386.4 1300.15,1386.4 1302.28,1386.4 1304.41,1386.4 1306.54,1386.4 1308.67,1386.4 \n",
       "  1310.8,1386.4 1312.93,1386.4 1315.06,1386.4 1317.19,1386.4 1319.32,1386.4 1321.45,1386.4 1323.58,1386.4 1325.72,1386.4 1327.85,1386.4 1329.98,1386.4 \n",
       "  1332.11,1386.4 1334.24,1386.4 1336.37,1386.4 1338.5,1386.4 1340.63,1386.4 1342.76,1386.4 1344.89,1386.4 1347.02,1386.4 1349.15,1386.4 1351.29,1386.4 \n",
       "  1353.42,1386.4 1355.55,1386.4 1357.68,1386.4 1359.81,1386.4 1361.94,1386.4 1364.07,1386.4 1366.2,1386.4 1368.33,1386.4 1370.46,1386.4 1372.59,1386.4 \n",
       "  1374.72,1386.4 1376.85,1386.4 1378.99,1386.4 1381.12,1386.4 1383.25,1386.4 1385.38,1386.4 1387.51,1386.4 1389.64,1386.4 1391.77,1386.4 1393.9,1386.4 \n",
       "  1396.03,1386.4 1398.16,1386.4 1400.29,1386.4 1402.42,1386.4 1404.55,1386.4 1406.69,1386.4 1408.82,1386.4 1410.95,1386.4 1413.08,1386.4 1415.21,1386.4 \n",
       "  1417.34,1386.4 1419.47,1386.4 1421.6,1386.4 1423.73,1386.4 1425.86,1386.4 1427.99,1386.4 1430.12,1386.4 1432.26,1386.4 1434.39,1386.4 1436.52,1386.4 \n",
       "  1438.65,1386.4 1440.78,1386.4 1442.91,1386.4 1445.04,1386.4 1447.17,1386.4 1449.3,1386.4 1451.43,1386.4 1453.56,1386.4 1455.69,1386.4 1457.82,1386.4 \n",
       "  1459.96,1386.4 1462.09,1386.4 1464.22,1386.4 1466.35,1386.4 1468.48,1386.4 1470.61,1386.4 1472.74,1386.4 1474.87,1386.4 1477,1386.4 1479.13,1386.4 \n",
       "  1481.26,1386.4 1483.39,1386.4 1485.52,1386.4 1487.66,1386.4 1489.79,1386.4 1491.92,1386.4 1494.05,1386.4 1496.18,1386.4 1498.31,1386.4 1500.44,1386.4 \n",
       "  1502.57,1386.4 1504.7,1386.4 1506.83,1386.4 1508.96,1386.4 1511.09,1386.4 1513.23,1386.4 1515.36,1386.4 1517.49,1386.4 1519.62,1386.4 1521.75,1386.4 \n",
       "  1523.88,1386.4 1526.01,1386.4 1528.14,1386.4 1530.27,1386.4 1532.4,1386.4 1534.53,1386.4 1536.66,1386.4 1538.79,1386.4 1540.93,1386.4 1543.06,1386.4 \n",
       "  1545.19,1386.4 1547.32,1386.4 1549.45,1386.4 1551.58,1386.4 1553.71,1386.4 1555.84,1386.4 1557.97,1386.4 1560.1,1386.4 1562.23,1386.4 1564.36,1386.4 \n",
       "  1566.49,1386.4 1568.63,1386.4 1570.76,1386.4 1572.89,1386.4 1575.02,1386.4 1577.15,1386.4 1579.28,1386.4 1581.41,1386.4 1583.54,1386.4 1585.67,1386.4 \n",
       "  1587.8,1386.4 1589.93,1386.4 1592.06,1386.4 1594.2,1386.4 1596.33,1386.4 1598.46,1386.4 1600.59,1386.4 1602.72,1386.4 1604.85,1386.4 1606.98,1386.4 \n",
       "  1609.11,1386.4 1611.24,1386.4 1613.37,1386.4 1615.5,1386.4 1617.63,1386.4 1619.76,1386.4 1621.9,1386.4 1624.03,1386.4 1626.16,1386.4 1628.29,1386.4 \n",
       "  1630.42,1386.4 1632.55,1386.4 1634.68,1386.4 1636.81,1386.4 1638.94,1386.4 1641.07,1386.4 1643.2,1386.4 1645.33,1386.4 1647.46,1386.4 1649.6,1386.4 \n",
       "  1651.73,1386.4 1653.86,1386.4 1655.99,1386.4 1658.12,1386.4 1660.25,1386.4 1662.38,1386.4 1664.51,1386.4 1666.64,1386.4 1668.77,1386.4 1670.9,1386.4 \n",
       "  1673.03,1386.4 1675.17,1386.4 1677.3,1386.4 1679.43,1386.4 1681.56,1386.4 1683.69,1386.4 1685.82,1386.4 1687.95,1386.4 1690.08,1386.4 1692.21,1386.4 \n",
       "  1694.34,1386.4 1696.47,1386.4 1698.6,1386.4 1700.73,1386.4 1702.87,1386.4 1705,1386.4 1707.13,1386.4 1709.26,1386.4 1711.39,1386.4 1713.52,1386.4 \n",
       "  1715.65,1386.4 1717.78,1386.4 1719.91,1386.4 1722.04,1386.4 1724.17,1386.4 1726.3,1386.4 1728.43,1386.4 1730.57,1386.4 1732.7,1386.4 1734.83,1386.4 \n",
       "  1736.96,1386.4 1739.09,1386.4 1741.22,1386.4 1743.35,1386.4 1745.48,1386.4 1747.61,1386.4 1749.74,1386.4 1751.87,1386.4 1754,1386.4 1756.14,1386.4 \n",
       "  1758.27,1386.4 1760.4,1386.4 1762.53,1386.4 1764.66,1386.4 1766.79,1386.4 1768.92,1386.4 1771.05,1386.4 1773.18,1386.4 1775.31,1386.4 1777.44,1386.4 \n",
       "  1779.57,1386.4 1781.7,1386.4 1783.84,1386.4 1785.97,1386.4 1788.1,1386.4 1790.23,1386.4 1792.36,1386.4 1794.49,1386.4 1796.62,1386.4 1798.75,1386.4 \n",
       "  1800.88,1386.4 1803.01,1386.4 1805.14,1386.4 1807.27,1386.4 1809.4,1386.4 1811.54,1386.4 1813.67,1386.4 1815.8,1386.4 1817.93,1386.4 1820.06,1386.4 \n",
       "  1822.19,1386.4 1824.32,1386.4 1826.45,1386.4 1828.58,1386.4 1830.71,1386.4 1832.84,1386.4 1834.97,1386.4 1837.11,1386.4 1839.24,1386.4 1841.37,1386.4 \n",
       "  1843.5,1386.4 1845.63,1386.4 1847.76,1386.4 1849.89,1386.4 1852.02,1386.4 1854.15,1386.4 1856.28,1386.4 1858.41,1386.4 1860.54,1386.4 1862.67,1386.4 \n",
       "  1864.81,1386.4 1866.94,1386.4 1869.07,1386.4 1871.2,1386.4 1873.33,1386.4 1875.46,1386.4 1877.59,1386.4 1879.72,1386.4 1881.85,1386.4 1883.98,1386.4 \n",
       "  1886.11,1386.4 1888.24,1386.4 1890.37,1386.4 1892.51,1386.4 1894.64,1386.4 1896.77,1386.4 1898.9,1386.4 1901.03,1386.4 1903.16,1386.4 1905.29,1386.4 \n",
       "  1907.42,1386.4 1909.55,1386.4 1911.68,1386.4 1913.81,1386.4 1915.94,1386.4 1918.07,1386.4 1920.21,1386.4 1922.34,1386.4 1924.47,1386.4 1926.6,1386.4 \n",
       "  1928.73,1386.4 1930.86,1386.4 1932.99,1386.4 1935.12,1386.4 1937.25,1386.4 1939.38,1386.4 1941.51,1386.4 1943.64,1386.4 1945.78,1386.4 1947.91,1386.4 \n",
       "  1950.04,1386.4 1952.17,1386.4 1954.3,1386.4 1956.43,1386.4 1958.56,1386.4 1960.69,1386.4 1962.82,1386.4 1964.95,1386.4 1967.08,1386.4 1969.21,1386.4 \n",
       "  1971.34,1386.4 1973.48,1386.4 1975.61,1386.4 1977.74,1386.4 1979.87,1386.4 1982,1386.4 1984.13,1386.4 1986.26,1386.4 1988.39,1386.4 1990.52,1386.4 \n",
       "  1992.65,1386.4 1994.78,1386.4 1996.91,1386.4 1999.04,1386.4 2001.18,1386.4 2003.31,1386.4 2005.44,1386.4 2007.57,1386.4 2009.7,1386.4 2011.83,1386.4 \n",
       "  2013.96,1386.4 2016.09,1386.4 2018.22,1386.4 2020.35,1386.4 2022.48,1386.4 2024.61,1386.4 2026.75,1386.4 2028.88,1386.4 2031.01,1386.4 2033.14,1386.4 \n",
       "  2035.27,1386.4 2037.4,1386.4 2039.53,1386.4 2041.66,1386.4 2043.79,1386.4 2045.92,1386.4 2048.05,1386.4 2050.18,1386.4 2052.31,1386.4 2054.45,1386.4 \n",
       "  2056.58,1386.4 2058.71,1386.4 2060.84,1386.4 2062.97,1386.4 2065.1,1386.4 2067.23,1386.4 2069.36,1386.4 2071.49,1386.4 2073.62,1386.4 2075.75,1386.4 \n",
       "  2077.88,1386.4 2080.01,1386.4 2082.15,1386.4 2084.28,1386.4 2086.41,1386.4 2088.54,1386.4 2090.67,1386.4 2092.8,1386.4 2094.93,1386.4 2097.06,1386.4 \n",
       "  2099.19,1386.4 2101.32,1386.4 2103.45,1386.4 2105.58,1386.4 2107.72,1386.4 2109.85,1386.4 2111.98,1386.4 2114.11,1386.4 2116.24,1386.4 2118.37,1386.4 \n",
       "  2120.5,1386.4 2122.63,1386.4 2124.76,1386.4 2126.89,1386.4 2129.02,1386.4 2131.15,1386.4 2133.28,1386.4 2135.42,1386.4 2137.55,1386.4 2139.68,1386.4 \n",
       "  2141.81,1386.4 2143.94,1386.4 2146.07,1386.4 2148.2,1386.4 2150.33,1386.4 2152.46,1386.4 2154.59,1386.4 2156.72,1386.4 2158.85,1386.4 2160.98,1386.4 \n",
       "  2163.12,1386.4 2165.25,1386.4 2167.38,1386.4 2169.51,1386.4 2171.64,1386.4 2173.77,1386.4 2175.9,1386.4 2178.03,1386.4 2180.16,1386.4 2182.29,1386.4 \n",
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     },
     "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 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",
    "    return xpoint\n",
    "end\n",
    "\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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Le contrôle du pas\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",
    "\n",
    "$$err = \\sqrt{\\frac{1}{n}\\sum_{i=1}^n\\left(\\frac{x_{1i}-\\hat{x}_{1i}}{sc_i}\\right)^2}<1.$$\n",
    "\n",
    "\n",
    "Référence : [Hairer, Nørsett, Wanner (2008) Solving Ordinary Differential Equations I Nonstiff Problems DOI: 10.1007/978-3-540-78862-1](https://link.springer.com/book/10.1007/978-3-540-78862-1)\n",
    "\n",
    "Dans `Julia`, on peut modifier les valeurs par défaut  : reltol = 1.e-3 et abstol = 1.e-6\n",
    "(lorsque ces valeurs sont des scalaires, on considère les mêmes quantités pour toutes les composantes).\n",
    "\n",
    "Réaliser l'intégration numérique pour \n",
    "- reltol = 1.e-3 et abstol = 1.e-6\n",
    "- reltol = 1.e-6 et abstol = 1.e-9\n",
    "- reltol = 1.e-10 et abstol = 1.e-15\n",
    "\n",
    "et afficher les résultats en ajoutant la solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Commentaires?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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       "  690.518,1034.42 692.625,1034.42 694.732,1034.42 696.839,1034.42 698.946,1034.42 701.052,1034.42 703.159,1034.42 705.266,1034.42 707.373,1034.42 709.479,1034.42 \n",
       "  711.586,1034.42 713.693,1034.42 715.8,1034.42 717.906,1034.42 720.013,1034.42 722.12,1034.42 724.227,1034.42 726.333,1034.42 728.44,1034.42 730.547,1034.42 \n",
       "  732.654,1034.42 734.761,1034.42 736.867,1034.42 738.974,1034.42 741.081,1034.42 743.188,1034.42 745.294,1034.42 747.401,1034.42 749.508,1034.42 751.615,1034.42 \n",
       "  753.721,1034.42 755.828,1034.42 757.935,1034.42 760.042,1034.42 762.148,1034.42 764.255,1034.42 766.362,1034.42 768.469,1034.42 770.576,1034.42 772.682,1034.42 \n",
       "  774.789,1034.42 776.896,1034.42 779.003,1034.42 781.109,1034.42 783.216,1034.42 785.323,1034.42 787.43,1034.42 789.536,1034.42 791.643,1034.42 793.75,1034.42 \n",
       "  795.857,1034.42 797.963,1034.42 800.07,1034.42 802.177,1034.42 804.284,1034.42 806.391,1034.42 808.497,1034.42 810.604,1034.42 812.711,1034.42 814.818,1034.42 \n",
       "  816.924,1034.42 819.031,1034.42 821.138,1034.42 823.245,1034.42 825.351,1034.42 827.458,1034.42 829.565,1034.42 831.672,1034.42 833.778,1034.42 835.885,1034.42 \n",
       "  837.992,1034.42 840.099,1034.42 842.206,1034.42 844.312,1034.42 846.419,1034.42 848.526,1034.42 850.633,1034.42 852.739,1034.42 854.846,1034.42 856.953,1034.42 \n",
       "  859.06,1034.42 861.166,1034.42 863.273,1034.42 865.38,1034.42 867.487,1034.42 869.593,1034.42 871.7,1034.42 873.807,1034.42 875.914,1034.42 878.021,1034.42 \n",
       "  880.127,1034.42 882.234,1034.42 884.341,1034.42 886.448,1034.42 888.554,1034.42 890.661,1034.42 892.768,1034.42 894.875,1034.42 896.981,1034.42 899.088,1034.42 \n",
       "  901.195,1034.42 903.302,1034.42 905.408,1034.42 907.515,1034.42 909.622,1034.42 911.729,1034.42 913.836,1034.42 915.942,1034.42 918.049,1034.42 920.156,1034.42 \n",
       "  922.263,1034.42 924.369,1034.42 926.476,1034.42 928.583,1034.42 930.69,1034.42 932.796,1034.42 934.903,1034.42 937.01,1034.42 939.117,1034.42 941.223,1034.42 \n",
       "  943.33,1034.42 945.437,1034.42 947.544,1034.42 949.651,1034.42 951.757,1034.42 953.864,1034.42 955.971,1034.42 958.078,1034.42 960.184,1034.42 962.291,1034.42 \n",
       "  964.398,1034.42 966.505,1034.42 968.611,1034.42 970.718,1034.42 972.825,1034.42 974.932,1034.42 977.038,1034.42 979.145,1034.42 981.252,1034.42 983.359,1034.42 \n",
       "  985.466,1034.42 987.572,1034.42 989.679,1034.42 991.786,1034.42 993.893,1034.42 995.999,1034.42 998.106,1034.42 1000.21,1034.42 1002.32,1034.42 1004.43,1034.42 \n",
       "  1006.53,1034.42 1008.64,1034.42 1010.75,1034.42 1012.85,1034.42 1014.96,1034.42 1017.07,1034.42 1019.17,1034.42 1021.28,1034.42 1023.39,1034.42 1025.49,1034.42 \n",
       "  1027.6,1034.42 1029.71,1034.42 1031.81,1034.42 1033.92,1034.42 1036.03,1034.42 1038.13,1034.42 1040.24,1034.42 1042.35,1034.42 1044.45,1034.42 1046.56,1034.42 \n",
       "  1048.67,1034.42 1050.78,1034.42 1052.88,1034.42 1054.99,1034.42 1057.1,1034.42 1059.2,1034.42 1061.31,1034.42 1063.42,1034.42 1065.52,1034.42 1067.63,1034.42 \n",
       "  1069.74,1034.42 1071.84,1034.42 1073.95,1034.42 1076.06,1034.42 1078.16,1034.42 1080.27,1034.42 1082.38,1034.42 1084.48,1034.42 1086.59,1034.42 1088.7,1034.42 \n",
       "  1090.8,1034.42 1092.91,1034.42 1095.02,1034.42 1097.12,1034.42 1099.23,1034.42 1101.34,1034.42 1103.44,1034.42 1105.55,1034.42 1107.66,1034.42 1109.76,1034.42 \n",
       "  1111.87,1034.42 1113.98,1034.42 1116.08,1034.42 1118.19,1034.42 1120.3,1034.42 1122.41,1034.42 1124.51,1034.42 1126.62,1034.42 1128.73,1034.42 1130.83,1034.42 \n",
       "  1132.94,1034.42 1135.05,1034.42 1137.15,1034.42 1139.26,1034.42 1141.37,1034.42 1143.47,1034.42 1145.58,1034.42 1147.69,1034.42 1149.79,1034.42 1151.9,1034.42 \n",
       "  1154.01,1034.42 1156.11,1034.42 1158.22,1034.42 1160.33,1034.42 1162.43,1034.42 1164.54,1034.42 1166.65,1034.42 1168.75,1034.42 1170.86,1034.42 1172.97,1034.42 \n",
       "  1175.07,1034.42 1177.18,1034.42 1179.29,1034.42 1181.39,1034.42 1183.5,1034.42 1185.61,1034.42 1187.71,1034.42 1189.82,1034.42 1191.93,1034.42 1194.04,1034.42 \n",
       "  1196.14,1034.42 1198.25,1034.42 1200.36,1034.42 1202.46,1034.42 1204.57,1034.42 1206.68,1034.42 1208.78,1034.42 1210.89,1034.42 1213,1034.42 1215.1,1034.42 \n",
       "  1217.21,1034.42 1219.32,1034.42 1221.42,1034.42 1223.53,1034.42 1225.64,1034.42 1227.74,1034.42 1229.85,1034.42 1231.96,1034.42 1234.06,1034.42 1236.17,1034.42 \n",
       "  1238.28,1034.42 1240.38,1034.42 1242.49,1034.42 1244.6,1034.42 1246.7,1034.42 1248.81,1034.42 1250.92,1034.42 1253.02,1034.42 1255.13,1034.42 1257.24,1034.42 \n",
       "  1259.34,1034.42 1261.45,1034.42 1263.56,1034.42 1265.67,1034.42 1267.77,1034.42 1269.88,1034.42 1271.99,1034.42 1274.09,1034.42 1276.2,1034.42 1278.31,1034.42 \n",
       "  1280.41,1034.42 1282.52,1034.42 1284.63,1034.42 1286.73,1034.42 1288.84,1034.42 1290.95,1034.42 1293.05,1034.42 1295.16,1034.42 1297.27,1034.42 1299.37,1034.42 \n",
       "  1301.48,1034.42 1303.59,1034.42 1305.69,1034.42 1307.8,1034.42 1309.91,1034.42 1312.01,1034.42 1314.12,1034.42 1316.23,1034.42 1318.33,1034.42 1320.44,1034.42 \n",
       "  1322.55,1034.42 1324.65,1034.42 1326.76,1034.42 1328.87,1034.42 1330.97,1034.42 1333.08,1034.42 1335.19,1034.42 1337.3,1034.42 1339.4,1034.42 1341.51,1034.42 \n",
       "  1343.62,1034.42 1345.72,1034.42 1347.83,1034.42 1349.94,1034.42 1352.04,1034.42 1354.15,1034.42 1356.26,1034.42 1358.36,1034.42 1360.47,1034.42 1362.58,1034.42 \n",
       "  1364.68,1034.42 1366.79,1034.42 1368.9,1034.42 1371,1034.42 1373.11,1034.42 1375.22,1034.42 1377.32,1034.42 1379.43,1034.42 1381.54,1034.42 1383.64,1034.42 \n",
       "  1385.75,1034.42 1387.86,1034.42 1389.96,1034.42 1392.07,1034.42 1394.18,1034.42 1396.28,1034.42 1398.39,1034.42 1400.5,1034.42 1402.6,1034.42 1404.71,1034.42 \n",
       "  1406.82,1034.42 1408.93,1034.42 1411.03,1034.42 1413.14,1034.42 1415.25,1034.42 1417.35,1034.42 1419.46,1034.42 1421.57,1034.42 1423.67,1034.42 1425.78,1034.42 \n",
       "  1427.89,1034.42 1429.99,1034.42 1432.1,1034.42 1434.21,1034.42 1436.31,1034.42 1438.42,1034.42 1440.53,1034.42 1442.63,1034.42 1444.74,1034.42 1446.85,1034.42 \n",
