In [1]:
import numpy as np
from scipy.integrate import solve_ivp
from scipy.integrate import odeint
import matplotlib.pyplot as plt
%matplotlib inline
import math
from scipy.optimize import broyden1
In [2]:
A11 = np.array([[0.45, 0.25], [0.15, 0.45]])
A12 = np.array([[0.25, 0.25]]).T
A21 = np.array([[0.15, 0.05]])
A22 = np.array([0.35])
B1 = np.array([[0.25, 0.35], [0.65, 0.15]])
B2 = np.array([[0.1, 0.05]]).T
x1_0 = np.array([[3550, 1750]]).T
def c1(t):
return np.array([850, 550]) * np.exp(0.025*t)
c2 = 75
In [3]:
A22_neg_inv = np.linalg.inv(np.eye(1) - A22)
A1 = A11 + np.dot(np.dot(A12, A22_neg_inv), A21)
B = B1 + np.dot(np.dot(B2, A22_neg_inv), A21)
def c(t):
return c1(t) - np.dot(A22_neg_inv.T, A12.T).reshape(A12.shape[0]) * c2
In [4]:
B_inv = np.linalg.inv(B)
A1_neg_inv = np.linalg.inv(np.eye(A1.shape[0]) - A1)
def x_der(t, x):
return np.dot(B_inv, x) - np.dot(np.dot(B_inv, A1), x) - np.dot(B_inv, c(t))
def x_der_change_params(x, t = 0):
return np.dot(B_inv, x) - np.dot(np.dot(B_inv, A1), x) - np.dot(B_inv, c(t))
In [5]:
t = np.linspace(1, 10, 10)
x1 = solve_ivp(x_der, [1, 10], np.array([3550, 1750]), t_eval=t).y
x2 = np.dot(A22_neg_inv, (np.dot(A21, x1) - c2))
print(x1)
[[3550. 3307.69463124 3427.90806514 3598.47070813 3811.71773205
4084.03487099 4445.7337662 4932.03559713 5597.89591791 6525.20833033]
[1750. 2474.46950298 2633.03501587 2783.51690952 2972.49637834
3220.56781044 3546.59911285 3993.32832852 4616.66649301 5494.80202871]]
In [6]:
plt.figure(figsize=(20, 10))
plt.plot(t, x1[0], 'ro', t, x1[1], 'bx', t, x2[0], 'go')
plt.show()
In [7]:
y1_0 = np.dot((np.eye(A1.shape[0]) - A1), x1_0)
y1_0
Out[7]:
array([[1276.53846154],
[ 191.53846154]])
In [8]:
p1 = broyden1(x_der_change_params, [1, 1])
print('Точка спокою загальної системи: {}'.format(p1))
Точка спокою загальної системи: [2805.24344386 2079.58801269]
In [9]:
def y_der(t, y):
return np.dot(np.dot(np.eye(A1.shape[0]) - A1, B_inv), y)
In [10]:
y1 = solve_ivp(y_der, [1, 10], [3550, 1750], t_eval=t).y
print(y1)
[[ 3550. 3538.22398939 5055.33190581 7356.02945239
10714.59153106 15597.64211781 22719.68443817 33115.08477008
48227.67763495 70212.96821049]
[ 1750. 4460.64311496 6611.25178432 9636.74597497
14034.04259497 20448.33660907 29778.83512322 43340.99079924
63125.82522697 91970.79653559]]
In [13]:
plt.figure(figsize=(20, 10))
plt.plot(t, y1[0], 'ro', t, y1[1], 'bx')
plt.show()
In [14]:
eigen_values, vector = np.linalg.eig(np.dot(A1_neg_inv, B))
print(np.dot(A1_neg_inv, B))
tech_growth = 1.0 / np.max(eigen_values)
print(tech_growth)
TIME_RANGE = 10
X1_tech_growth = [[3550, 1750]]
st = 1e-3
eps = 1e-3
while st < TIME_RANGE + 2.0:
new_X1 = np.dot(X1_tech_growth[0], np.exp(tech_growth * st))
X1_tech_growth.append(new_X1)
st += eps
print( X1_tech_growth)
[[1.57288966 1.12604437]
[1.86185537 0.73048113]]
0.375989906283497
[[3550, 1750], array([3551.33501513, 1750.65810605]), array([3552.6705323 , 1751.31645959]), array([3554.00655171, 1751.9750607 ]), array([3555.34307355, 1752.63390949]), array([3556.68009799, 1753.29300605]), array([3558.01762524, 1753.95235047]), array([3559.35565548, 1754.61194284]), array([3560.6941889 , 1755.27178326]), array([3562.03322569, 1755.93187182]), array([3563.37276604, 1756.59220861]), array([3564.71281014, 1757.25279373]), array([3566.05335818, 1757.91362727]), array([3567.39441034, 1758.57470932]), array([3568.73596682, 1759.23603998]), array([3570.07802781, 1759.89761934]), array([3571.42059349, 1760.5594475 ]), array([3572.76366406, 1761.22152454]), array([3574.1072397 , 1761.88385056]), array([3575.45132062, 1762.54642566]), array([3576.79590698, 1763.20924992]), array([3578.14099899, 1763.87232345]), array([3579.48659684, 1764.53564633]), array([3580.83270072, 1765.19921866]), array([3582.17931081, 1765.86304054]), array([3583.52642731, 1766.52711205]), array([3584.8740504, 