       "  1448.95,1034.42 1451.06,1034.42 1453.17,1034.42 1455.27,1034.42 1457.38,1034.42 1459.49,1034.42 1461.59,1034.42 1463.7,1034.42 1465.81,1034.42 1467.91,1034.42 \n",
       "  1470.02,1034.42 1472.13,1034.42 1474.23,1034.42 1476.34,1034.42 1478.45,1034.42 1480.56,1034.42 1482.66,1034.42 1484.77,1034.42 1486.88,1034.42 1488.98,1034.42 \n",
       "  1491.09,1034.42 1493.2,1034.42 1495.3,1034.42 1497.41,1034.42 1499.52,1034.42 1501.62,1034.42 1503.73,1034.42 1505.84,1034.42 1507.94,1034.42 1510.05,1034.42 \n",
       "  1512.16,1034.42 1514.26,1034.42 1516.37,1034.42 1518.48,1034.42 1520.58,1034.42 1522.69,1034.42 1524.8,1034.42 1526.9,1034.42 1529.01,1034.42 1531.12,1034.42 \n",
       "  1533.22,1034.42 1535.33,1034.42 1537.44,1034.42 1539.54,1034.42 1541.65,1034.42 1543.76,1034.42 1545.86,1034.42 1547.97,1034.42 1550.08,1034.42 1552.19,1034.42 \n",
       "  1554.29,1034.42 1556.4,1034.42 1558.51,1034.42 1560.61,1034.42 1562.72,1034.42 1564.83,1034.42 1566.93,1034.42 1569.04,1034.42 1571.15,1034.42 1573.25,1034.42 \n",
       "  1575.36,1034.42 1577.47,1034.42 1579.57,1034.42 1581.68,1034.42 1583.79,1034.42 1585.89,1034.42 1588,1034.42 1590.11,1034.42 1592.21,1034.42 1594.32,1034.42 \n",
       "  1596.43,1034.42 1598.53,1034.42 1600.64,1034.42 1602.75,1034.42 1604.85,1034.42 1606.96,1034.42 1609.07,1034.42 1611.17,1034.42 1613.28,1034.42 1615.39,1034.42 \n",
       "  1617.49,1034.42 1619.6,1034.42 1621.71,1034.42 1623.82,1034.42 1625.92,1034.42 1628.03,1034.42 1630.14,1034.42 1632.24,1034.42 1634.35,1034.42 1636.46,1034.42 \n",
       "  1638.56,1034.42 1640.67,1034.42 1642.78,1034.42 1644.88,1034.42 1646.99,1034.42 1649.1,1034.42 1651.2,1034.42 1653.31,1034.42 1655.42,1034.42 1657.52,1034.42 \n",
       "  1659.63,1034.42 1661.74,1034.42 1663.84,1034.42 1665.95,1034.42 1668.06,1034.42 1670.16,1034.42 1672.27,1034.42 1674.38,1034.42 1676.48,1034.42 1678.59,1034.42 \n",
       "  1680.7,1034.42 1682.8,1034.42 1684.91,1034.42 1687.02,1034.42 1689.12,1034.42 1691.23,1034.42 1693.34,1034.42 1695.45,1034.42 1697.55,1034.42 1699.66,1034.42 \n",
       "  1701.77,1034.42 1703.87,1034.42 1705.98,1034.42 1708.09,1034.42 1710.19,1034.42 1712.3,1034.42 1714.41,1034.42 1716.51,1034.42 1718.62,1034.42 1720.73,1034.42 \n",
       "  1722.83,1034.42 1724.94,1034.42 1727.05,1034.42 1729.15,1034.42 1731.26,1034.42 1733.37,1034.42 1735.47,1034.42 1737.58,1034.42 1739.69,1034.42 1741.79,1034.42 \n",
       "  1743.9,1034.42 1746.01,1034.42 1748.11,1034.42 1750.22,1034.42 1752.33,1034.42 1754.43,1034.42 1756.54,1034.42 1758.65,1034.42 1760.75,1034.42 1762.86,1034.42 \n",
       "  1764.97,1034.42 1767.08,1034.42 1769.18,1034.42 1771.29,1034.42 1773.4,1034.42 1775.5,1034.42 1777.61,1034.42 1779.72,1034.42 1781.82,1034.42 1783.93,1034.42 \n",
       "  1786.04,1034.42 1788.14,1034.42 1790.25,1034.42 1792.36,1034.42 1794.46,1034.42 1796.57,1034.42 1798.68,1034.42 1800.78,1034.42 1802.89,1034.42 1805,1034.42 \n",
       "  1807.1,1034.42 1809.21,1034.42 1811.32,1034.42 1813.42,1034.42 1815.53,1034.42 1817.64,1034.42 1819.74,1034.42 1821.85,1034.42 1823.96,1034.42 1826.06,1034.42 \n",
       "  1828.17,1034.42 1830.28,1034.42 1832.39,1034.42 1834.49,1034.42 1836.6,1034.42 1838.71,1034.42 1840.81,1034.42 1842.92,1034.42 1845.03,1034.42 1847.13,1034.42 \n",
       "  1849.24,1034.42 1851.35,1034.42 1853.45,1034.42 1855.56,1034.42 1857.67,1034.42 1859.77,1034.42 1861.88,1034.42 1863.99,1034.42 1866.09,1034.42 1868.2,1034.42 \n",
       "  1870.31,1034.42 1872.41,1034.42 1874.52,1034.42 1876.63,1034.42 1878.73,1034.42 1880.84,1034.42 1882.95,1034.42 1885.05,1034.42 1887.16,1034.42 1889.27,1034.42 \n",
       "  1891.37,1034.42 1893.48,1034.42 1895.59,1034.42 1897.69,1034.42 1899.8,1034.42 1901.91,1034.42 1904.02,1034.42 1906.12,1034.42 1908.23,1034.42 1910.34,1034.42 \n",
       "  1912.44,1034.42 1914.55,1034.42 1916.66,1034.42 1918.76,1034.42 1920.87,1034.42 1922.98,1034.42 1925.08,1034.42 1927.19,1034.42 1929.3,1034.42 1931.4,1034.42 \n",
       "  1933.51,1034.42 1935.62,1034.42 1937.72,1034.42 1939.83,1034.42 1941.94,1034.42 1944.04,1034.42 1946.15,1034.42 1948.26,1034.42 1950.36,1034.42 1952.47,1034.42 \n",
       "  1954.58,1034.42 1956.68,1034.42 1958.79,1034.42 1960.9,1034.42 1963,1034.42 1965.11,1034.42 1967.22,1034.42 1969.32,1034.42 1971.43,1034.42 1973.54,1034.42 \n",
       "  1975.65,1034.42 1977.75,1034.42 1979.86,1034.42 1981.97,1034.42 1984.07,1034.42 1986.18,1034.42 1988.29,1034.42 1990.39,1034.42 1992.5,1034.42 1994.61,1034.42 \n",
       "  1996.71,1034.42 1998.82,1034.42 2000.93,1034.42 2003.03,1034.42 2005.14,1034.42 2007.25,1034.42 2009.35,1034.42 2011.46,1034.42 2013.57,1034.42 2015.67,1034.42 \n",
       "  2017.78,1034.42 2019.89,1034.42 2021.99,1034.42 2024.1,1034.42 2026.21,1034.42 2028.31,1034.42 2030.42,1034.42 2032.53,1034.42 2034.63,1034.42 2036.74,1034.42 \n",
       "  2038.85,1034.42 2040.95,1034.42 2043.06,1034.42 2045.17,1034.42 2047.28,1034.42 2049.38,1034.42 2051.49,1034.42 2053.6,1034.42 2055.7,1034.42 2057.81,1034.42 \n",
       "  2059.92,1034.42 2062.02,1034.42 2064.13,1034.42 2066.24,1034.42 2068.34,1034.42 2070.45,1034.42 2072.56,1034.42 2074.66,1034.42 2076.77,1034.42 2078.88,1034.42 \n",
       "  2080.98,1034.42 2083.09,1034.42 2085.2,1034.42 2087.3,1034.42 2089.41,1034.42 2091.52,1034.42 2093.62,1034.42 2095.73,1034.42 2097.84,1034.42 2099.94,1034.42 \n",
       "  2102.05,1034.42 2104.16,1034.42 2106.26,1034.42 2108.37,1034.42 2110.48,1034.42 2112.58,1034.42 2114.69,1034.42 2116.8,1034.42 2118.91,1034.42 2121.01,1034.42 \n",
       "  2123.12,1034.42 2125.23,1034.42 2127.33,1034.42 2129.44,1034.42 2131.55,1034.42 2133.65,1034.42 2135.76,1034.42 2137.87,1034.42 2139.97,1034.42 2142.08,1034.42 \n",
       "  2144.19,1034.42 2146.29,1034.42 2148.4,1034.42 2150.51,1034.42 2152.61,1034.42 2154.72,1034.42 2156.83,1034.42 2158.93,1034.42 2161.04,1034.42 2163.15,1034.42 \n",
       "  2165.25,1034.42 2167.36,1034.42 2169.47,1034.42 2171.57,1034.42 2173.68,1034.42 2175.79,1034.42 2177.89,1034.42 2180,1034.42 2182.11,1034.42 2184.21,1034.42 \n",
       "  2186.32,1034.42 2188.43,1034.42 2190.54,1034.42 2192.64,1034.42 2194.75,1034.42 2196.86,1034.42 2198.96,1034.42 2201.07,1034.42 2203.18,1034.42 2205.28,1034.42 \n",
       "  2207.39,1034.42 2209.5,1034.42 2211.6,1034.42 2213.71,1034.42 2215.82,1034.42 2217.92,1034.42 2220.03,1034.42 2222.14,1034.42 2224.24,1034.42 2226.35,1034.42 \n",
       "  2228.46,1034.42 2230.56,1034.42 2232.67,1034.42 2234.78,1034.42 2236.88,1034.42 2238.99,1034.42 2241.1,1034.42 2243.2,1034.42 2245.31,1034.42 2247.42,1034.42 \n",
       "  2249.52,1034.42 2251.63,1034.42 2253.74,1034.42 2255.84,1034.42 2257.95,1034.42 2260.06,1034.42 2262.17,1034.42 2264.27,1034.42 2266.38,1034.42 2268.49,1034.42 \n",
       "  2270.59,1034.42 2272.7,1034.42 2274.81,1034.42 2276.91,1034.42 2279.02,1034.42 2281.13,1034.42 2283.23,1034.42 2285.34,1034.42 2287.45,1034.42 2289.55,1034.42 \n",
       "  2291.66,1034.42 2293.77,1034.42 2295.87,1034.42 2297.98,1034.42 2300.09,1034.42 2302.19,1034.42 2304.3,1034.42 2306.41,1034.42 2308.51,1034.42 2310.62,1034.42 \n",
       "  2312.73,1034.42 2314.83,1034.42 2316.94,1034.42 2319.05,1034.42 2321.15,1034.42 2323.26,1034.42 2325.37,1034.42 2327.47,1034.42 2329.58,1034.42 2331.69,1034.42 \n",
       "  2333.8,1034.42 2335.9,1034.42 2338.01,1034.42 2340.12,1034.42 2342.22,1034.42 2344.33,1034.42 2346.44,1034.42 2348.54,1034.42 2350.65,1034.42 2352.76,1034.42 \n",
       "  \n",
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       "  269.166,135.325 271.272,135.325 273.379,135.325 275.486,135.325 277.593,135.325 279.699,135.325 281.806,135.325 283.913,135.325 286.02,135.325 288.126,135.325 \n",
       "  290.233,135.325 292.34,135.325 294.447,135.325 296.553,135.325 298.66,135.325 300.767,135.325 302.874,135.325 304.981,135.325 307.087,135.325 309.194,135.325 \n",
       "  311.301,135.325 313.408,135.325 315.514,135.325 317.621,135.325 319.728,135.325 321.835,135.325 323.941,135.325 326.048,135.325 328.155,135.325 330.262,135.325 \n",
       "  332.368,135.325 334.475,135.325 336.582,135.325 338.689,135.325 340.796,135.325 342.902,135.325 345.009,135.325 347.116,135.325 349.223,135.325 351.329,135.325 \n",
       "  353.436,135.325 355.543,135.325 357.65,135.325 359.756,135.325 361.863,135.325 363.97,135.325 366.077,135.325 368.183,135.325 370.29,135.325 372.397,135.325 \n",
       "  374.504,135.325 376.611,135.325 378.717,135.325 380.824,135.325 382.931,135.325 385.038,135.325 387.144,135.325 389.251,135.325 391.358,135.325 393.465,135.325 \n",
       "  395.571,135.325 397.678,135.325 399.785,135.325 401.892,135.325 403.998,135.325 406.105,135.325 408.212,135.325 410.319,135.325 412.426,135.325 414.532,135.325 \n",
       "  416.639,135.325 418.746,135.325 420.853,135.325 422.959,135.325 425.066,135.325 427.173,135.325 429.28,135.325 431.386,135.325 433.493,135.325 435.6,135.325 \n",
       "  437.707,135.325 439.813,135.325 441.92,135.325 444.027,135.325 446.134,135.325 448.241,135.325 450.347,135.325 452.454,135.325 454.561,135.325 456.668,135.325 \n",
       "  458.774,135.325 460.881,135.325 462.988,135.325 465.095,135.325 467.201,135.325 469.308,135.325 471.415,135.325 473.522,135.325 475.628,135.325 477.735,135.325 \n",
       "  479.842,135.325 481.949,135.325 484.056,135.325 486.162,135.325 488.269,135.325 490.376,135.325 492.483,135.325 494.589,135.325 496.696,135.325 498.803,135.325 \n",
       "  500.91,135.325 503.016,135.325 505.123,135.325 507.23,135.325 509.337,135.325 511.443,135.325 513.55,135.325 515.657,135.325 517.764,135.325 519.871,135.325 \n",
       "  521.977,135.325 524.084,135.325 526.191,135.325 528.298,135.325 530.404,135.325 532.511,135.325 534.618,135.325 536.725,135.325 538.831,135.325 540.938,135.325 \n",
       "  543.045,135.325 545.152,135.325 547.258,135.325 549.365,135.325 551.472,135.325 553.579,135.325 555.686,135.325 557.792,135.325 559.899,135.325 562.006,135.325 \n",
       "  564.113,135.325 566.219,135.325 568.326,135.325 570.433,135.325 572.54,135.325 574.646,135.325 576.753,135.325 578.86,135.325 580.967,135.325 583.073,135.325 \n",
       "  585.18,135.325 587.287,135.325 589.394,135.325 591.501,135.325 593.607,135.325 595.714,135.325 597.821,135.325 599.928,135.325 602.034,135.325 604.141,135.325 \n",
       "  606.248,135.325 608.355,135.325 610.461,135.325 612.568,135.325 614.675,135.325 616.782,135.325 618.888,135.325 620.995,135.325 623.102,135.325 625.209,135.325 \n",
       "  627.316,135.325 629.422,135.325 631.529,135.325 633.636,135.325 635.743,135.325 637.849,135.325 639.956,135.325 642.063,135.325 644.17,135.325 646.276,135.325 \n",
       "  648.383,135.325 650.49,135.325 652.597,135.325 654.703,135.325 656.81,135.325 658.917,135.325 661.024,135.325 663.131,135.325 665.237,135.325 667.344,135.325 \n",
       "  669.451,135.325 671.558,135.325 673.664,135.325 675.771,135.325 677.878,135.325 679.985,135.325 682.091,135.325 684.198,135.325 686.305,135.325 688.412,135.325 \n",
       "  690.518,135.325 692.625,135.325 694.732,135.325 696.839,135.325 698.946,135.325 701.052,135.325 703.159,135.325 705.266,135.325 707.373,135.325 709.479,135.325 \n",
       "  711.586,135.325 713.693,135.325 715.8,135.325 717.906,135.325 720.013,135.325 722.12,135.325 724.227,135.325 726.333,135.325 728.44,135.325 730.547,135.325 \n",
       "  732.654,135.325 734.761,135.325 736.867,135.325 738.974,135.325 741.081,135.325 743.188,135.325 745.294,135.325 747.401,135.325 749.508,135.325 751.615,135.325 \n",
       "  753.721,135.325 755.828,135.325 757.935,135.325 760.042,135.325 762.148,135.325 764.255,135.325 766.362,135.325 768.469,135.325 770.576,135.325 772.682,135.325 \n",
       "  774.789,135.325 776.896,135.325 779.003,135.325 781.109,135.325 783.216,135.325 785.323,135.325 787.43,135.325 789.536,135.325 791.643,135.325 793.75,135.325 \n",
       "  795.857,135.325 797.963,135.325 800.07,135.325 802.177,135.325 804.284,135.325 806.391,135.325 808.497,135.325 810.604,135.325 812.711,135.325 814.818,135.325 \n",
       "  816.924,135.325 819.031,135.325 821.138,135.325 823.245,135.325 825.351,135.325 827.458,135.325 829.565,135.325 831.672,135.325 833.778,135.325 835.885,135.325 \n",
       "  837.992,135.325 840.099,135.325 842.206,135.325 844.312,135.325 846.419,135.325 848.526,135.325 850.633,135.325 852.739,135.325 854.846,135.325 856.953,135.325 \n",
       "  859.06,135.325 861.166,135.325 863.273,135.325 865.38,135.325 867.487,135.325 869.593,135.325 871.7,135.325 873.807,135.325 875.914,135.325 878.021,135.325 \n",
       "  880.127,135.325 882.234,135.325 884.341,135.325 886.448,135.325 888.554,135.325 890.661,135.325 892.768,135.325 894.875,135.325 896.981,135.325 899.088,135.325 \n",
       "  901.195,135.325 903.302,135.325 905.408,135.325 907.515,135.325 909.622,135.325 911.729,135.325 913.836,135.325 915.942,135.325 918.049,135.325 920.156,135.325 \n",
       "  922.263,135.325 924.369,135.325 926.476,135.325 928.583,135.325 930.69,135.325 932.796,135.325 934.903,135.325 937.01,135.325 939.117,135.325 941.223,135.325 \n",
       "  943.33,135.325 945.437,135.325 947.544,135.325 949.651,135.325 951.757,135.325 953.864,135.325 955.971,135.325 958.078,135.325 960.184,135.325 962.291,135.325 \n",
       "  964.398,135.325 966.505,135.325 968.611,135.325 970.718,135.325 972.825,135.325 974.932,135.325 977.038,135.325 979.145,135.325 981.252,135.325 983.359,135.325 \n",
       "  985.466,135.325 987.572,135.325 989.679,135.325 991.786,135.325 993.893,135.325 995.999,135.325 998.106,135.325 1000.21,135.325 1002.32,135.325 1004.43,135.325 \n",
       "  1006.53,135.325 1008.64,135.325 1010.75,135.325 1012.85,135.325 1014.96,135.325 1017.07,135.325 1019.17,135.325 1021.28,135.325 1023.39,135.325 1025.49,135.325 \n",
       "  1027.6,135.325 1029.71,135.325 1031.81,135.325 1033.92,135.325 1036.03,135.325 1038.13,135.325 1040.24,135.325 1042.35,135.325 1044.45,135.325 1046.56,135.325 \n",
       "  1048.67,135.325 1050.78,135.325 1052.88,135.325 1054.99,135.325 1057.1,135.325 1059.2,135.325 1061.31,135.325 1063.42,135.325 1065.52,135.325 1067.63,135.325 \n",
       "  1069.74,135.325 1071.84,135.325 1073.95,135.325 1076.06,135.325 1078.16,135.325 1080.27,135.325 1082.38,135.325 1084.48,135.325 1086.59,135.325 1088.7,135.325 \n",
       "  1090.8,135.325 1092.91,135.325 1095.02,135.325 1097.12,135.325 1099.23,135.325 1101.34,135.325 1103.44,135.325 1105.55,135.325 1107.66,135.325 1109.76,135.325 \n",
       "  1111.87,135.325 1113.98,135.325 1116.08,135.325 1118.19,135.325 1120.3,135.325 1122.41,135.325 1124.51,135.325 1126.62,135.325 1128.73,135.325 1130.83,135.325 \n",
       "  1132.94,135.325 1135.05,135.325 1137.15,135.325 1139.26,135.325 1141.37,135.325 1143.47,135.325 1145.58,135.325 1147.69,135.325 1149.79,135.325 1151.9,135.325 \n",
       "  1154.01,135.325 1156.11,135.325 1158.22,135.325 1160.33,135.325 1162.43,135.325 1164.54,135.325 1166.65,135.325 1168.75,135.325 1170.86,135.325 1172.97,135.325 \n",
       "  1175.07,135.325 1177.18,135.325 1179.29,135.325 1181.39,135.325 1183.5,135.325 1185.61,135.325 1187.71,135.325 1189.82,135.325 1191.93,135.325 1194.04,135.325 \n",
       "  1196.14,135.325 1198.25,135.325 1200.36,135.325 1202.46,135.325 1204.57,135.325 1206.68,135.325 1208.78,135.325 1210.89,135.325 1213,135.325 1215.1,135.325 \n",
       "  1217.21,135.325 1219.32,135.325 1221.42,135.325 1223.53,135.325 1225.64,135.325 1227.74,135.325 1229.85,135.325 1231.96,135.325 1234.06,135.325 1236.17,135.325 \n",
       "  1238.28,135.325 1240.38,135.325 1242.49,135.325 1244.6,135.325 1246.7,135.325 1248.81,135.325 1250.92,135.325 1253.02,135.325 1255.13,135.325 1257.24,135.325 \n",
       "  1259.34,135.325 1261.45,135.325 1263.56,135.325 1265.67,135.325 1267.77,135.325 1269.88,135.325 1271.99,135.325 1274.09,135.325 1276.2,135.325 1278.31,135.325 \n",
       "  1280.41,135.325 1282.52,135.325 1284.63,135.325 1286.73,135.325 1288.84,135.325 1290.95,135.325 1293.05,135.325 1295.16,135.325 1297.27,135.325 1299.37,135.325 \n",
       "  1301.48,135.325 1303.59,135.325 1305.69,135.325 1307.8,135.325 1309.91,135.325 1312.01,135.325 1314.12,135.325 1316.23,135.325 1318.33,135.325 1320.44,135.325 \n",
       "  1322.55,135.325 1324.65,135.325 1326.76,135.325 1328.87,135.325 1330.97,135.325 1333.08,135.325 1335.19,135.325 1337.3,135.325 1339.4,135.325 1341.51,135.325 \n",
       "  1343.62,135.325 1345.72,135.325 1347.83,135.325 1349.94,135.325 1352.04,135.325 1354.15,135.325 1356.26,135.325 1358.36,135.325 1360.47,135.325 1362.58,135.325 \n",
       "  1364.68,135.325 1366.79,135.325 1368.9,135.325 1371,135.325 1373.11,135.325 1375.22,135.325 1377.32,135.325 1379.43,135.325 1381.54,135.325 1383.64,135.325 \n",
       "  1385.75,135.325 1387.86,135.325 1389.96,135.325 1392.07,135.325 1394.18,135.325 1396.28,135.325 1398.39,135.325 1400.5,135.325 1402.6,135.325 1404.71,135.325 \n",
       "  1406.82,135.325 1408.93,135.325 1411.03,135.325 1413.14,135.325 1415.25,135.325 1417.35,135.325 1419.46,135.325 1421.57,135.325 1423.67,135.325 1425.78,135.325 \n",
       "  1427.89,135.325 1429.99,135.325 1432.1,135.325 1434.21,135.325 1436.31,135.325 1438.42,135.325 1440.53,135.325 1442.63,135.325 1444.74,135.325 1446.85,135.325 \n",
       "  1448.95,135.325 1451.06,135.325 1453.17,135.325 1455.27,135.325 1457.38,135.325 1459.49,135.325 1461.59,135.325 1463.7,135.325 1465.81,135.325 1467.91,135.325 \n",
       "  1470.02,135.325 1472.13,135.325 1474.23,135.325 1476.34,135.325 1478.45,135.325 1480.56,135.325 1482.66,135.325 1484.77,135.325 1486.88,135.325 1488.98,135.325 \n",
       "  1491.09,135.325 1493.2,135.325 1495.3,135.325 1497.41,135.325 1499.52,135.325 1501.62,135.325 1503.73,135.325 1505.84,135.325 1507.94,135.325 1510.05,135.325 \n",
       "  1512.16,135.325 1514.26,135.325 1516.37,135.325 1518.48,135.325 1520.58,135.325 1522.69,135.325 1524.8,135.325 1526.9,135.325 1529.01,135.325 1531.12,135.325 \n",
       "  1533.22,135.325 1535.33,135.325 1537.44,135.325 1539.54,135.325 1541.65,135.325 1543.76,135.325 1545.86,135.325 1547.97,135.325 1550.08,135.325 1552.19,135.325 \n",
       "  1554.29,135.325 1556.4,135.325 1558.51,135.325 1560.61,135.325 1562.72,135.325 1564.83,135.325 1566.93,135.325 1569.04,135.325 1571.15,135.325 1573.25,135.325 \n",
       "  1575.36,135.325 1577.47,135.325 1579.57,135.325 1581.68,135.325 1583.79,135.325 1585.89,135.325 1588,135.325 1590.11,135.325 1592.21,135.325 1594.32,135.325 \n",
       "  1596.43,135.325 1598.53,135.325 1600.64,135.325 1602.75,135.325 1604.85,135.325 1606.96,135.325 1609.07,135.325 1611.17,135.325 1613.28,135.325 1615.39,135.325 \n",
       "  1617.49,135.325 1619.6,135.325 1621.71,135.325 1623.82,135.325 1625.92,135.325 1628.03,135.325 1630.14,135.325 1632.24,135.325 1634.35,135.325 1636.46,135.325 \n",
       "  1638.56,135.325 1640.67,135.325 1642.78,135.325 1644.88,135.325 1646.99,135.325 1649.1,135.325 1651.2,135.325 1653.31,135.325 1655.42,135.325 1657.52,135.325 \n",
       "  1659.63,135.325 1661.74,135.325 1663.84,135.325 1665.95,135.325 1668.06,135.325 1670.16,135.325 1672.27,135.325 1674.38,135.325 1676.48,135.325 1678.59,135.325 \n",
       "  1680.7,135.325 1682.8,135.325 1684.91,135.325 1687.02,135.325 1689.12,135.325 1691.23,135.325 1693.34,135.325 1695.45,135.325 1697.55,135.325 1699.66,135.325 \n",
       "  1701.77,135.325 1703.87,135.325 1705.98,135.325 1708.09,135.325 1710.19,135.325 1712.3,135.325 1714.41,135.325 1716.51,135.325 1718.62,135.325 1720.73,135.325 \n",
       "  1722.83,135.325 1724.94,135.325 1727.05,135.325 1729.15,135.325 1731.26,135.325 1733.37,135.325 1735.47,135.325 1737.58,135.325 1739.69,135.325 1741.79,135.325 \n",
       "  1743.9,135.325 1746.01,135.325 1748.11,135.325 1750.22,135.325 1752.33,135.325 1754.43,135.325 1756.54,135.325 1758.65,135.325 1760.75,135.325 1762.86,135.325 \n",
       "  1764.97,135.325 1767.08,135.325 1769.18,135.325 1771.29,135.325 1773.4,135.325 1775.5,135.325 1777.61,135.325 1779.72,135.325 1781.82,135.325 1783.93,135.325 \n",
       "  1786.04,135.325 1788.14,135.325 1790.25,135.325 1792.36,135.325 1794.46,135.325 1796.57,135.325 1798.68,135.325 1800.78,135.325 1802.89,135.325 1805,135.325 \n",
       "  1807.1,135.325 1809.21,135.325 1811.32,135.325 1813.42,135.325 1815.53,135.325 1817.64,135.325 1819.74,135.325 1821.85,135.325 1823.96,135.325 1826.06,135.325 \n",
       "  1828.17,135.325 1830.28,135.325 1832.39,135.325 1834.49,135.325 1836.6,135.325 1838.71,135.325 1840.81,135.325 1842.92,135.325 1845.03,135.325 1847.13,135.325 \n",
       "  1849.24,135.325 1851.35,135.325 1853.45,135.325 1855.56,135.325 1857.67,135.325 1859.77,135.325 1861.88,135.325 1863.99,135.325 1866.09,135.325 1868.2,135.325 \n",
       "  1870.31,135.325 1872.41,135.325 1874.52,135.325 1876.63,135.325 1878.73,135.325 1880.84,135.325 1882.95,135.325 1885.05,135.325 1887.16,135.325 1889.27,135.325 \n",
       "  1891.37,135.325 1893.48,135.325 1895.59,135.325 1897.69,135.325 1899.8,135.325 1901.91,135.325 1904.02,135.325 1906.12,135.325 1908.23,135.325 1910.34,135.325 \n",
       "  1912.44,135.325 1914.55,135.325 1916.66,135.325 1918.76,135.325 1920.87,135.325 1922.98,135.325 1925.08,135.325 1927.19,135.325 1929.3,135.325 1931.4,135.325 \n",
       "  1933.51,135.325 1935.62,135.325 1937.72,135.325 1939.83,135.325 1941.94,135.325 1944.04,135.325 1946.15,135.325 1948.26,135.325 1950.36,135.325 1952.47,135.325 \n",
       "  1954.58,135.325 1956.68,135.325 1958.79,135.325 1960.9,135.325 1963,135.325 1965.11,135.325 1967.22,135.325 1969.32,135.325 1971.43,135.325 1973.54,135.325 \n",
       "  1975.65,135.325 1977.75,135.325 1979.86,135.325 1981.97,135.325 1984.07,135.325 1986.18,135.325 1988.29,135.325 1990.39,135.325 1992.5,135.325 1994.61,135.325 \n",
       "  1996.71,135.325 1998.82,135.325 2000.93,135.325 2003.03,135.325 2005.14,135.325 2007.25,135.325 2009.35,135.325 2011.46,135.325 2013.57,135.325 2015.67,135.325 \n",
       "  2017.78,135.325 2019.89,135.325 2021.99,135.325 2024.1,135.325 2026.21,135.325 2028.31,135.325 2030.42,135.325 2032.53,135.325 2034.63,135.325 2036.74,135.325 \n",
       "  2038.85,135.325 2040.95,135.325 2043.06,135.325 2045.17,135.325 2047.28,135.325 2049.38,135.325 2051.49,135.325 2053.6,135.325 2055.7,135.325 2057.81,135.325 \n",
       "  2059.92,135.325 2062.02,135.325 2064.13,135.325 2066.24,135.325 2068.34,135.325 2070.45,135.325 2072.56,135.325 2074.66,135.325 2076.77,135.325 2078.88,135.325 \n",
       "  2080.98,135.325 2083.09,135.325 2085.2,135.325 2087.3,135.325 2089.41,135.325 2091.52,135.325 2093.62,135.325 2095.73,135.325 2097.84,135.325 2099.94,135.325 \n",
       "  2102.05,135.325 2104.16,135.325 2106.26,135.325 2108.37,135.325 2110.48,135.325 2112.58,135.325 2114.69,135.325 2116.8,135.325 2118.91,135.325 2121.01,135.325 \n",
       "  2123.12,135.325 2125.23,135.325 2127.33,135.325 2129.44,135.325 2131.55,135.325 2133.65,135.325 2135.76,135.325 2137.87,135.325 2139.97,135.325 2142.08,135.325 \n",
       "  2144.19,135.325 2146.29,135.325 2148.4,135.325 2150.51,135.325 2152.61,135.325 2154.72,135.325 2156.83,135.325 2158.93,135.325 2161.04,135.325 2163.15,135.325 \n",
       "  2165.25,135.325 2167.36,135.325 2169.47,135.325 2171.57,135.325 2173.68,135.325 2175.79,135.325 2177.89,135.325 2180,135.325 2182.11,135.325 2184.21,135.325 \n",
       "  2186.32,135.325 2188.43,135.325 2190.54,135.325 2192.64,135.325 2194.75,135.325 2196.86,135.325 2198.96,135.325 2201.07,135.325 2203.18,135.325 2205.28,135.325 \n",
       "  2207.39,135.325 2209.5,135.325 2211.6,135.325 2213.71,135.325 2215.82,135.325 2217.92,135.325 2220.03,135.325 2222.14,135.325 2224.24,135.325 2226.35,135.325 \n",
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "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",
    "# A COMPLETER/MODIFIER\n",
    "p2 = plot()\n",
    "p3 = plot()\n",
    "\n",
    "#\n",
    "T = t0:(tf-t0)/100:tf\n",
    "sinT = sin.(T)                                # opération vectorielle\n",
    "cosT = cos.(T)\n",
    "p4 = plot(T,[(-1/25)*(13*sinT+9*cosT) (-1/25)*(3*sinT+4*cosT)], label = [\"x_1(t)\" \"x_2(t)\"], title = \"Solution exacte\")\n",
    "plot(p1,p2,p3,p4, size=(1200, 600))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pendule simple\n",
    "### Introduction\n",
    "On s'intéresse ici au pendule simple. Les principes physiques de la mécanique classique donnent comme équation qui régit l'évolution du mouvement\n",
    "\n",
    "$$ ml^2\\ddot{\\alpha}(t)+mlg\\sin(\\alpha(t)) +k\\dot{\\alpha}(t)=0,$$\n",
    "où $\\ddot{\\alpha}(t)$ désigne la dérivée seconde de l'angle $\\alpha$ par rapport au temps $t$. \n",
    "\n",
    "- En prenant comme variable d'état qui décrit le système $x(t)=(x_1(t),x_2(t))=(\\alpha(t), \\dot{\\alpha}(t))$, écrire la fonction $f$ qui permet d'écrire l'équation différentielle sous la forme $\\dot{x}(t) = f(t,x(t))$\n",
    "- Coder cette fonction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "function pendule(x,p,t)\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": 9,
   "metadata": {},
   "outputs": [
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       "  1942.88,86.1857 1945.05,86.1857 1947.22,86.1857 1949.39,86.1857 1951.56,86.1857 1953.72,86.1857 1955.89,86.1857 1958.06,86.1857 1960.23,86.1857 1962.4,86.1857 \n",
       "  1964.57,86.1857 1966.74,86.1857 1968.9,86.1857 1971.07,86.1857 1973.24,86.1857 1975.41,86.1857 1977.58,86.1857 1979.75,86.1857 1981.92,86.1857 1984.09,86.1857 \n",
       "  1986.25,86.1857 1988.42,86.1857 1990.59,86.1857 1992.76,86.1857 1994.93,86.1857 1997.1,86.1857 1999.27,86.1857 2001.43,86.1857 2003.6,86.1857 2005.77,86.1857 \n",
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       "  1942.88,1384.24 1945.05,1384.24 1947.22,1384.24 1949.39,1384.24 1951.56,1384.24 1953.72,1384.24 1955.89,1384.24 1958.06,1384.24 1960.23,1384.24 1962.4,1384.24 \n",
       "  1964.57,1384.24 1966.74,1384.24 1968.9,1384.24 1971.07,1384.24 1973.24,1384.24 1975.41,1384.24 1977.58,1384.24 1979.75,1384.24 1981.92,1384.24 1984.09,1384.24 \n",
       "  1986.25,1384.24 1988.42,1384.24 1990.59,1384.24 1992.76,1384.24 1994.93,1384.24 1997.1,1384.24 1999.27,1384.24 2001.43,1384.24 2003.6,1384.24 2005.77,1384.24 \n",
       "  2007.94,1384.24 2010.11,1384.24 2012.28,1384.24 2014.45,1384.24 2016.62,1384.24 2018.78,1384.24 2020.95,1384.24 2023.12,1384.24 2025.29,1384.24 2027.46,1384.24 \n",
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Main\n",
    "#\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 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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Diagrammes de phases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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   "source": [
    "# Main\n",
    "#\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, 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, color=\"blue\")  # lw = linewidth\n",
    "end\n",
    "\n",
    "theta0 = pi-10*eps()\n",
    "x0 = [theta0 0]\n",
    "tf = 50                              # problem for tf=50 1/4 of the period!\n",
    "tspan = (0.0,tf)\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, color=\"green\")  # lw = linewidth\n",
    "\n",
    "theta0 = pi+10*eps()\n",
    "x0 = [theta0 0]\n",
    "tf = 50\n",
    "tspan = (0.0,tf)\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, color=\"green\")  # lw = linewidth\n",
    "\n",
    "# circulation case\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, 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, color=\"red\")  # lw = linewidth\n",
    "end\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, 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, 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), size=(900, 600))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Interprétez ces graphiques\n",
    "- On considère le cas où $k=0.15$ (on introduit un frotement). Que se passe-t-til?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modèle de Van der Pol\n",
    "### Exemple de cycle limite\n",
    "L'équation différentielle considérée est l'équation de Van der Pol\n",
    "$$(IVP)\\left\\{\\begin{array}{l}\n",
    "\\dot{y}_1(t)=y_2(t)\\\\\n",
    "\\dot{y}_2(t)=(1-y_1^2(t))y_2(t)-y_1(t)\\\\\n",
    "y_1(0)=2.00861986087484313650940188\\\\\n",
    "y_2(0)=0\n",
    "\\end{array}\\right.\n",
    "$$\n",
    "$t_f=T=6.6632868593231301896996820305$\n",
    "\n",
    "\n",
    "La solution de ce problème de Cauchy est périodique de période $T$.\n",
    "\n",
    "Résoudre et de visualiser la solution pour les points de départ :\n",
    "- x0 = [2.00861986087484313650940188,0]\n",
    "- [x01 , 0] pour x01 dans -2:0.4:0\n",
    "- [0 , x02] pour x02 dans 2:1:4\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "function vdp(x,p,t)\n",
    "# Van der Pol model\n",
    "# second member of the IVP\n",
    "# x : state\n",
    "#     real(2)\n",
    "# p : parameter vector\n",
    "# t : time, variable not use here\n",
    "#     real\n",