1767.1914333]), array([3586.22218029, 1767.85600437]), array([3587.57081715, 1768.52082536]), array([3588.91996118, 1769.18589636]), array([3590.26961257, 1769.85121747]), array([3591.61977152, 1770.51678878]), array([3592.9704382 , 1771.18261038]), array([3594.32161282, 1771.84868237]), array([3595.67329556, 1772.51500485]), array([3597.02548661, 1773.18157791]), array([3598.37818617, 1773.84840163]), array([3599.73139443, 1774.51547613]), array([3601.08511157, 1775.18280148]), array([3602.4393378 , 1775.85037779]), array([3603.7940733 , 1776.51820515]), array([3605.14931826, 1777.18628365]), array([3606.50507287, 1777.85461339]), array([3607.86133733, 1778.52319446]), array([3609.21811183, 1779.19202696]), array([3610.57539655, 1779.86111098]), array([3611.9331917 , 1780.53044661]), array([3613.29149746, 1781.20003396]), array([3614.65031403, 1781.86987311]), array([3616.00964159, 1782.53996416]), array([3617.36948034, 1783.21030721]), array([3618.72983048, 1783.88090235]), array([3620.09069219, 1784.55174967]), array([3621.45206566, 1785.22284927]), array([3622.8139511 , 1785.89420124]), array([3624.17634868, 1786.56580569]), array([3625.53925861, 1787.2376627 ]), array([3626.90268108, 1787.90977236]), array([3628.26661627, 1788.58213478]), array([3629.63106439, 1789.25475005]), array([3630.99602563, 1789.92761827]), array([3632.36150017, 1790.60073952]), array([3633.72748821, 1791.27411391]), array([3635.09398995, 1791.94774152]), array([3636.46100557, 1792.62162247]), array([3637.82853528, 1793.29575683]), array([3639.19657926, 1793.9701447 ]), array([3640.5651377 , 1794.64478619]), array([3641.93421081, 1795.31968139]), array([3643.30379877, 1795.99483038]), array([3644.67390178, 1796.67023327]), array([3646.04452004, 1797.34589016]), array([3647.41565372, 1798.02180113]), array([3648.78730304, 1798.69796629]), array([3650.15946818, 1799.37438572]), array([3651.53214934, 1800.05105953]), array([3652.90534671, 1800.72798781]), array([3654.27906048, 1801.40517066]), array([3655.65329085, 1802.08260817]), array([3657.02803802, 1802.76030043]), array([3658.40330218, 1803.43824755]), array([3659.77908352, 1804.11644962]), array([3661.15538223, 1804.79490673]), array([3662.53219852, 1805.47361899]), array([3663.90953257, 1806.15258648]), array([3665.28738459, 1806.8318093 ]), array([3666.66575476, 1807.51128756]), array([3668.04464328, 1808.19102133]), array([3669.42405035, 1808.87101073]), array([3670.80397615, 1809.55125585]), array([3672.1844209 , 1810.23175678]), array([3673.56538477, 1810.91251362]), array([3674.94686797, 1811.59352646]), array([3676.32887069, 1812.27479541]), array([3677.71139313, 1812.95632056]), array([3679.09443548, 1813.638102 ]), array([3680.47799794, 1814.32013983]), array([3681.8620807 , 1815.00243415]), array([3683.24668396, 1815.68498505]), array([3684.63180792, 1816.36779264]), array([3686.01745276, 1817.050857 ]), array([3687.4036187 , 1817.73417823]), array([3688.79030591, 1818.41775644]), array([3690.17751461, 1819.10159171]), array([3691.56524497, 1819.78568414]), array([3692.95349721, 1820.47003384]), array([3694.34227152, 1821.15464089]), array([3695.73156809, 1821.83950539]), array([3697.12138711, 1822.52462745]), array([3698.5117288 , 1823.21000715]), array([3699.90259334, 1823.8956446 ]), array([3701.29398092, 1824.58153989]), array([3702.68589176, 1825.26769312]), array([3704.07832603, 1825.95410438]), array([3705.47128395, 1826.64077378]), array([3706.8647657, 1827.3277014]), array([3708.25877148, 1828.01488735]), array([3709.6533015 , 1828.70233173]), array([3711.04835595, 1829.39003462]), array([3712.44393501, 1830.07799613]), array([3713.84003891, 1830.76621636]), array([3715.23666782, 1831.4546954 ]), array([3716.63382194, 1832.14343335]), array([3718.03150149, 1832.83243031]), array([3719.42970664, 