    "# Output\n",
    "# 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",
    "    return xpoint\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x0 = [2.0086198608748433, 0.0], p = [1], tspan = (0.0, 6.66328685932313)\n"
     ]
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     "execution_count": 16,
     "metadata": {},
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    }
   ],
   "source": [
    "# Main\n",
    "#\n",
    "t0 = 0.;\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, 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",
    "\n",
    "# Trajecctoires extérieures\n",
    "for x02 in 2.5:0.5:4\n",
    "    x0 = [0.,x02]\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), size=(900, 600))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Flots en dimension 2\n",
    "\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",
    "sous les contraintes\n",
    "\n",
    "$$ \\dot{x}(t) = -x(t) + u(t)$$\n",
    "\n",
    "$$ x(0)=x_0=-1,\\quad x(1)=0. $$\n",
    "\n",
    "Le but est ici de tracer les extrémales et le flot associé pour des points initiaux $z(0) = (x_0, p(0))$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plot_traj (generic function with 1 method)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function plot_traj(plt,f,Z0,tf; par=[])\n",
    "    \"\"\" \n",
    "    Plot the trajectories in the phase plan of the ode dot{x}(t)=f(t,z(t))\n",
    "    The initial points are in the matrix `Z0`\n",
    "    -------\n",
    "    \n",
    "    parameters (input)\n",
    "    ------------------\n",
    "    plt : plot object\n",
    "    f  : function f(x,p,t)\n",
    "    Z0 : array of float(n,nbpoints)\n",
    "         Z0[:,i] = initial point number i\n",
    "    returns\n",
    "    -------\n",
    "    nothing\n",
    "    \"\"\"\n",
    "    nbpoints = size(Z0,2)\n",
    "    tspan = (0. , tf)\n",
    "    for i in 1:nbpoints                      \n",
    "        prob = ODEProblem(f,Z0[:,i],tspan,par)          # défini le problème en Julia\n",
    "        sol = solve(prob)\n",
    "        plot!(plt, sol, vars = (1,2),legend = false, color = :blue) # lw = linewidth\n",
    "     end\n",
    "    return nothing\n",
    "end   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "function plot_flow(plt,f,Z0,tf; par=[])\n",
    "    \"\"\" \n",
    "    plot the flows of the ode dot{x}(t)=f(t,z(t))\n",
    "    The initial points are in the matrix `Z0`\n",
    "    -------\n",
    "    \n",
    "    parameters (input)\n",
    "    ------------------\n",
    "    plt : plot object\n",
    "    f  : function f(x,p,t)\n",
    "    Z0 : array of float(n,nbpoints)\n",
    "         Z0[:,i] = initial point number i\n",
    "    returns\n",
    "    -------\n",
    "    nothing\n",
    "    \"\"\"\n",
    "    nbpoints = size(Z0,2)\n",
    "    tspan = (0. , tf)\n",
    "    T = range(0.,tf,length=4)\n",
    "    for i in 1:nbpoints                      \n",
    "        # TO COMPLETE\n",
    "        \n",
    "    end\n",
    "    return nothing\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Compléter la fonction suivante pour tracer le flot pour 4 valeurs de $t$\n",
    "\n",
    "Remarque : `sol(t)` donne la valeur de l'approximation de la solution à l'instant $t$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
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       "  876.716,953.744 876.836,953.931 876.956,954.118 877.075,954.306 877.193,954.494 877.31,954.682 877.427,954.87 877.543,955.058 877.658,955.247 877.773,955.435 \n",
       "  877.887,955.624 878,955.813 878.112,956.002 878.224,956.192 878.334,956.382 878.445,956.571 878.554,956.761 878.663,956.952 878.77,957.142 878.878,957.333 \n",
       "  878.984,957.523 879.09,957.714 879.195,957.905 879.299,958.097 879.403,958.288 879.505,958.48 879.607,958.672 879.709,958.864 879.809,959.056 879.909,959.249 \n",
       "  880.008,959.442 880.107,959.634 880.204,959.828 880.301,960.021 880.398,960.214 880.493,960.408 880.588,960.602 880.682,960.796 880.775,960.99 880.868,961.185 \n",
       "  880.959,961.379 881.05,961.574 881.141,961.769 881.23,961.964 881.319,962.16 881.407,962.355 881.495,962.551 881.582,962.747 881.668,962.943 881.753,963.14 \n",
       "  881.837,963.336 881.921,963.533 882.004,963.73 882.086,963.927 882.168,964.125 882.249,964.322 882.329,964.52 882.408,964.718 882.487,964.916 882.565,965.114 \n",
       "  882.642,965.313 882.719,965.512 882.794,965.711 882.87,965.91 882.944,966.109 883.017,966.309 883.09,966.509 883.162,966.708 883.234,966.909 883.304,967.109 \n",
       "  883.374,967.31 883.443,967.51 883.512,967.711 883.58,967.912 883.647,968.114 883.713,968.315 883.778,968.517 883.843,968.719 883.907,968.921 883.971,969.124 \n",
       "  884.033,969.326 884.095,969.529 884.157,969.732 884.217,969.935 884.277,970.139 884.336,970.342 884.394,970.546 884.452,970.75 884.508,970.954 884.564,971.159 \n",
       "  884.62,971.363 884.674,971.568 884.728,971.773 884.782,971.978 884.834,972.184 884.886,972.389 884.937,972.595 884.987,972.801 885.037,973.008 885.085,973.214 \n",
       "  885.133,973.421 885.181,973.628 885.227,973.835 885.273,974.042 885.319,974.25 885.363,974.457 885.407,974.665 885.45,974.873 885.492,975.082 885.534,975.29 \n",
       "  885.574,975.499 885.614,975.708 885.654,975.917 885.693,976.127 885.73,976.336 885.768,976.546 885.804,976.756 885.84,976.966 885.875,977.177 885.909,977.387 \n",
       "  885.943,977.598 885.976,977.809 886.008,978.02 886.039,978.232 886.07,978.444 886.1,978.656 886.129,978.868 886.157,979.08 886.185,979.293 886.212,979.505 \n",
       "  886.238,979.718 886.264,979.932 886.289,980.145 886.313,980.359 886.336,980.572 886.359,980.786 886.381,981.001 886.402,981.215 886.423,981.43 886.443,981.645 \n",
       "  886.462,981.86 886.48,982.075 886.498,982.291 886.514,982.507 886.531,982.723 886.546,982.939 886.561,983.155 886.575,983.372 886.588,983.589 886.6,983.806 \n",
       "  886.612,984.023 886.623,984.24 886.634,984.458 886.643,984.676 886.652,984.894 886.66,985.113 886.668,985.331 886.674,985.55 886.68,985.769 886.686,985.988 \n",
       "  886.69,986.208 886.694,986.428 886.697,986.647 886.7,986.868 886.701,987.088 886.702,987.309 886.702,987.529 886.702,987.75 886.701,987.972 886.699,988.193 \n",
       "  886.696,988.415 886.692,988.637 886.688,988.859 886.683,989.081 886.678,989.304 886.671,989.526 886.664,989.749 886.657,989.973 886.648,990.196 886.639,990.42 \n",
       "  886.629,990.644 886.618,990.868 886.607,991.092 886.595,991.317 886.582,991.542 886.568,991.767 886.554,991.992 886.539,992.217 886.523,992.443 886.507,992.669 \n",
       "  886.489,992.895 886.471,993.122 886.453,993.348 886.433,993.575 886.413,993.802 886.392,994.029 886.371,994.257 886.349,994.485 886.326,994.713 886.302,994.941 \n",
       "  886.277,995.169 886.252,995.398 886.226,995.627 886.2,995.856 886.172,996.085 886.144,996.315 886.115,996.545 886.086,996.775 886.055,997.005 886.024,997.236 \n",
       "  885.993,997.466 885.96,997.697 885.927,997.928 885.893,998.16 885.859,998.392 885.823,998.623 885.787,998.856 885.75,999.088 885.713,999.321 885.675,999.553 \n",
       "  885.636,999.786 885.596,1000.02 885.555,1000.25 885.514,1000.49 885.472,1000.72 885.43,1000.96 885.386,1001.19 885.342,1001.42 885.298,1001.66 885.252,1001.89 \n",
       "  885.206,1002.13 885.159,1002.37 885.111,1002.6 885.063,1002.84 885.013,1003.07 884.964,1003.31 884.913,1003.55 884.862,1003.78 884.81,1004.02 884.757,1004.26 \n",
       "  884.703,1004.5 884.649,1004.73 884.594,1004.97 884.538,1005.21 884.482,1005.45 884.425,1005.69 884.367,1005.93 884.308,1006.17 884.249,1006.41 884.189,1006.65 \n",
       "  884.128,1006.89 884.067,1007.13 884.004,1007.37 883.941,1007.61 883.878,1007.85 883.813,1008.09 883.748,1008.33 883.682,1008.58 883.615,1008.82 883.548,1009.06 \n",
       "  883.48,1009.3 883.411,1009.55 883.342,1009.79 883.271,1010.03 883.2,1010.28 883.129,1010.52 883.056,1010.76 882.983,1011.01 882.909,1011.25 882.835,1011.5 \n",
       "  882.759,1011.74 882.683,1011.99 882.606,1012.23 882.529,1012.48 882.45,1012.72 882.371,1012.97 882.292,1013.22 882.211,1013.46 882.13,1013.71 882.048,1013.96 \n",
       "  881.965,1014.21 881.882,1014.45 881.798,1014.7 881.713,1014.95 881.628,1015.2 881.541,1015.45 881.454,1015.7 881.366,1015.94 881.278,1016.19 881.189,1016.44 \n",
       "  881.099,1016.69 881.008,1016.94 880.917,1017.19 880.824,1017.45 880.732,1017.7 880.638,1017.95 880.544,1018.2 880.449,1018.45 880.353,1018.7 880.256,1018.96 \n",
       "  880.159,1019.21 880.061,1019.46 879.962,1019.71 879.863,1019.97 879.762,1020.22 879.662,1020.47 879.56,1020.73 879.457,1020.98 879.354,1021.24 879.251,1021.49 \n",
       "  879.146,1021.75 879.041,1022 878.935,1022.26 878.828,1022.51 878.72,1022.77 878.612,1023.03 878.503,1023.28 878.393,1023.54 878.283,1023.8 878.172,1024.05 \n",
       "  878.06,1024.31 877.947,1024.57 877.834,1024.83 877.719,1025.09 877.605,1025.34 877.489,1025.6 877.373,1025.86 877.256,1026.12 877.138,1026.38 877.019,1026.64 \n",
       "  876.9,1026.9 876.78,1027.16 876.659,1027.42 876.538,1027.68 876.416,1027.94 876.293,1028.21 876.169,1028.47 876.045,1028.73 875.919,1028.99 875.793,1029.25 \n",
       "  875.667,1029.52 875.54,1029.78 875.411,1030.04 875.283,1030.31 875.153,1030.57 875.023,1030.84 874.892,1031.1 874.76,1031.36 874.627,1031.63 874.494,1031.89 \n",
       "  874.36,1032.16 874.225,1032.43 874.09,1032.69 873.954,1032.96 873.817,1033.22 873.679,1033.49 873.541,1033.76 873.402,1034.03 873.262,1034.29 873.121,1034.56 \n",
       "  872.98,1034.83 872.837,1035.1 872.695,1035.37 872.551,1035.63 872.407,1035.9 872.262,1036.17 872.116,1036.44 871.969,1036.71 871.822,1036.98 871.674,1037.25 \n",
       "  871.525,1037.52 871.376,1037.8 871.225,1038.07 871.074,1038.34 870.923,1038.61 870.77,1038.88 870.617,1039.16 870.463,1039.43 870.308,1039.7 870.153,1039.97 \n",
       "  869.997,1040.25 869.84,1040.52 869.682,1040.8 869.524,1041.07 869.365,1041.34 869.205,1041.62 869.044,1041.89 868.883,1042.17 868.721,1042.45 868.558,1042.72 \n",
       "  868.394,1043 868.23,1043.28 868.065,1043.55 867.899,1043.83 867.732,1044.11 867.565,1044.38 867.397,1044.66 867.228,1044.94 867.059,1045.22 866.888,1045.5 \n",
       "  866.717,1045.78 866.545,1046.06 866.373,1046.34 866.2,1046.62 866.025,1046.9 865.851,1047.18 865.675,1047.46 865.499,1047.74 865.322,1048.02 865.144,1048.3 \n",
       "  864.966,1048.58 864.786,1048.86 864.606,1049.15 864.426,1049.43 864.244,1049.71 864.062,1050 863.879,1050.28 863.695,1050.56 863.511,1050.85 863.325,1051.13 \n",
       "  863.139,1051.42 862.953,1051.7 862.765,1051.99 862.577,1052.27 862.388,1052.56 862.198,1052.84 862.008,1053.13 861.817,1053.42 861.625,1053.71 861.432,1053.99 \n",
       "  861.238,1054.28 861.044,1054.57 860.849,1054.86 860.653,1055.14 860.457,1055.43 860.26,1055.72 860.062,1056.01 859.863,1056.3 859.663,1056.59 859.463,1056.88 \n",
       "  859.262,1057.17 859.06,1057.46 858.858,1057.75 858.655,1058.04 858.451,1058.34 858.246,1058.63 858.04,1058.92 857.834,1059.21 857.627,1059.51 857.419,1059.8 \n",
       "  857.211,1060.09 857.001,1060.39 856.791,1060.68 856.581,1060.97 856.369,1061.27 856.157,1061.56 855.944,1061.86 855.73,1062.15 855.515,1062.45 855.3,1062.75 \n",
       "  855.084,1063.04 854.867,1063.34 854.649,1063.64 854.431,1063.93 854.212,1064.23 853.992,1064.53 853.771,1064.83 853.55,1065.13 853.328,1065.42 853.105,1065.72 \n",
       "  852.881,1066.02 852.657,1066.32 852.432,1066.62 852.206,1066.92 851.979,1067.22 851.751,1067.52 851.523,1067.82 851.294,1068.13 851.064,1068.43 850.834,1068.73 \n",
       "  850.603,1069.03 850.371,1069.33 850.138,1069.64 849.904,1069.94 849.67,1070.24 849.435,1070.55 849.199,1070.85 848.962,1071.16 848.725,1071.46 848.487,1071.77 \n",
       "  848.248,1072.07 848.008,1072.38 847.768,1072.68 847.527,1072.99 847.285,1073.3 847.042,1073.6 846.799,1073.91 846.554,1074.22 846.309,1074.53 846.064,1074.83 \n",
       "  845.817,1075.14 845.57,1075.45 845.322,1075.76 845.073,1076.07 844.823,1076.38 844.573,1076.69 844.322,1077 844.07,1077.31 843.817,1077.62 843.564,1077.93 \n",
       "  843.309,1078.24 843.054,1078.56 842.799,1078.87 842.542,1079.18 842.285,1079.49 842.027,1079.81 841.768,1080.12 841.508,1080.43 841.248,1080.75 840.987,1081.06 \n",
       "  840.725,1081.38 840.462,1081.69 840.199,1082.01 839.934,1082.32 839.669,1082.64 839.404,1082.96 839.137,1083.27 838.87,1083.59 838.602,1083.91 838.333,1084.22 \n",
       "  838.063,1084.54 837.793,1084.86 837.521,1085.18 837.249,1085.5 836.977,1085.82 836.703,1086.14 836.429,1086.46 836.154,1086.78 835.878,1087.1 835.601,1087.42 \n",
       "  835.324,1087.74 835.046,1088.06 834.767,1088.38 834.487,1088.7 834.207,1089.02 833.925,1089.35 833.643,1089.67 833.36,1089.99 833.077,1090.32 832.792,1090.64 \n",
       "  832.507,1090.97 832.221,1091.29 831.935,1091.62 831.647,1091.94 831.359,1092.27 831.07,1092.59 830.78,1092.92 830.489,1093.24 830.198,1093.57 829.905,1093.9 \n",
       "  829.612,1094.23 829.319,1094.55 829.024,1094.88 828.729,1095.21 828.433,1095.54 828.136,1095.87 827.838,1096.2 827.54,1096.53 827.24,1096.86 826.94,1097.19 \n",
       "  826.64,1097.52 826.338,1097.85 826.035,1098.18 825.732,1098.51 825.428,1098.85 825.124,1099.18 824.818,1099.51 824.512,1099.85 824.205,1100.18 823.897,1100.51 \n",
       "  823.588,1100.85 823.279,1101.18 822.968,1101.52 822.657,1101.85 822.345,1102.19 822.033,1102.52 821.719,1102.86 821.405,1103.2 821.09,1103.53 820.774,1103.87 \n",
       "  820.458,1104.21 820.14,1104.54 819.822,1104.88 819.503,1105.22 819.183,1105.56 818.863,1105.9 818.541,1106.24 818.219,1106.58 817.896,1106.92 817.573,1107.26 \n",
       "  817.248,1107.6 816.923,1107.94 816.597,1108.28 816.27,1108.63 815.942,1108.97 815.614,1109.31 815.284,1109.65 814.954,1110 814.623,1110.34 814.292,1110.68 \n",
       "  813.959,1111.03 813.626,1111.37 813.292,1111.72 812.957,1112.06 812.621,1112.41 812.285,1112.76 811.947,1113.1 811.609,1113.45 811.27,1113.8 810.931,1114.14 \n",
       "  810.59,1114.49 810.249,1114.84 809.907,1115.19 809.564,1115.54 809.22,1115.89 808.876,1116.24 808.531,1116.59 808.184,1116.94 807.838,1117.29 807.49,1117.64 \n",
       "  807.141,1117.99 806.792,1118.34 806.442,1118.69 806.091,1119.04 805.739,1119.4 805.387,1119.75 805.033,1120.1 804.679,1120.46 804.324,1120.81 803.969,1121.17 \n",