1833.52168637]), array([3720.82843761, 1834.21120164]), array([3722.22769458, 1834.9009762 ]), array([3723.62747776, 1835.59101016]), array([3725.02778734, 1836.28130362]), array([3726.42862352, 1836.97185666]), array([3727.8299865, 1837.6626694]), array([3729.23187648, 1838.35374193]), array([3730.63429365, 1839.04507434]), array([3732.03723822, 1839.73666673]), array([3733.44071038, 1840.4285192 ]), array([3734.84471034, 1841.12063186]), array([3736.24923828, 1841.81300478]), array([3737.6542944 , 1842.50563809]), array([3739.05987892, 1843.19853186]), array([3740.46599202, 1843.89168621]), array([3741.8726339 , 1844.58510122]), array([3743.27980476, 1845.278777 ]), array([3744.68750481, 1845.97271364]), array([3746.09573424, 1846.66691124]), array([3747.50449325, 1847.36136991]), array([3748.91378203, 1848.05608973]), array([3750.3236008 , 1848.75107081]), array([3751.73394974, 1849.44631325]), array([3753.14482905, 1850.14181714]), array([3754.55623895, 1850.83758258]), array([3755.96817962, 1851.53360967]), array([3757.38065126, 1852.22989851]), array([3758.79365408, 1852.9264492 ]), array([3760.20718828, 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array([3797.14624603, 1871.83265649]), array([3798.57420312, 1872.536579 ]), array([3800.00269721, 1873.24076623]), array([3801.4317285 , 1873.94521828]), array([3802.8612972 , 1874.64993524]), array([3804.2914035 , 1875.35491722]), array([3805.7220476 , 1876.06016431]), array([3807.15322972, 1876.76567662]), array([3808.58495004, 1877.47145425]), array([3810.01720878, 1878.17749729]), array([3811.45000614, 1878.88380584]), array([3812.88334231, 1879.59038001]), array([3814.3172175, 1880.2972199]), array([3815.75163192, 1881.0043256 ]), array([3817.18658577, 1881.71169721]), array([3818.62207924, 1882.41933484]), array([3820.05811255, 1883.12723858]), array([3821.49468589, 1883.83540854]), array([3822.93179948, 1884.54384481]), array([3824.3694535, 1885.2525475]), array([3825.80764817, 1885.9615167 ]), array([3827.24638369, 1886.67075252]), array([3828.68566026, 1887.38025506]), array([3830.12547808, 1888.09002441]), array([3831.56583736, 1888.80006067]), array([3833.00673831, 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array([3870.66095526, 1908.07230189]), array([3872.11655834, 1908.7898527 ]), array([3873.57270882, 1909.50767336]), array([3875.02940689, 1910.22576396]), array([3876.48665277, 1910.94412461]), array([3877.94444667, 1911.6627554 ]), array([3879.40278878, 1912.38165644]), array([3880.86167932, 1913.10082783]), array([3882.32111849, 1913.82026968]), array([3883.78110649, 1914.53998207]), array([3885.24164354, 1915.25996513]), array([3886.70272984, 1915.98021894]), array([3888.1643656 , 1916.70074361]), array([3889.62655102, 1917.42153924]), array([3891.08928632, 1918.14260593]), array([3892.55257168, 1918.86394379]), array([3894.01640734, 1919.58555291]), array([3895.48079348, 1920.30743341]), array([3896.94573032, 1921.02958537]), array([3898.41121807, 1921.75200891]), array([3899.87725693, 1922.47470412]), array([3901.34384711, 1923.19767111]), array([3902.81098881, 1923.92090998]), array([3904.27868225, 1924.64442083]), array([3905.74692763, 1925.36820376]), array([3907.21572517, 1926.09225888]), array([3908.68507505, 1926.81658629]), array([3910.1549775 , 1927.54118609]), array([3911.62543273, 1928.26605839]), array([3913.09644093, 1928.99120328]), array([3914.56800233, 1929.71662087]), array([3916.04011712, 1930.44231125]), array([3917.51278551, 1931.16827455]), array([3918.98600772, 1931.89451085]), array([3920.45978394, 1932.62102025]), array([3921.9341144 , 1933.34780287]), array([3923.40899929, 1934.07485881]), array([3924.88443883, 1934.80218816]), array([3926.36043323, 1935.52979103]), array([3927.83698268, 1936.25766752]), array([3929.31408741, 1936.98581774]), array([3930.79174762, 