       "  803.612,1121.52 803.255,1121.88 802.896,1122.23 802.537,1122.59 802.178,1122.94 801.817,1123.3 801.456,1123.66 801.093,1124.02 800.73,1124.37 800.366,1124.73 \n",
       "  800.002,1125.09 799.636,1125.45 799.27,1125.81 798.903,1126.17 798.535,1126.53 798.166,1126.89 797.797,1127.25 797.426,1127.61 797.055,1127.97 796.683,1128.33 \n",
       "  796.31,1128.69 795.937,1129.06 795.562,1129.42 795.187,1129.78 794.811,1130.15 794.434,1130.51 794.056,1130.87 793.678,1131.24 793.298,1131.6 792.918,1131.97 \n",
       "  792.537,1132.33 792.155,1132.7 791.773,1133.07 791.389,1133.43 791.005,1133.8 790.62,1134.17 790.234,1134.54 789.847,1134.9 789.46,1135.27 789.071,1135.64 \n",
       "  788.682,1136.01 788.292,1136.38 787.901,1136.75 787.51,1137.12 787.117,1137.49 786.724,1137.86 786.33,1138.23 785.935,1138.61 785.539,1138.98 785.142,1139.35 \n",
       "  784.745,1139.72 784.347,1140.1 783.948,1140.47 783.548,1140.85 783.147,1141.22 782.745,1141.6 782.343,1141.97 781.94,1142.35 781.536,1142.72 781.131,1143.1 \n",
       "  780.725,1143.48 780.318,1143.85 779.911,1144.23 779.503,1144.61 779.094,1144.99 778.684,1145.37 778.273,1145.74 777.861,1146.12 777.449,1146.5 777.036,1146.88 \n",
       "  776.622,1147.27 776.207,1147.65 775.791,1148.03 775.374,1148.41 774.957,1148.79 774.539,1149.17 774.12,1149.56 773.7,1149.94 773.279,1150.32 772.857,1150.71 \n",
       "  772.435,1151.09 772.012,1151.48 771.588,1151.86 771.163,1152.25 770.737,1152.63 770.31,1153.02 769.883,1153.41 769.454,1153.79 769.025,1154.18 768.595,1154.57 \n",
       "  768.164,1154.96 767.733,1155.35 767.3,1155.74 766.867,1156.13 766.432,1156.51 765.997,1156.91 765.561,1157.3 765.125,1157.69 764.687,1158.08 764.248,1158.47 \n",
       "  763.809,1158.86 763.369,1159.25 762.928,1159.65 762.486,1160.04 762.043,1160.44 761.6,1160.83 761.155,1161.22 760.71,1161.62 760.264,1162.01 759.817,1162.41 \n",
       "  759.369,1162.81 758.921,1163.2 758.471,1163.6 758.021,1164 757.569,1164.39 757.117,1164.79 756.664,1165.19 756.211,1165.59 755.756,1165.99 755.3,1166.39 \n",
       "  754.844,1166.79 754.387,1167.19 753.929,1167.59 753.47,1167.99 753.01,1168.39 752.549,1168.8 752.088,1169.2 751.625,1169.6 751.162,1170.01 750.698,1170.41 \n",
       "  750.233,1170.81 749.767,1171.22 749.301,1171.62 748.833,1172.03 748.365,1172.43 747.895,1172.84 747.425,1173.25 746.954,1173.65 746.482,1174.06 746.01,1174.47 \n",
       "  745.536,1174.88 745.061,1175.29 744.586,1175.69 744.11,1176.1 743.633,1176.51 743.155,1176.92 742.676,1177.34 742.196,1177.75 741.716,1178.16 741.234,1178.57 \n",
       "  740.752,1178.98 740.269,1179.39 739.785,1179.81 739.3,1180.22 738.814,1180.64 738.328,1181.05 737.84,1181.46 737.352,1181.88 736.862,1182.3 736.372,1182.71 \n",
       "  735.881,1183.13 735.389,1183.55 734.897,1183.96 734.403,1184.38 733.908,1184.8 733.413,1185.22 732.917,1185.64 732.419,1186.06 731.921,1186.48 731.422,1186.9 \n",
       "  730.923,1187.32 730.422,1187.74 729.92,1188.16 729.418,1188.58 728.915,1189 728.41,1189.43 727.905,1189.85 727.399,1190.27 726.892,1190.7 726.385,1191.12 \n",
       "  725.876,1191.55 725.366,1191.97 724.856,1192.4 724.345,1192.82 723.832,1193.25 723.319,1193.68 722.805,1194.1 722.29,1194.53 721.775,1194.96 721.258,1195.39 \n",
       "  720.741,1195.82 720.222,1196.25 719.703,1196.68 719.183,1197.11 718.661,1197.54 718.139,1197.97 717.617,1198.4 717.093,1198.84 716.568,1199.27 716.043,1199.7 \n",
       "  715.516,1200.13 714.989,1200.57 714.46,1201 713.931,1201.44 713.401,1201.87 712.87,1202.31 712.338,1202.74 711.806,1203.18 711.272,1203.62 710.737,1204.06 \n",
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       "  891.112,638.587 892.252,638.459 893.393,638.33 894.532,638.201 895.671,638.073 896.809,637.944 897.946,637.815 899.082,637.685 900.218,637.556 901.353,637.426 \n",
       "  902.488,637.297 903.622,637.167 904.755,637.037 905.887,636.907 907.018,636.777 908.149,636.647 909.279,636.516 910.409,636.386 911.538,636.255 912.666,636.124 \n",
       "  913.793,635.994 914.92,635.862 916.046,635.731 917.171,635.6 918.296,635.469 919.42,635.337 920.543,635.205 921.665,635.073 922.787,634.941 923.908,634.809 \n",
       "  925.029,634.677 926.148,634.545 927.267,634.412 928.386,634.28 929.504,634.147 930.621,634.014 931.737,633.881 932.853,633.748 933.968,633.614 935.082,633.481 \n",
       "  936.195,633.347 937.308,633.214 938.421,633.08 939.532,632.946 940.643,632.812 941.753,632.678 942.863,632.543 943.972,632.409 945.08,632.274 946.188,632.139 \n",
       "  947.295,632.004 948.401,631.869 949.507,631.734 950.612,631.599 951.716,631.463 952.819,631.328 953.922,631.192 955.025,631.056 956.126,630.92 957.227,630.784 \n",
       "  958.328,630.648 959.427,630.511 960.526,630.375 961.625,630.238 962.722,630.101 963.819,629.965 964.916,629.827 966.012,629.69 967.107,629.553 968.201,629.415 \n",
       "  969.295,629.278 970.388,629.14 971.481,629.002 972.573,628.864 973.664,628.726 974.755,628.587 975.845,628.449 976.934,628.31 978.023,628.172 979.111,628.033 \n",
       "  980.198,627.894 981.285,627.755 982.371,627.615 983.457,627.476 984.542,627.336 985.626,627.197 986.71,627.057 987.793,626.917 988.875,626.777 989.957,626.636 \n",
       "  991.038,626.496 992.119,626.355 993.199,626.215 994.278,626.074 995.357,625.933 996.435,625.792 997.513,625.65 998.589,625.509 999.666,625.367 1000.74,625.226 \n",
       "  1001.82,625.084 1002.89,624.942 1003.96,624.8 1005.04,624.658 1006.11,624.515 1007.18,624.373 1008.25,624.23 1009.32,624.087 1010.39,623.944 1011.46,623.801 \n",
       "  1012.53,623.658 1013.6,623.514 1014.67,623.371 1015.74,623.227 1016.8,623.083 1017.87,622.939 1018.93,622.795 1020,622.651 1021.06,622.506 1022.13,622.362 \n",
       "  1023.19,622.217 1024.25,622.072 1025.31,621.927 1026.38,621.782 1027.44,621.637 1028.5,621.492 1029.56,621.346 1030.62,621.2 1031.67,621.054 1032.73,620.908 \n",
       "  1033.79,620.762 1034.85,620.616 1035.9,620.469 1036.96,620.323 1038.01,620.176 1039.07,620.029 1040.12,619.882 1041.17,619.735 1042.23,619.588 1043.28,619.44 \n",
       "  1044.33,619.293 1045.38,619.145 1046.43,618.997 1047.48,618.849 1048.53,618.701 1049.58,618.552 1050.63,618.404 1051.67,618.255 1052.72,618.106 1053.77,617.957 \n",
       "  1054.81,617.808 1055.86,617.659 1056.9,617.509 1057.95,617.36 1058.99,617.21 1060.03,617.06 1061.08,616.91 1062.12,616.76 1063.16,616.61 1064.2,616.459 \n",
       "  1065.24,616.309 1066.28,616.158 1067.32,616.007 1068.36,615.856 1069.4,615.705 1070.43,615.553 1071.47,615.402 1072.51,615.25 1073.54,615.098 1074.58,614.946 \n",
       "  1075.61,614.794 1076.65,614.642 1077.68,614.49 1078.72,614.337 1079.75,614.184 1080.78,614.031 1081.81,613.878 1082.84,613.725 1083.87,613.572 1084.91,613.418 \n",
       "  1085.93,613.265 1086.96,613.111 1087.99,612.957 1089.02,612.803 1090.05,612.648 1091.07,612.494 1092.1,612.339 1093.13,612.185 1094.15,612.03 1095.18,611.875 \n",
       "  1096.2,611.72 1097.23,611.564 1098.25,611.409 1099.27,611.253 1100.29,611.097 1101.32,610.941 1102.34,610.785 1103.36,610.629 1104.38,610.472 1105.4,610.316 \n",
       "  1106.42,610.159 1107.44,610.002 1108.45,609.845 1109.47,609.688 1110.49,609.53 1111.51,609.373 1112.52,609.215 1113.54,609.057 1114.55,608.899 1115.57,608.741 \n",
       "  1116.58,608.583 1117.6,608.424 1118.61,608.265 1119.62,608.107 1120.64,607.948 1121.65,607.789 1122.66,607.629 1123.67,607.47 1124.68,607.31 1125.69,607.15 \n",
       "  1126.7,606.99 1127.71,606.83 1128.72,606.67 1129.72,606.51 1130.73,606.349 1131.74,606.188 1132.74,606.027 1133.75,605.866 1134.76,605.705 1135.76,605.544 \n",
       "  1136.77,605.382 1137.77,605.221 1138.77,605.059 1139.78,604.897 1140.78,604.734 1141.78,604.572 1142.78,604.41 1143.78,604.247 1144.79,604.084 1145.79,603.921 \n",
       "  1146.79,603.758 1147.79,603.595 1148.78,603.431 1149.78,603.267 1150.78,603.104 1151.78,602.94 1152.78,602.775 1153.77,602.611 1154.77,602.447 1155.76,602.282 \n",
       "  1156.76,602.117 1157.75,601.952 1158.75,601.787 1159.74,601.622 1160.74,601.456 1161.73,601.291 1162.72,601.125 1163.71,600.959 1164.71,600.793 1165.7,600.626 \n",
       "  1166.69,600.46 1167.68,600.293 1168.67,600.127 1169.66,599.96 1170.65,599.792 1171.63,599.625 1172.62,599.458 1173.61,599.29 1174.6,599.122 1175.58,598.954 \n",
       "  1176.57,598.786 1177.56,598.618 1178.54,598.449 1179.53,598.281 1180.51,598.112 1181.5,597.943 1182.48,597.774 1183.46,597.604 1184.45,597.435 1185.43,597.265 \n",
       "  1186.41,597.095 1187.39,596.925 1188.37,596.755 1189.35,596.585 1190.33,596.414 1191.31,596.244 1192.29,596.073 1193.27,595.902 1194.25,595.731 1195.23,595.559 \n",
       "  1196.21,595.388 1197.19,595.216 1198.16,595.044 1199.14,594.872 1200.12,594.7 1201.09,594.527 1202.07,594.355 1203.04,594.182 1204.02,594.009 1204.99,593.836 \n",
       "  1205.96,593.663 1206.94,593.489 1207.91,593.316 1208.88,593.142 1209.86,592.968 1210.83,592.794 1211.8,592.619 1212.77,592.445 1213.74,592.27 1214.71,592.095 \n",
       "  1215.68,591.92 1216.65,591.745 1217.62,591.57 1218.59,591.394 1219.55,591.219 1220.52,591.043 1221.49,590.867 1222.46,590.69 1223.42,590.514 1224.39,590.337 \n",
       "  1225.36,590.161 1226.32,589.984 1227.29,589.807 1228.25,589.629 1229.22,589.452 1230.18,589.274 1231.14,589.096 1232.11,588.918 1233.07,588.74 1234.03,588.562 \n",
       "  1234.99,588.383 1235.95,588.204 1236.92,588.025 1237.88,587.846 1238.84,587.667 1239.8,587.488 1240.76,587.308 1241.72,587.128 1242.68,586.948 1243.63,586.768 \n",
       "  1244.59,586.588 1245.55,586.407 1246.51,586.226 1247.47,586.046 1248.42,585.865 1249.38,585.683 1250.33,585.502 1251.29,585.32 1252.25,585.138 1253.2,584.956 \n",
       "  1254.16,584.774 1255.11,584.592 1256.06,584.409 1257.02,584.227 1257.97,584.044 1258.92,583.861 1259.88,583.677 1260.83,583.494 1261.78,583.31 1262.73,583.127 \n",
       "  1263.68,582.943 1264.63,582.758 1265.58,582.574 1266.53,582.389 1267.48,582.205 1268.43,582.02 1269.38,581.835 1270.33,581.649 1271.28,581.464 1272.23,581.278 \n",
       "  1273.18,581.092 1274.12,580.906 1275.07,580.72 1276.02,580.534 1276.96,580.347 1277.91,580.16 1278.86,579.973 1279.8,579.786 1280.75,579.599 1281.69,579.411 \n",
       "  1282.63,579.224 1283.58,579.036 1284.52,578.848 1285.47,578.659 1286.41,578.471 1287.35,578.282 1288.29,578.093 1289.24,577.904 1290.18,577.715 1291.12,577.526 \n",
       "  1292.06,577.336 1293,577.146 1293.94,576.956 1294.88,576.766 1295.82,576.576 1296.76,576.385 1297.7,576.195 1298.64,576.004 1299.58,575.812 1300.52,575.621 \n",
       "  1301.45,575.43 1302.39,575.238 1303.33,575.046 1304.27,574.854 1305.2,574.662 1306.14,574.469 1307.08,574.277 1308.01,574.084 1308.95,573.891 1309.88,573.697 \n",
       "  1310.82,573.504 1311.75,573.31 1312.69,573.116 1313.62,572.922 1314.55,572.728 1315.49,572.534 1316.42,572.339 1317.35,572.144 1318.29,571.949 1319.22,571.754 \n",
       "  1320.15,571.559 1321.08,571.363 1322.01,571.168 1322.95,570.972 1323.88,570.775 1324.81,570.579 1325.74,570.382 1326.67,570.186 1327.6,569.989 1328.53,569.792 \n",
       "  1329.46,569.594 1330.38,569.397 1331.31,569.199 1332.24,569.001 1333.17,568.803 1334.1,568.605 1335.02,568.406 1335.95,568.207 1336.88,568.008 1337.8,567.809 \n",
       "  1338.73,567.61 1339.66,567.41 1340.58,567.211 1341.51,567.011 1342.43,566.811 1343.36,566.61 1344.28,566.41 1345.21,566.209 1346.13,566.008 1347.06,565.807 \n",
       "  1347.98,565.605 1348.9,565.404 1349.83,565.202 1350.75,565 1351.67,564.798 1352.59,564.596 1353.52,564.393 1354.44,564.19 1355.36,563.987 1356.28,563.784 \n",
       "  1357.2,563.581 1358.12,563.377 1359.04,563.174 1359.96,562.97 1360.88,562.765 1361.8,562.561 1362.72,562.356 1363.64,562.152 1364.56,561.947 1365.48,561.741 \n",
       "  1366.4,561.536 1367.32,561.33 1368.23,561.125 1369.15,560.919 1370.07,560.712 1370.99,560.506 1371.9,560.299 1372.82,560.092 1373.74,559.885 1374.65,559.678 \n",
       "  1375.57,559.471 1376.49,559.263 1377.4,559.055 1378.32,558.847 1379.23,558.639 1380.15,558.43 1381.06,558.221 1381.98,558.012 1382.89,557.803 1383.81,557.594 \n",
       "  1384.72,557.384 1385.63,557.175 1386.55,556.965 1387.46,556.754 1388.37,556.544 1389.29,556.333 1390.2,556.123 1391.11,555.911 1392.02,555.7 1392.93,555.489 \n",
       "  1393.85,555.277 1394.76,555.065 1395.67,554.853 1396.58,554.641 1397.49,554.428 1398.4,554.216 1399.31,554.003 1400.22,553.789 1401.13,553.576 1402.04,553.362 \n",
       "  1402.95,553.149 1403.86,552.935 1404.77,552.72 1405.68,552.506 1406.59,552.291 1407.5,552.076 1408.4,551.861 1409.31,551.646 1410.22,551.431 1411.13,551.215 \n",
       "  1412.03,550.999 1412.94,550.783 1413.85,550.566 1414.76,550.35 1415.66,550.133 1416.57,549.916 1417.48,549.699 1418.38,549.481 1419.29,549.263 1420.19,549.045 \n",
       "  1421.1,548.827 1422,548.609 1422.91,548.39 1423.81,548.172 1424.72,547.953 1425.62,547.733 1426.53,547.514 1427.43,547.294 1428.34,547.074 1429.24,546.854 \n",
       "  1430.14,546.634 1431.05,546.413 1431.95,546.193 1432.85,545.972 1433.76,545.751 1434.66,545.529 1435.56,545.307 1436.46,545.086 1437.37,544.863 1438.27,544.641 \n",
       "  1439.17,544.419 1440.07,544.196 1440.97,543.973 1441.87,543.75 1442.78,543.526 1443.68,543.303 1444.58,543.079 1445.48,542.855 1446.38,542.63 1447.28,542.406 \n",
       "  1448.18,542.181 1449.08,541.956 1449.98,541.731 1450.88,541.505 1451.78,541.28 1452.68,541.054 1453.58,540.828 1454.48,540.601 1455.38,540.375 1456.27,540.148 \n",
       "  1457.17,539.921 1458.07,539.693 1458.97,539.466 1459.87,539.238 1460.77,539.01 1461.66,538.782 1462.56,538.554 1463.46,538.325 1464.36,538.096 1465.25,537.867 \n",
       "  1466.15,537.638 1467.05,537.408 1467.95,537.178 1468.84,536.948 1469.74,536.718 1470.64,536.488 1471.53,536.257 1472.43,536.026 1473.32,535.795 1474.22,535.563 \n",
       "  1475.12,535.332 1476.01,535.1 1476.91,534.868 1477.8,534.636 1478.7,534.403 1479.59,534.17 1480.49,533.937 1481.38,533.704 1482.28,533.47 1483.17,533.237 \n",
       "  1484.07,533.003 1484.96,532.769 1485.85,532.534 1486.75,532.299 1487.64,532.065 1488.54,531.829 1489.43,531.594 1490.32,531.358 1491.22,531.123 1492.11,530.887 \n",
       "  1493,530.65 1493.9,530.414 1494.79,530.177 1495.68,529.94 1496.57,529.703 1497.47,529.465 1498.36,529.227 1499.25,528.989 1500.14,528.751 1501.04,528.513 \n",
       "  1501.93,528.274 1502.82,528.035 1503.71,527.796 1504.6,527.557 1505.5,527.317 1506.39,527.077 1507.28,526.837 1508.17,526.596 1509.06,526.356 1509.95,526.115 \n",
       "  1510.84,525.874 1511.73,525.633 1512.63,525.391 1513.52,525.149 1514.41,524.907 1515.3,524.665 1516.19,524.422 1517.08,524.179 1517.97,523.936 1518.86,523.693 \n",
       "  1519.75,523.45 1520.64,523.206 1521.53,522.962 1522.42,522.718 1523.31,522.473 1524.2,522.228 1525.09,521.983 1525.98,521.738 1526.87,521.493 1527.75,521.247 \n",
       "  1528.64,521.001 1529.53,520.755 1530.42,520.508 1531.31,520.262 1532.2,520.015 1533.09,519.767 1533.98,519.52 1534.87,519.272 1535.75,519.024 1536.64,518.776 \n",
       "  1537.53,518.528 1538.42,518.279 1539.31,518.03 1540.2,517.781 1541.08,517.531 1541.97,517.282 1542.86,517.032 1543.75,516.782 1544.64,516.531 1545.52,516.281 \n",
       "  1546.41,516.03 1547.3,515.778 1548.19,515.527 1549.07,515.275 1549.96,515.023 1550.85,514.771 1551.74,514.519 1552.62,514.266 1553.51,514.013 1554.4,513.76 \n",
       "  1555.29,513.506 1556.17,513.253 1557.06,512.999 1557.95,512.744 1558.83,512.49 1559.72,512.235 1560.61,511.98 1561.49,511.725 1562.38,511.469 1563.27,511.214 \n",
       "  1564.15,510.958 1565.04,510.701 1565.93,510.445 1566.81,510.188 1567.7,509.931 1568.59,509.674 1569.47,509.416 1570.36,509.158 1571.25,508.9 1572.13,508.642 \n",
       "  1573.02,508.383 1573.9,508.124 1574.79,507.865 1575.68,507.606 1576.56,507.346 1577.45,507.086 1578.33,506.826 1579.22,506.566 1580.11,506.305 1580.99,506.044 \n",
       "  1581.88,505.783 1582.76,505.522 1583.65,505.26 1584.54,504.998 1585.42,504.736 1586.31,504.473 1587.19,504.21 1588.08,503.947 1588.96,503.684 1589.85,503.421 \n",
       "  1590.73,503.157 1591.62,502.893 1592.51,502.628 1593.39,502.364 1594.28,502.099 1595.16,501.834 1596.05,501.568 1596.93,501.303 1597.82,501.037 1598.7,500.77 \n",
       "  1599.59,500.504 1600.48,500.237 1601.36,499.97 1602.25,499.703 1603.13,499.435 1604.02,499.168 1604.9,498.9 1605.79,498.631 1606.67,498.363 1607.56,498.094 \n",
       "  1608.44,497.825 1609.33,497.555 1610.21,497.285 1611.1,497.015 1611.99,496.745 1612.87,496.475 1613.76,496.204 1614.64,495.933 1615.53,495.662 1616.41,495.39 \n",
       "  1617.3,495.118 1618.18,494.846 1619.07,494.574 1619.95,494.301 1620.84,494.028 1621.72,493.755 1622.61,493.481 1623.5,493.207 1624.38,492.933 1625.27,492.659 \n",
       "  1626.15,492.384 1627.04,492.11 1627.92,491.834 1628.81,491.559 1629.69,491.283 1630.58,491.007 1631.47,490.731 1632.35,490.455 1633.24,490.178 1634.12,489.901 \n",
       "  1635.01,489.623 1635.89,489.346 1636.78,489.068 1637.67,488.79 1638.55,488.511 1639.44,488.232 1640.32,487.953 1641.21,487.674 1642.09,487.394 1642.98,487.115 \n",
       "  1643.87,486.834 1644.75,486.554 1645.64,486.273 1646.52,485.992 1647.41,485.711 1648.3,485.429 1649.18,485.148 1650.07,484.866 1650.96,484.583 1651.84,484.3 \n",
       "  1652.73,484.017 1653.61,483.734 1654.5,483.451 1655.39,483.167 1656.27,482.883 1657.16,482.598 1658.05,482.314 1658.93,482.029 1659.82,481.743 1660.71,481.458 \n",
       "  1661.59,481.172 1662.48,480.886 1663.37,480.6 1664.25,480.313 1665.14,480.026 1666.03,479.739 1666.92,479.451 1667.8,479.163 1668.69,478.875 1669.58,478.587 \n",
       "  1670.47,478.298 1671.35,478.009 1672.24,477.72 1673.13,477.43 1674.02,477.141 1674.9,476.85 1675.79,476.56 1676.68,476.269 1677.57,475.978 1678.45,475.687 \n",
       "  1679.34,475.395 1680.23,475.103 1681.12,474.811 1682.01,474.519 1682.9,474.226 1683.78,473.933 1684.67,473.64 1685.56,473.346 1686.45,473.052 1687.34,472.758 \n",
       "  1688.23,472.463 1689.12,472.168 1690,471.873 1690.89,471.578 1691.78,471.282 1692.67,470.986 1693.56,470.69 1694.45,470.393 1695.34,470.096 1696.23,469.799 \n",
       "  1697.12,469.502 1698.01,469.204 1698.9,468.906 1699.79,468.607 1700.68,468.309 1701.57,468.01 1702.46,467.71 1703.35,467.411 1704.24,467.111 1705.13,466.811 \n",
       "  1706.02,466.51 1706.91,466.21 1707.8,465.908 1708.69,465.607 1709.58,465.305 1710.47,465.003 1711.36,464.701 1712.25,464.398 1713.15,464.096 1714.04,463.792 \n",
       "  1714.93,463.489 1715.82,463.185 1716.71,462.881 1717.6,462.577 1718.5,462.272 1719.39,461.967 1720.28,461.661 1721.17,461.356 1722.06,461.05 1722.96,460.744 \n",
       "  1723.85,460.437 1724.74,460.13 1725.64,459.823 1726.53,459.515 1727.42,459.208 1728.31,458.9 1729.21,458.591 1730.1,458.282 1730.99,457.973 1731.89,457.664 \n",
       "  1732.78,457.354 1733.68,457.044 1734.57,456.734 1735.46,456.424 1736.36,456.113 1737.25,455.802 1738.15,455.49 1739.04,455.178 1739.94,454.866 1740.83,454.554 \n",
       "  1741.73,454.241 1742.62,453.928 1743.52,453.614 1744.41,453.301 1745.31,452.987 1746.2,452.672 1747.1,452.358 1747.99,452.043 1748.89,451.727 1749.79,451.412 \n",
       "  1750.68,451.096 1751.58,450.78 1752.48,450.463 1753.37,450.146 1754.27,449.829 1755.17,449.512 1756.06,449.194 1756.96,448.876 1757.86,448.557 1758.76,448.238 \n",
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       "  1779.48,378.734 1780.59,378.346 1781.69,377.957 1782.8,377.567 1783.9,377.177 1785.01,376.787 1786.12,376.396 1787.22,376.005 1788.33,375.614 1789.44,375.222 \n",
       "  1790.54,374.83 1791.65,374.437 1792.76,374.044 1793.86,373.651 1794.97,373.257 1796.08,372.863 1797.19,372.468 1798.3,372.073 1799.4,371.678 1800.51,371.282 \n",
       "  1801.62,370.886 1802.73,370.49 1803.84,370.093 1804.95,369.695 1806.06,369.297 1807.17,368.899 1808.28,368.501 1809.39,368.102 1810.5,367.702 1811.61,367.303 \n",
       "  1812.72,366.903 1813.83,366.502 1814.94,366.101 1816.05,365.7 1817.16,365.298 1818.28,364.896 1819.39,364.493 1820.5,364.09 1821.61,363.687 1822.72,363.283 \n",
       "  1823.84,362.879 1824.95,362.475 1826.06,362.07 1827.18,361.664 1828.29,361.258 1829.4,360.852 1830.52,360.446 1831.63,360.039 1832.75,359.631 1833.86,359.223 \n",
       "  1834.98,358.815 1836.09,358.406 1837.21,357.997 1838.32,357.588 1839.44,357.178 1840.55,356.768 1841.67,356.357 1842.79,355.946 1843.9,355.534 1845.02,355.123 \n",
       "  1846.14,354.71 1847.26,354.297 1848.37,353.884 1849.49,353.471 1850.61,353.057 1851.73,352.642 1852.85,352.227 1853.97,351.812 1855.09,351.397 1856.2,350.98 \n",
       "  1857.32,350.564 1858.44,350.147 1859.56,349.73 1860.68,349.312 1861.81,348.894 1862.93,348.475 1864.05,348.056 1865.17,347.637 1866.29,347.217 1867.41,346.797 \n",
       "  1868.53,346.376 1869.66,345.955 1870.78,345.533 1871.9,345.111 1873.03,344.689 1874.15,344.266 1875.27,343.843 1876.4,343.419 1877.52,342.995 1878.65,342.571 \n",
       "  1879.77,342.146 1880.9,341.721 1882.02,341.295 1883.15,340.869 1884.27,340.442 1885.4,340.015 1886.53,339.588 1887.65,339.16 1888.78,338.731 1889.91,338.303 \n",
       "  1891.03,337.873 1892.16,337.444 1893.29,337.014 1894.42,336.583 1895.55,336.152 1896.68,335.721 1897.81,335.289 1898.94,334.857 1900.07,334.424 1901.2,333.991 \n",
       "  1902.33,333.558 1903.46,333.124 1904.59,332.689 1905.72,332.255 1906.85,331.819 1907.98,331.384 1909.11,330.948 1910.25,330.511 1911.38,330.074 1912.51,329.637 \n",
       "  1913.65,329.199 1914.78,328.76 1915.91,328.322 1917.05,327.882 1918.18,327.443 1919.32,327.003 1920.45,326.562 1921.59,326.121 1922.72,325.68 1923.86,325.238 \n",
       "  1925,324.796 1926.13,324.353 1927.27,323.91 1928.41,323.466 1929.55,323.022 1930.68,322.578 1931.82,322.133 1932.96,321.687 1934.1,321.242 1935.24,320.795 \n",
       "  1936.38,320.349 1937.52,319.902 1938.66,319.454 1939.8,319.006 1940.94,318.557 1942.08,318.108 1943.23,317.659 1944.37,317.209 1945.51,316.759 1946.65,316.308 \n",
       "  1947.8,315.857 1948.94,315.405 1950.08,314.953 1951.23,314.5 1952.37,314.047 1953.52,313.594 1954.66,313.14 1955.81,312.686 1956.95,312.231 1958.1,311.775 \n",
       "  1959.24,311.32 1960.39,310.863 1961.54,310.407 1962.69,309.95 1963.83,309.492 1964.98,309.034 1966.13,308.576 1967.28,308.117 1968.43,307.657 1969.58,307.197 \n",
       "  1970.73,306.737 1971.88,306.276 1973.03,305.815 1974.18,305.353 1975.33,304.891 1976.48,304.428 1977.64,303.965 1978.79,303.502 1979.94,303.038 1981.09,302.573 \n",
       "  1982.25,302.108 1983.4,301.643 1984.56,301.177 1985.71,300.71 1986.87,300.244 1988.02,299.776 1989.18,299.309 1990.33,298.84 1991.49,298.372 1992.65,297.902 \n",
       "  1993.81,297.433 1994.96,296.963 1996.12,296.492 1997.28,296.021 1998.44,295.549 1999.6,295.077 2000.76,294.605 2001.92,294.132 2003.08,293.659 2004.24,293.185 \n",
       "  2005.4,292.71 2006.56,292.235 2007.73,291.76 2008.89,291.284 2010.05,290.808 2011.21,290.331 2012.38,289.854 2013.54,289.376 2014.71,288.898 2015.87,288.42 \n",
       "  \n",
       "  \"/>\n",
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       "  882.966,538.28 884.444,538.051 885.922,537.821 887.399,537.592 888.875,537.363 890.35,537.133 891.825,536.903 893.299,536.672 894.772,536.442 896.245,536.211 \n",
       "  897.717,535.98 899.188,535.749 900.658,535.517 902.128,535.286 903.597,535.054 905.066,534.822 906.533,534.589 908,534.357 909.466,534.124 910.932,533.891 \n",
       "  912.397,533.658 913.861,533.424 915.324,533.19 916.787,532.956 918.249,532.722 919.71,532.487 921.171,532.253 922.631,532.018 924.09,531.783 925.549,531.547 \n",
       "  927.006,531.312 928.464,531.076 929.92,530.84 931.376,530.603 932.831,530.367 934.286,530.13 935.739,529.893 937.192,529.655 938.645,529.418 940.097,529.18 \n",
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       "  1125.01,495.879 1126.38,495.608 1127.76,495.336 1129.13,495.064 1130.51,494.792 1131.88,494.519 1133.25,494.247 1134.63,493.974 1136,493.7 1137.37,493.427 \n",
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       "  1152.43,490.4 1153.79,490.123 1155.16,489.846 1156.52,489.568 1157.89,489.29 1159.25,489.012 1160.61,488.734 1161.98,488.456 1163.34,488.177 1164.7,487.898 \n",
       "  1166.07,487.618 1167.43,487.339 1168.79,487.059 1170.15,486.779 1171.51,486.498 1172.87,486.217 1174.23,485.936 1175.59,485.655 1176.94,485.373 1178.3,485.091 \n",
       "  1179.66,484.809 1181.02,484.527 1182.37,484.244 1183.73,483.961 1185.09,483.678 1186.44,483.394 1187.8,483.11 1189.15,482.826 1190.5,482.542 1191.86,482.257 \n",
       "  1193.21,481.972 1194.56,481.687 1195.92,481.401 1197.27,481.115 1198.62,480.829 1199.97,480.543 1201.32,480.256 1202.67,479.969 1204.02,479.682 1205.37,479.394 \n",
       "  1206.72,479.106 1208.07,478.818 1209.42,478.529 1210.77,478.241 1212.11,477.952 1213.46,477.662 1214.81,477.373 1216.15,477.083 1217.5,476.793 1218.85,476.502 \n",
       "  1220.19,476.211 1221.54,475.92 1222.88,475.629 1224.22,475.337 1225.57,475.045 1226.91,474.753 1228.25,474.46 1229.6,474.168 1230.94,473.875 1232.28,473.581 \n",
       "  1233.62,473.287 1234.96,472.993 1236.3,472.699 1237.64,472.405 1238.98,472.11 1240.32,471.814 1241.66,471.519 1243,471.223 1244.34,470.927 1245.68,470.631 \n",
       "  1247.01,470.334 1248.35,470.037 1249.69,469.74 1251.03,469.442 1252.36,469.145 1253.7,468.846 1255.03,468.548 1256.37,468.249 1257.7,467.95 1259.04,467.651 \n",
       "  1260.37,467.351 1261.7,467.051 1263.04,466.751 1264.37,466.45 1265.7,466.15 1267.04,465.848 1268.37,465.547 1269.7,465.245 1271.03,464.943 1272.36,464.641 \n",
       "  1273.69,464.338 1275.02,464.035 1276.35,463.732 1277.68,463.428 1279.01,463.124 1280.34,462.82 1281.67,462.516 1283,462.211 1284.32,461.906 1285.65,461.601 \n",
       "  1286.98,461.295 1288.31,460.989 1289.63,460.682 1290.96,460.376 1292.29,460.069 1293.61,459.762 1294.94,459.454 1296.26,459.146 1297.59,458.838 1298.91,458.53 \n",
       "  1300.23,458.221 1301.56,457.912 1302.88,457.602 1304.21,457.293 1305.53,456.983 1306.85,456.672 1308.17,456.362 1309.49,456.051 1310.82,455.739 1312.14,455.428 \n",
       "  1313.46,455.116 1314.78,454.804 1316.1,454.491 1317.42,454.179 1318.74,453.865 1320.06,453.552 1321.38,453.238 1322.7,452.924 1324.02,452.61 1325.34,452.295 \n",
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       "  1339.82,448.812 1341.13,448.494 1342.45,448.175 1343.76,447.856 1345.08,447.536 1346.39,447.217 1347.71,446.896 1349.02,446.576 1350.33,446.255 1351.65,445.934 \n",
       "  1352.96,445.613 1354.27,445.291 1355.58,444.969 1356.89,444.647 1358.21,444.324 1359.52,444.001 1360.83,443.678 1362.14,443.354 1363.45,443.03 1364.76,442.706 \n",
       "  1366.07,442.381 1367.38,442.056 1368.69,441.731 1370,441.405 1371.31,441.079 1372.62,440.753 1373.93,440.426 1375.24,440.1 1376.54,439.772 1377.85,439.445 \n",
       "  1379.16,439.117 1380.47,438.789 1381.77,438.46 1383.08,438.131 1384.39,437.802 1385.69,437.472 1387,437.143 1388.31,436.812 1389.61,436.482 1390.92,436.151 \n",
       "  1392.22,435.82 1393.53,435.488 1394.83,435.156 1396.14,434.824 1397.44,434.492 1398.75,434.159 1400.05,433.826 1401.36,433.492 1402.66,433.158 1403.96,432.824 \n",
       "  1405.27,432.49 1406.57,432.155 1407.87,431.82 1409.18,431.484 1410.48,431.148 1411.78,430.812 1413.08,430.475 1414.39,430.139 1415.69,429.801 1416.99,429.464 \n",
       "  1418.29,429.126 1419.59,428.788 1420.89,428.449 1422.19,428.11 1423.49,427.771 1424.79,427.431 1426.1,427.091 1427.4,426.751 1428.7,426.411 1429.99,426.07 \n",
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       "  1444.28,422.297 1445.58,421.952 1446.87,421.606 1448.17,421.26 1449.47,420.914 1450.77,420.568 1452.06,420.221 1453.36,419.874 1454.66,419.526 1455.95,419.179 \n",
       "  1457.25,418.83 1458.54,418.482 1459.84,418.133 1461.14,417.784 1462.43,417.434 1463.73,417.084 1465.02,416.734 1466.32,416.383 1467.61,416.032 1468.91,415.681 \n",
       "  1470.2,415.329 1471.5,414.977 1472.79,414.625 1474.09,414.272 1475.38,413.919 1476.67,413.565 1477.97,413.212 1479.26,412.858 1480.55,412.503 1481.85,412.148 \n",
       "  1483.14,411.793 1484.44,411.437 1485.73,411.081 1487.02,410.725 1488.31,410.368 1489.61,410.011 1490.9,409.654 1492.19,409.296 1493.48,408.938 1494.78,408.58 \n",
       "  1496.07,408.221 1497.36,407.862 1498.65,407.502 1499.94,407.142 1501.24,406.782 1502.53,406.422 1503.82,406.061 1505.11,405.699 1506.4,405.338 1507.69,404.976 \n",
       "  1508.98,404.613 1510.27,404.25 1511.57,403.887 1512.86,403.524 1514.15,403.16 1515.44,402.796 1516.73,402.431 1518.02,402.066 1519.31,401.701 1520.6,401.335 \n",
       "  1521.89,400.969 1523.18,400.603 1524.47,400.236 1525.76,399.869 1527.05,399.501 1528.34,399.133 1529.63,398.765 1530.92,398.396 1532.21,398.027 1533.5,397.658 \n",
       "  1534.79,397.288 1536.07,396.918 1537.36,396.548 1538.65,396.177 1539.94,395.805 1541.23,395.434 1542.52,395.062 1543.81,394.69 1545.1,394.317 1546.39,393.944 \n",