1937.71424179]), array([3932.26996352, 1938.44293977]), array([3933.74873532, 1939.17191178]), array([3935.22806323, 1939.90115793]), array([3936.70794746, 1940.63067832]), array([3938.18838821, 1941.36047306]), array([3939.66938569, 1942.09054224]), array([3941.15094012, 1942.82088598]), array([3942.63305171, 1943.55150436]), array([3944.11572065, 1944.28239751]), array([3945.59894718, 1945.01356551]), array([3947.08273148, 1945.74500848]), array([3948.56707378, 1946.47672651]), array([3950.05197428, 1947.20871972]), array([3951.53743319, 1947.94098819]), array([3953.02345073, 1948.67353205]), array([3954.5100271 , 1949.40635139]), array([3955.99716251, 1950.13944631]), array([3957.48485717, 1950.87281692]), array([3958.9731113 , 1951.60646332]), array([3960.4619251 , 1952.34038561]), array([3961.95129878, 1953.07458391]), array([3963.44123256, 1953.80905831]), array([3964.93172665, 1954.54380891]), array([3966.42278125, 1955.27883583]), array([3967.91439658, 1956.01413916]), array([3969.40657285, 1956.74971901]), array([3970.89931026, 1957.48557548]), array([3972.39260904, 1958.22170868]), array([3973.88646938, 1958.95811871]), array([3975.38089151, 1959.69480567]), array([3976.87587563, 1960.43176968]), array([3978.37142195, 1961.16901082]), array([3979.8675307 , 1961.90652922]), array([3981.36420206, 1962.64432496]), array([3982.86143627, 1963.38239816]), array([3984.35923353, 1964.12074892]), array([3985.85759405, 1964.85937735]), array([3987.35651805, 1965.59828354]), array([3988.85600573, 1966.33746761]), array([3990.35605731, 1967.07692966]), array([3991.856673 , 1967.81666979]), array([3993.35785301, 1968.55668811]), array([3994.85959756, 1969.29698471]), array([3996.36190686, 1970.03755972]), array([3997.86478111, 1970.77841322]), array([3999.36822054, 1971.51954533]), array([4000.87222535, 1972.26095616]), array([4002.37679575, 1973.00264579]), array([4003.88193197, 1973.74461435]), array([4005.38763421, 1974.48686193]), array([4006.89390268, 1975.22938865]), array([4008.40073761, 1975.97219459]), array([4009.90813919, 1976.71527988]), array([4011.41610765, 1977.45864461]), array([4012.92464319, 1978.2022889 ]), array([4014.43374604, 1978.94621284]), array([4015.9434164 , 1979.69041654]), array([4017.45365449, 1980.4349001 ]), array([4018.96446052, 1981.17966364]), array([4020.4758347 , 1981.92470725]), 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array([320715.58455511, 158099.23182294]), array([320836.19305006, 158158.68671482]), array([320956.84690111, 158218.16396534]), array([321077.54612532, 158277.6635829 ]), array([321198.29073975, 158337.18557593]), array([321319.08076148, 158396.72995284]), array([321439.91620757, 158456.29672204]), array([321560.79709511, 158515.88589196]), array([321681.7234412 , 158575.49747101]), array([321802.69526291, 158635.13146763]), array([321923.71257736, 158694.78789025]), array([322044.77540166, 158754.4667473 ]), array([322165.88375292, 158814.16804721]), array([322287.03764825, 158873.89179843]), array([322408.23710479, 158933.6380094 ]), array([322529.48213967, 158993.40668857]), array([322650.77277003, 159053.19784438]), array([322772.10901302, 159113.01148529]), array([322893.49088579, 159172.84761975]), array([323014.91840549, 159232.70625623]), array([323136.39158931, 159292.58740318]), array([323257.9104544 , 159352.49106907]), array([323379.47501795, 159412.41726237])]
In [25]:
plt.plot(x1[0], x1[1], 'g',
[x1_tg[0] for x1_tg in X1_tech_growth], [x1_tg[1] for x1_tg in X1_tech_growth], 'r',
y1[0], y1[1], 'b')
plt.axis([0, 20000, 0, 50000])
print(x1[0])
plt.show()
[3550. 3307.69463124 3427.90806514 3598.47070813 3811.71773205
4084.03487099 4445.7337662 4932.03559713 5597.89591791 6525.20833033]
In [ ]:
In [ ]:
Content source: lionell/laboratories
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