       "  1547.67,393.57 1548.96,393.196 1550.25,392.822 1551.54,392.448 1552.83,392.073 1554.12,391.697 1555.4,391.322 1556.69,390.946 1557.98,390.569 1559.27,390.192 \n",
       "  1560.56,389.815 1561.84,389.437 1563.13,389.059 1564.42,388.681 1565.71,388.302 1567,387.923 1568.28,387.544 1569.57,387.164 1570.86,386.784 1572.15,386.403 \n",
       "  1573.43,386.022 1574.72,385.641 1576.01,385.259 1577.3,384.877 1578.58,384.494 1579.87,384.111 1581.16,383.728 1582.45,383.344 1583.73,382.96 1585.02,382.576 \n",
       "  1586.31,382.191 1587.59,381.806 1588.88,381.42 1590.17,381.034 1591.46,380.648 1592.74,380.261 1594.03,379.874 1595.32,379.486 1596.6,379.098 1597.89,378.71 \n",
       "  1599.18,378.321 1600.47,377.932 1601.75,377.542 1603.04,377.153 1604.33,376.762 1605.61,376.372 1606.9,375.981 1608.19,375.589 1609.47,375.197 1610.76,374.805 \n",
       "  1612.05,374.412 1613.34,374.019 1614.62,373.626 1615.91,373.232 1617.2,372.838 1618.48,372.443 1619.77,372.048 1621.06,371.653 1622.34,371.257 1623.63,370.861 \n",
       "  1624.92,370.464 1626.21,370.067 1627.49,369.67 1628.78,369.272 1630.07,368.874 1631.35,368.476 1632.64,368.077 1633.93,367.677 1635.22,367.277 1636.5,366.877 \n",
       "  1637.79,366.477 1639.08,366.076 1640.37,365.674 1641.65,365.273 1642.94,364.87 1644.23,364.468 1645.52,364.065 1646.8,363.661 1648.09,363.258 1649.38,362.853 \n",
       "  1650.67,362.449 1651.95,362.044 1653.24,361.638 1654.53,361.233 1655.82,360.826 1657.1,360.42 1658.39,360.013 1659.68,359.605 1660.97,359.197 1662.26,358.789 \n",
       "  1663.54,358.38 1664.83,357.971 1666.12,357.562 1667.41,357.152 1668.7,356.742 1669.98,356.331 1671.27,355.92 1672.56,355.508 1673.85,355.096 1675.14,354.684 \n",
       "  1676.43,354.271 1677.72,353.858 1679.01,353.444 1680.29,353.03 1681.58,352.616 1682.87,352.201 1684.16,351.786 1685.45,351.37 1686.74,350.954 1688.03,350.537 \n",
       "  1689.32,350.121 1690.61,349.703 1691.9,349.285 1693.19,348.867 1694.48,348.449 1695.77,348.03 1697.06,347.61 1698.35,347.19 1699.64,346.77 1700.93,346.349 \n",
       "  1702.22,345.928 1703.51,345.507 1704.8,345.085 1706.09,344.662 1707.38,344.239 1708.67,343.816 1709.96,343.393 1711.25,342.968 1712.54,342.544 1713.83,342.119 \n",
       "  1715.12,341.694 1716.42,341.268 1717.71,340.842 1719,340.415 1720.29,339.988 1721.58,339.56 1722.87,339.132 1724.17,338.704 1725.46,338.275 1726.75,337.846 \n",
       "  1728.04,337.416 1729.33,336.986 1730.63,336.556 1731.92,336.125 1733.21,335.694 1734.51,335.262 1735.8,334.83 1737.09,334.397 1738.39,333.964 1739.68,333.53 \n",
       "  1740.97,333.096 1742.27,332.662 1743.56,332.227 1744.85,331.792 1746.15,331.356 1747.44,330.92 1748.74,330.483 1750.03,330.046 1751.33,329.609 1752.62,329.171 \n",
       "  1753.92,328.733 1755.21,328.294 1756.51,327.855 1757.8,327.415 1759.1,326.975 1760.39,326.534 1761.69,326.093 1762.98,325.652 1764.28,325.21 1765.58,324.768 \n",
       "  1766.87,324.325 1768.17,323.882 1769.47,323.438 1770.76,322.994 1772.06,322.55 1773.36,322.105 1774.65,321.659 1775.95,321.213 1777.25,320.767 1778.55,320.32 \n",
       "  1779.85,319.873 1781.14,319.426 1782.44,318.977 1783.74,318.529 1785.04,318.08 1786.34,317.63 1787.64,317.181 1788.94,316.73 1790.24,316.279 1791.54,315.828 \n",
       "  1792.84,315.376 1794.14,314.924 1795.44,314.472 1796.74,314.019 1798.04,313.565 1799.34,313.111 1800.64,312.657 1801.94,312.202 1803.24,311.747 1804.54,311.291 \n",
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       "  1831.92,301.613 1833.23,301.147 1834.53,300.681 1835.84,300.214 1837.15,299.747 1838.45,299.279 1839.76,298.811 1841.07,298.342 1842.38,297.873 1843.68,297.403 \n",
       "  1844.99,296.933 1846.3,296.462 1847.61,295.991 1848.92,295.52 1850.23,295.048 1851.54,294.575 1852.84,294.102 1854.15,293.629 1855.46,293.155 1856.77,292.68 \n",
       "  1858.09,292.205 1859.4,291.73 1860.71,291.254 1862.02,290.778 1863.33,290.301 1864.64,289.824 1865.95,289.346 1867.27,288.868 1868.58,288.389 1869.89,287.91 \n",
       "  1871.2,287.431 1872.52,286.95 1873.83,286.47 1875.15,285.989 1876.46,285.507 1877.77,285.025 1879.09,284.542 1880.4,284.059 1881.72,283.576 1883.03,283.092 \n",
       "  1884.35,282.607 1885.67,282.123 1886.98,281.637 1888.3,281.151 1889.62,280.665 1890.93,280.178 1892.25,279.69 1893.57,279.203 1894.89,278.714 1896.2,278.225 \n",
       "  1897.52,277.736 1898.84,277.246 1900.16,276.756 1901.48,276.265 1902.8,275.774 1904.12,275.282 1905.44,274.79 1906.76,274.297 1908.08,273.803 1909.4,273.31 \n",
       "  1910.72,272.815 1912.05,272.321 1913.37,271.825 1914.69,271.33 1916.01,270.833 1917.34,270.337 1918.66,269.839 1919.98,269.342 1921.31,268.843 1922.63,268.345 \n",
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       "  1950.51,257.755 1951.85,257.245 1953.18,256.734 1954.51,256.224 1955.84,255.712 1957.17,255.2 1958.51,254.688 1959.84,254.175 1961.17,253.661 1962.51,253.148 \n",
       "  1963.84,252.633 1965.18,252.118 1966.51,251.603 1967.85,251.087 1969.19,250.57 1970.52,250.053 1971.86,249.535 1973.2,249.017 1974.53,248.499 1975.87,247.98 \n",
       "  1977.21,247.46 1978.55,246.94 1979.89,246.419 1981.23,245.898 1982.57,245.376 1983.91,244.854 1985.25,244.331 1986.59,243.808 1987.93,243.284 1989.27,242.76 \n",
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     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fun1(z,p,t)\n",
    "    zpoint = similar(z)\n",
    "    zpoint[1] = -z[1] + z[2]\n",
    "    zpoint[2] = z[2]\n",
    "    return zpoint\n",
    "end\n",
    "\n",
    "t0 = 0; tf = 1.;\n",
    "\n",
    "seg = range(-1.,1.,length=10)\n",
    "Z0 = transpose([-ones(size(seg)) seg])\n",
    "plt1 = plot()\n",
    "plot_traj(plt1,fun1,Z0,tf)\n",
    "\n",
    "seg = range(-1.,1.,length=200)\n",
    "Z0 = transpose([-ones(size(seg)) seg])\n",
    "plot_flow(plt1,fun1,Z0,tf)\n",
    "plot!(plt1, xlims = (-2.,1))\n",
    "plot!(plt1,[0.,0.],[-3.,3], linestyle = :dot, color = :black)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Retrouvez graphiquement la solution du problème de contrôle optimal\n",
    "- La droite de départ se transforme par le plot en une droite, pourquoi?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Flot des extrémales pour le problème de conrôle 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",
    "sous les contraintes\n",
    "\n",
    "$$ \\dot{x}(t) = -x(t) + \\alpha x^2(t) + u(t)$$\n",
    "\n",
    "$$ x(0)=x_0=-1,\\quad x(1)=0. $$\n",
    "\n",
    "Le but est ici de tracer les extrémales et le flot associé pour des points initiaux $z(0) = (x_0, p(0))$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
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       "  191.808,919.243 192.939,917.366 194.067,915.487 195.193,913.608 196.317,911.727 197.439,909.846 198.559,907.963 199.676,906.08 200.792,904.195 201.905,902.31 \n",
       "  203.016,900.423 204.126,898.536 205.233,896.648 206.338,894.759 207.441,892.869 208.543,890.978 209.642,889.086 210.739,887.194 211.835,885.301 212.928,883.408 \n",
       "  214.02,881.513 215.11,879.618 216.198,877.723 217.284,875.826 218.369,873.93 219.451,872.032 220.532,870.135 221.611,868.236 222.689,866.338 223.764,864.438 \n",
       "  224.838,862.539 225.91,860.639 226.981,858.738 228.05,856.838 229.117,854.937 230.183,853.036 231.247,851.134 232.31,849.233 233.371,847.331 234.43,845.429 \n",
       "  235.488,843.527 236.544,841.625 237.599,839.722 238.652,837.82 239.704,835.918 240.754,834.016 241.803,832.113 242.851,830.211 243.897,828.309 244.941,826.407 \n",
       "  245.984,824.506 247.026,822.604 248.067,820.703 249.106,818.802 250.144,816.901 251.18,815.001 252.215,813.101 253.249,811.201 254.282,809.302 255.313,807.403 \n",
       "  256.343,805.504 257.372,803.606 258.399,801.709 259.426,799.812 260.451,797.916 261.475,796.02 262.498,794.125 263.519,792.231 264.54,790.337 265.559,788.445 \n",
       "  266.577,786.552 267.594,784.661 268.61,782.771 269.625,780.881 270.639,778.992 271.652,777.105 272.663,775.218 273.674,773.332 274.684,771.447 275.692,769.564 \n",
       "  276.7,767.681 277.706,765.799 278.712,763.919 279.716,762.04 280.72,760.162 281.723,758.285 282.725,756.41 283.725,754.536 284.725,752.663 285.724,750.791 \n",
       "  286.722,748.921 287.719,747.053 288.716,745.186 289.711,743.32 290.706,741.456 291.699,739.594 292.692,737.733 293.684,735.874 294.675,734.016 295.666,732.16 \n",
       "  296.655,730.306 297.644,728.454 298.632,726.603 299.619,724.755 300.605,722.908 301.591,721.063 302.576,719.22 303.56,717.379 304.544,715.54 305.526,713.703 \n",
       "  306.508,711.868 307.49,710.035 308.47,708.205 309.45,706.376 310.43,704.55 311.408,702.726 312.386,700.904 313.363,699.085 314.34,697.268 315.316,695.453 \n",
       "  316.291,693.641 317.266,691.831 318.24,690.024 319.214,688.219 320.186,686.417 321.159,684.617 322.13,682.82 323.102,681.026 324.072,679.234 325.042,677.445 \n",
       "  326.012,675.659 326.981,673.876 327.949,672.095 328.917,670.317 329.884,668.543 330.851,666.771 331.817,665.002 332.783,663.236 333.748,661.474 334.713,659.714 \n",
       "  335.677,657.957 336.641,656.204 337.605,654.454 338.568,652.707 339.53,650.963 340.492,649.222 341.454,647.485 342.415,645.752 343.375,644.021 344.336,642.294 \n",
       "  345.296,640.571 346.255,638.851 347.214,637.134 348.173,635.421 349.131,633.712 350.089,632.007 351.046,630.305 352.003,628.606 352.96,626.912 353.917,625.221 \n",
       "  354.873,623.534 355.828,621.851 356.784,620.172 357.738,618.497 358.693,616.825 359.647,615.158 360.601,613.495 361.555,611.835 362.508,610.18 363.461,608.529 \n",
       "  364.414,606.882 365.367,605.24 366.319,603.601 367.271,601.967 368.222,600.337 369.174,598.712 370.125,597.09 371.075,595.474 372.026,593.861 372.976,592.253 \n",
       "  373.926,590.65 374.876,589.051 375.825,587.457 376.774,585.867 377.723,584.282 378.672,582.702 379.621,581.126 380.569,579.556 381.517,577.99 382.465,576.428 \n",
       "  383.413,574.872 384.36,573.32 385.307,571.774 386.254,570.232 387.201,568.696 388.148,567.164 389.094,565.638 390.041,564.116 390.987,562.6 391.933,561.089 \n",
       "  392.878,559.583 393.824,558.082 394.769,556.587 395.715,555.096 396.66,553.612 397.605,552.132 398.549,550.658 399.494,549.19 400.438,547.726 401.383,546.269 \n",
       "  402.327,544.817 403.271,543.37 404.215,541.93 405.159,540.494 406.102,539.065 407.046,537.641 407.989,536.223 408.932,534.81 409.876,533.404 410.819,532.003 \n",
       "  411.762,530.608 412.704,529.219 413.647,527.836 414.59,526.458 415.532,525.087 416.475,523.722 417.417,522.363 418.359,521.01 419.302,519.663 420.244,518.322 \n",
       "  421.186,516.987 422.128,515.659 423.07,514.337 424.011,513.021 424.953,511.711 425.895,510.408 426.836,509.112 427.778,507.821 428.719,506.537 429.66,505.26 \n",
       "  430.602,503.989 431.543,502.724 432.484,501.467 433.425,500.215 434.366,498.971 435.307,497.733 436.248,496.501 437.189,495.277 438.13,494.059 439.071,492.848 \n",
       "  440.012,491.644 440.952,490.447 441.893,489.256 442.834,488.073 443.774,486.896 444.715,485.726 445.655,484.563 446.596,483.408 447.536,482.259 448.477,481.117 \n",
       "  449.417,479.983 450.358,478.855 451.298,477.735 452.238,476.622 453.179,475.516 454.119,474.418 455.059,473.326 456,472.242 456.94,471.166 457.88,470.096 \n",
       "  458.821,469.034 459.761,467.979 460.701,466.932 461.641,465.892 462.581,464.86 463.522,463.835 464.462,462.818 465.402,461.808 466.342,460.806 467.282,459.811 \n",
       "  468.223,458.824 469.163,457.845 470.103,456.873 471.043,455.909 471.983,454.953 472.924,454.004 473.864,453.063 474.804,452.13 475.744,451.205 476.685,450.288 \n",
       "  477.625,449.378 478.565,448.477 479.505,447.583 480.446,446.697 481.386,445.819 482.326,444.949 483.266,444.087 484.207,443.234 485.147,442.388 486.088,441.55 \n",
       "  487.028,440.72 487.968,439.898 488.909,439.085 489.849,438.279 490.79,437.482 491.73,436.693 492.67,435.912 493.611,435.139 494.551,434.375 495.492,433.619 \n",
       "  496.433,432.871 497.373,432.131 498.314,431.4 499.254,430.677 500.195,429.962 501.136,429.256 502.076,428.558 503.017,427.869 503.958,427.188 504.898,426.515 \n",
       "  505.839,425.851 506.78,425.196 507.721,424.548 508.662,423.91 509.603,423.28 510.543,422.658 511.484,422.045 512.425,421.441 513.366,420.845 514.307,420.258 \n",
       "  515.248,419.679 516.189,419.109 517.13,418.548 518.071,417.995 519.012,417.451 519.953,416.916 520.894,416.39 521.836,415.872 522.777,415.363 523.718,414.863 \n",
       "  524.659,414.372 525.6,413.889 526.541,413.415 527.483,412.95 528.424,412.494 529.365,412.047 530.307,411.608 531.248,411.179 532.189,410.758 533.13,410.346 \n",
       "  534.072,409.943 535.013,409.549 535.955,409.164 536.896,408.788 537.837,408.421 538.779,408.063 539.72,407.713 540.662,407.373 541.603,407.042 542.545,406.719 \n",
       "  543.486,406.406 544.428,406.101 545.369,405.806 546.311,405.52 547.253,405.242 548.194,404.974 549.136,404.715 550.077,404.464 551.019,404.223 551.961,403.991 \n",
       "  552.902,403.768 553.844,403.554 554.786,403.349 555.727,403.153 556.669,402.966 557.611,402.789 558.552,402.62 559.494,402.46 560.436,402.31 561.378,402.169 \n",
       "  562.319,402.037 563.261,401.914 564.203,401.8 565.144,401.695 566.086,401.599 567.028,401.513 567.97,401.435 568.911,401.367 569.853,401.308 570.795,401.258 \n",
       "  571.737,401.217 572.679,401.185 573.62,401.162 574.562,401.149 575.504,401.144 576.446,401.149 577.387,401.163 578.329,401.186 579.271,401.218 580.213,401.26 \n",
       "  581.154,401.31 582.096,401.37 583.038,401.439 583.979,401.517 584.921,401.604 585.863,401.7 586.805,401.805 587.746,401.919 588.688,402.043 589.63,402.175 \n",
       "  590.571,402.317 591.513,402.468 592.455,402.628 593.396,402.797 594.338,402.975 595.28,403.162 596.221,403.359 597.163,403.564 598.105,403.779 599.046,404.002 \n",
       "  599.988,404.235 600.93,404.477 601.871,404.727 602.813,404.987 603.754,405.256 604.696,405.534 605.637,405.821 606.579,406.117 607.52,406.421 608.462,406.735 \n",
       "  609.403,407.058 610.345,407.39 611.286,407.731 612.228,408.081 613.169,408.44 614.111,408.807 615.052,409.184 615.993,409.57 616.935,409.964 617.876,410.367 \n",
       "  618.817,410.78 619.759,411.201 620.7,411.631 621.641,412.07 622.583,412.518 623.524,412.974 624.465,413.44 625.406,413.914 626.347,414.397 627.289,414.889 \n",
       "  628.23,415.39 629.171,415.899 630.112,416.417 631.053,416.944 631.994,417.48 632.935,418.024 633.876,418.577 634.817,419.139 635.758,419.709 636.699,420.288 \n",
       "  637.64,420.876 638.581,421.472 639.522,422.077 640.463,422.69 641.404,423.312 642.345,423.943 643.286,424.582 644.226,425.23 645.167,425.886 646.108,426.55 \n",
       "  647.049,427.223 647.989,427.905 648.93,428.595 649.871,429.293 650.811,430 651.752,430.715 652.693,431.438 653.633,432.17 654.574,432.911 655.514,433.659 \n",
       "  656.455,434.416 657.395,435.181 658.336,435.954 659.276,436.736 660.217,437.526 661.157,438.324 662.098,439.13 663.038,439.945 663.979,440.767 664.919,441.598 \n",
       "  665.86,442.436 666.8,443.283 667.74,444.138 668.681,445 669.621,445.871 670.561,446.75 671.502,447.637 672.442,448.531 673.382,449.434 674.323,450.344 \n",
       "  675.263,451.262 676.203,452.188 677.143,453.122 678.084,454.063 679.024,455.013 679.964,455.97 680.904,456.934 681.845,457.907 682.785,458.887 683.725,459.874 \n",
       "  684.665,460.87 685.605,461.873 686.546,462.883 687.486,463.901 688.426,464.926 689.366,465.959 690.306,466.999 691.247,468.047 692.187,469.102 693.127,470.165 \n",
       "  694.067,471.235 695.008,472.312 695.948,473.396 696.888,474.488 697.828,475.587 698.769,476.693 699.709,477.806 700.649,478.927 701.59,480.054 702.53,481.189 \n",
       "  703.471,482.331 704.411,483.479 705.351,484.635 706.292,485.798 707.232,486.968 708.173,488.144 709.113,489.328 710.054,490.518 710.995,491.716 711.935,492.92 \n",
       "  712.876,494.131 713.817,495.348 714.758,496.573 715.699,497.804 716.64,499.042 717.58,500.286 718.521,501.537 719.463,502.795 720.404,504.059 721.345,505.33 \n",
       "  722.286,506.607 723.227,507.891 724.169,509.181 725.11,510.477 726.052,511.78 726.993,513.09 727.935,514.405 728.877,515.727 729.819,517.055 730.76,518.39 \n",
       "  731.702,519.73 732.645,521.077 733.587,522.43 734.529,523.789 735.471,525.154 736.414,526.525 737.356,527.902 738.299,529.285 739.242,530.674 740.185,532.069 \n",
       "  741.128,533.47 742.071,534.876 743.014,536.289 743.957,537.707 744.901,539.131 745.845,540.561 746.788,541.996 747.732,543.437 748.676,544.884 749.62,546.336 \n",
       "  750.565,547.794 751.509,549.258 752.454,550.726 753.399,552.201 754.343,553.681 755.289,555.166 756.234,556.656 757.179,558.152 758.125,559.653 759.071,561.16 \n",
       "  760.017,562.671 760.963,564.188 761.909,565.71 762.855,567.237 763.802,568.769 764.749,570.306 765.696,571.848 766.643,573.396 767.591,574.948 768.539,576.505 \n",
       "  769.487,578.067 770.435,579.633 771.383,581.205 772.332,582.781 773.281,584.362 774.23,585.948 775.179,587.539 776.129,589.134 777.079,590.733 778.029,592.337 \n",
       "  778.979,593.946 779.93,595.559 780.88,597.177 781.832,598.799 782.783,600.425 783.735,602.056 784.687,603.691 785.639,605.331 786.591,606.974 787.544,608.622 \n",
       "  788.498,610.274 789.451,611.93 790.405,613.59 791.359,615.254 792.313,616.922 793.268,618.594 794.223,620.27 795.179,621.949 796.134,623.633 797.09,625.321 \n",
       "  798.047,627.012 799.004,628.707 799.961,630.405 800.918,632.108 801.876,633.814 802.835,635.523 803.793,637.236 804.753,638.953 805.712,640.673 806.672,642.397 \n",
       "  807.633,644.124 808.594,645.854 809.555,647.588 810.517,649.325 811.479,651.066 812.442,652.809 813.405,654.556 814.369,656.306 815.333,658.06 816.297,659.816 \n",
       "  817.263,661.575 818.228,663.338 819.194,665.103 820.161,666.872 821.128,668.643 822.096,670.418 823.064,672.195 824.033,673.975 825.002,675.758 825.972,677.544 \n",
       "  826.942,679.332 827.913,681.123 828.884,682.917 829.856,684.713 830.829,686.512 831.802,688.314 832.776,690.118 833.75,691.925 834.725,693.734 835.701,695.546 \n",
       "  836.677,697.36 837.654,699.176 838.631,700.995 839.609,702.816 840.588,704.639 841.568,706.464 842.548,708.292 843.528,710.122 844.51,711.954 845.492,713.788 \n",
       "  846.475,715.624 847.458,717.462 848.443,719.302 849.428,721.145 850.413,722.989 851.4,724.835 852.387,726.683 853.375,728.532 854.364,730.384 855.353,732.237 \n",
       "  856.344,734.092 857.335,735.949 858.327,737.808 859.319,739.668 860.313,741.53 861.307,743.393 862.303,745.258 863.299,747.124 864.296,748.992 865.294,750.861 \n",
       "  866.293,752.732 867.292,754.604 868.293,756.478 869.294,758.353 870.297,760.229 871.3,762.106 872.305,763.985 873.31,765.865 874.316,767.746 875.324,769.628 \n",
       "  876.332,771.511 877.341,773.396 878.352,775.281 879.363,777.167 880.375,779.055 881.389,780.943 882.404,782.833 883.419,784.723 884.436,786.614 885.454,788.506 \n",
       "  886.473,790.398 887.493,792.292 888.514,794.186 889.537,796.081 890.561,797.976 891.585,799.873 892.611,801.769 893.639,803.667 894.667,805.565 895.697,807.463 \n",
       "  896.728,809.362 897.76,811.262 898.794,813.162 899.829,815.062 900.865,816.963 901.902,818.864 902.941,820.765 903.982,822.667 905.023,824.569 906.066,826.471 \n",
       "  907.111,828.373 908.156,830.276 909.204,832.178 910.252,834.081 911.302,835.984 912.354,837.887 913.407,839.79 914.462,841.693 915.518,843.596 916.576,845.498 \n",
       "  917.635,847.401 918.696,849.304 919.758,851.206 920.822,853.109 921.888,855.011 922.955,856.912 924.024,858.814 925.094,860.715 926.167,862.616 927.241,864.517 \n",
       "  928.316,866.417 929.394,868.317 930.473,870.216 931.554,872.115 932.637,874.014 933.721,875.912 934.808,877.809 935.896,879.706 936.986,881.602 938.078,883.498 \n",
       "  939.172,885.392 940.267,887.287 941.365,889.18 942.465,891.073 943.566,892.965 944.67,894.856 945.775,896.747 946.883,898.636 947.993,900.525 949.104,902.412 \n",
       "  950.218,904.299 951.334,906.185 952.452,908.07 953.572,909.954 954.694,911.837 955.819,913.718 956.945,915.599 958.074,917.479 959.205,919.357 960.339,921.234 \n",
       "  961.474,923.11 962.612,924.985 963.752,926.858 964.895,928.731 966.04,930.602 967.187,932.471 968.337,934.339 969.489,936.206 970.643,938.072 971.8,939.936 \n",
       "  972.96,941.798 974.122,943.659 975.286,945.519 976.453,947.377 977.623,949.233 978.795,951.088 979.97,952.941 981.147,954.793 982.327,956.643 983.51,958.491 \n",
       "  984.695,960.337 985.883,962.182 987.074,964.025 988.268,965.866 989.464,967.706 990.664,969.543 991.866,971.379 993.072,973.213 994.281,975.045 995.492,976.876 \n",
       "  996.707,978.704 997.925,980.531 999.146,982.356 1000.37,984.179 1001.6,986 1002.83,987.819 1004.06,989.636 1005.3,991.451 1006.54,993.264 1007.78,995.075 \n",
       "  1009.03,996.884 1010.28,998.691 1011.53,1000.5 1012.79,1002.3 1014.05,1004.1 1015.31,1005.9 1016.58,1007.69 1017.85,1009.49 1019.12,1011.28 1020.4,1013.07 \n",
       "  1021.68,1014.86 1022.97,1016.64 1024.25,1018.42 1025.55,1020.2 1026.84,1021.98 1028.14,1023.76 1029.44,1025.53 1030.75,1027.3 1032.06,1029.07 1033.37,1030.84 \n",
       "  1034.69,1032.6 1036.01,1034.36 1037.34,1036.12 1038.67,1037.88 1040,1039.63 1041.34,1041.38 1042.68,1043.13 1044.03,1044.88 1045.38,1046.62 1046.73,1048.36 \n",
       "  1048.09,1050.1 1049.45,1051.84 1050.82,1053.57 1052.19,1055.3 1053.57,1057.03 1054.95,1058.75 1056.33,1060.48 1057.72,1062.19 1059.11,1063.91 1060.51,1065.63 \n",
       "  1061.91,1067.34 1063.32,1069.05 1064.73,1070.75 1066.15,1072.45 1067.57,1074.15 1069,1075.85 1070.43,1077.55 1071.87,1079.24 1073.31,1080.93 1074.75,1082.61 \n",
       "  1076.2,1084.3 1077.66,1085.98 1079.12,1087.65 1080.59,1089.33 1082.06,1091 1083.54,1092.67 1085.02,1094.33 1086.51,1095.99 1088,1097.65 1089.5,1099.31 \n",
       "  1091.01,1100.96 1092.52,1102.61 1094.04,1104.26 1095.56,1105.9 1097.09,1107.54 1098.62,1109.18 1100.16,1110.81 1101.7,1112.44 1103.26,1114.07 1104.81,1115.69 \n",
       "  1106.38,1117.32 1107.95,1118.93 1109.52,1120.55 1111.11,1122.16 1112.7,1123.77 1114.29,1125.37 1115.89,1126.98 1117.5,1128.58 1119.12,1130.17 1120.74,1131.76 \n",
       "  1122.37,1133.35 1124.01,1134.94 1125.65,1136.52 1127.3,1138.1 1128.95,1139.67 1130.62,1141.24 1132.29,1142.81 1133.97,1144.38 1135.65,1145.94 1137.35,1147.5 \n",
       "  1139.05,1149.05 1140.75,1150.6 1142.47,1152.15 1144.19,1153.7 1145.92,1155.24 1147.66,1156.78 1149.41,1158.31 1151.16,1159.84 1152.93,1161.37 1154.7,1162.89 \n",
       "  1156.48,1164.41 1158.26,1165.93 1160.06,1167.44 1161.86,1168.95 1163.67,1170.46 1165.49,1171.96 1167.32,1173.46 1169.16,1174.95 1171.01,1176.44 1172.86,1177.93 \n",
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       "  1040.87,828.674 1042.41,831.34 1043.95,834.002 1045.5,836.662 1047.06,839.319 1048.62,841.973 1050.18,844.624 1051.74,847.273 1053.32,849.918 1054.89,852.561 \n",
       "  1056.47,855.2 1058.06,857.836 1059.64,860.47 1061.24,863.1 1062.84,865.727 1064.44,868.351 1066.05,870.971 1067.66,873.588 1069.28,876.202 1070.9,878.813 \n",
       "  1072.52,881.42 1074.16,884.024 1075.79,886.624 1077.44,889.221 1079.08,891.814 1080.74,894.404 1082.39,896.99 1084.06,899.572 1085.72,902.151 1087.4,904.726 \n",
       "  1089.08,907.297 1090.76,909.864 1092.45,912.428 1094.15,914.988 1095.85,917.543 1097.56,920.095 1099.27,922.643 1100.99,925.187 1102.72,927.727 1104.45,930.263 \n",
       "  1106.19,932.795 1107.94,935.323 1109.69,937.846 1111.44,940.365 1113.21,942.881 1114.98,945.391 1116.75,947.898 1118.54,950.4 1120.33,952.898 1122.12,955.392 \n",
       "  1123.93,957.881 1125.74,960.365 1127.55,962.846 1129.38,965.321 1131.21,967.792 1133.05,970.259 1134.89,972.721 1136.75,975.178 1138.61,977.631 1140.48,980.079 \n",
       "  1142.35,982.522 1144.24,984.961 1146.13,987.395 1148.03,989.824 1149.94,992.248 1151.85,994.667 1153.77,997.081 1155.71,999.491 1157.65,1001.9 1159.59,1004.29 \n",
       "  1161.55,1006.69 1163.52,1009.08 1165.49,1011.46 1167.47,1013.84 1169.46,1016.22 1171.46,1018.59 1173.47,1020.95 1175.49,1023.31 1177.52,1025.66 1179.55,1028.01 \n",
       "  1181.6,1030.35 1183.65,1032.69 1185.72,1035.02 1187.79,1037.35 1189.87,1039.67 1191.97,1041.99 1194.07,1044.3 1196.18,1046.61 1198.31,1048.91 1200.44,1051.2 \n",
       "  1202.58,1053.49 1204.74,1055.77 1206.9,1058.05 1209.08,1060.32 1211.26,1062.59 1213.46,1064.85 1215.66,1067.11 1217.88,1069.35 1220.11,1071.6 1222.35,1073.84 \n",
       "  1224.6,1076.07 1226.86,1078.3 1229.14,1080.52 1231.42,1082.73 1233.72,1084.94 1236.03,1087.15 1238.35,1089.34 1240.69,1091.54 1243.04,1093.72 1245.4,1095.91 \n",
       "  1247.78,1098.08 1250.17,1100.25 1252.57,1102.42 1254.99,1104.58 1257.43,1106.73 1259.87,1108.88 1262.33,1111.03 1264.81,1113.16 1267.29,1115.3 1269.8,1117.42 \n",
       "  1272.31,1119.54 1274.85,1121.66 1277.39,1123.76 1279.95,1125.87 1282.53,1127.96 1285.12,1130.05 1287.72,1132.14 1290.34,1134.22 1292.97,1136.29 1295.62,1138.35 \n",
       "  1298.29,1140.41 1300.97,1142.47 1303.66,1144.52 1306.37,1146.56 1309.1,1148.59 1311.84,1150.62 1314.6,1152.64 1317.38,1154.66 1320.17,1156.67 1322.98,1158.67 \n",
       "  1325.8,1160.67 1328.65,1162.66 1331.51,1164.64 1334.38,1166.62 1337.28,1168.59 1340.19,1170.55 1343.12,1172.51 1346.07,1174.46 1349.04,1176.4 1352.03,1178.34 \n",
       "  1355.03,1180.27 1358.06,1182.19 1361.1,1184.11 1364.17,1186.02 1367.25,1187.92 1370.36,1189.82 1373.48,1191.71 1376.63,1193.59 1379.79,1195.47 1382.98,1197.33 \n",
       "  1386.19,1199.2 1389.43,1201.05 1392.68,1202.9 1395.96,1204.74 1399.26,1206.58 1402.58,1208.4 1405.93,1210.22 1409.3,1212.04 1412.7,1213.84 1416.12,1215.64 \n",
       "  1419.57,1217.44 1423.04,1219.22 1426.53,1221 1430.06,1222.77 1433.61,1224.54 1437.18,1226.3 1440.79,1228.05 1444.42,1229.79 1448.08,1231.53 1451.76,1233.26 \n",
       "  1455.48,1234.98 1459.23,1236.69 1463,1238.4 1466.81,1240.1 1470.64,1241.8 1474.51,1243.49 1478.41,1245.17 1482.34,1246.84 1486.3,1248.51 1490.3,1250.17 \n",
       "  1494.33,1251.82 1498.39,1253.46 1502.49,1255.1 1506.62,1256.73 1510.79,1258.36 1514.99,1259.97 1519.23,1261.59 1523.5,1263.19 1527.81,1264.79 1532.16,1266.37 \n",
       "  1536.55,1267.96 1540.98,1269.53 1545.45,1271.1 1549.95,1272.66 1554.5,1274.22 1559.09,1275.77 1563.71,1277.31 1568.39,1278.84 1573.1,1280.37 1577.86,1281.89 \n",
       "  1582.66,1283.4 1587.5,1284.91 1592.39,1286.41 1597.33,1287.91 1602.31,1289.39 1607.33,1290.87 1612.41,1292.35 1617.53,1293.81 1622.7,1295.27 1627.92,1296.73 \n",
       "  1633.2,1298.17 1638.52,1299.61 1643.89,1301.05 1649.31,1302.48 1654.79,1303.9 1660.32,1305.31 1665.9,1306.72 1671.54,1308.12 1677.23,1309.52 1682.98,1310.9 \n",
       "  1688.78,1312.29 1694.64,1313.66 1700.57,1315.03 1706.55,1316.4 1712.6,1317.75 1718.71,1319.1 1724.89,1320.44 1731.13,1321.78 1737.44,1323.1 1743.81,1324.43 \n",
       "  1750.25,1325.74 1756.77,1327.05 1763.35,1328.35 1770,1329.64 1776.73,1330.92 1783.53,1332.2 1790.4,1333.47 1797.35,1334.74 1804.38,1335.99 1811.48,1337.24 \n",
       "  1818.67,1338.49 1825.93,1339.72 1833.28,1340.95 1840.72,1342.17 1848.23,1343.39 1855.84,1344.59 1863.54,1345.79 1871.32,1346.99 1879.2,1348.17 1887.18,1349.35 \n",
       "  1895.25,1350.52 1903.41,1351.68 1911.68,1352.84 1920.05,1353.99 1928.53,1355.13 1937.11,1356.27 1945.8,1357.39 1954.6,1358.51 1963.52,1359.63 1972.55,1360.73 \n",
       "  1981.69,1361.83 1990.96,1362.92 2000.35,1364.01 2009.87,1365.09 2019.51,1366.16 2029.29,1367.22 2039.19,1368.27 2049.24,1369.32 2059.42,1370.36 2069.74,1371.4 \n",
       "  2080.21,1372.43 2090.82,1373.45 2101.59,1374.46 2112.5,1375.46 2123.58,1376.46 2134.81,1377.46 2146.2,1378.44 2157.76,1379.42 2169.49,1380.39 2181.39,1381.35 \n",
       "  \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip142)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" stroke-dasharray=\"2, 4\" points=\"\n",
       "  467.732,1445.72 467.732,87.9763 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fun2(z,p,t)\n",
    "    α = p\n",
    "    zpoint = similar(z)\n",
    "    zpoint[1] = -z[1] + α*z[1]^2 + z[2]\n",
    "    zpoint[2] = (1. - 2*α*z[1])*z[2]\n",
    "    return zpoint\n",
    "end\n",
    "\n",
    "t0 = 0; tf = 1.;\n",
    "α = 1.5\n",
    "seg = range(-0.,2,length=10)\n",
    "Z0 = transpose([-ones(size(seg)) seg])\n",
    "plt1 = plot()\n",
    "plot_traj(plt1,fun2,Z0,tf,par=α)\n",
    "\n",
    "seg = range(-0.,2,length=200)\n",
    "Z0 = transpose([-ones(size(seg)) seg])\n",
    "plot_flow(plt1,fun2,Z0,tf,par=α)\n",
    "#plot!(plt1, xlims = (-2.,1))\n",
    "plot!(plt1,[0.,0.],[0.,4], linestyle = :dot, color = :black)\n",
    "plot(plt1)"
   ]
  }
 ],
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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