Solve an ODE: the Van der Pol problem

from __future__ import annotations

import matplotlib.pyplot as plt
from numpy import array
from numpy import zeros

from gemseo.algos.ode.ode_problem import ODEProblem
from gemseo.algos.ode.ode_solvers_factory import ODESolversFactory
from gemseo.problems.ode.van_der_pol import VanDerPol

This tutorial describes how to solve an ordinary differential equation (ODE) problem with GEMSEO. A first-order ODE is a differential equation that can be written as

\[\frac{ds(t)}{dt} = f(t, s(t))\]

where the right-hand side of the equation \(f(t, s(t))\) is a function of time \(t\) and of a state \(s(t)\) that returns another state \(\frac{ds(t)}{dt}\) (see Hartman, Philip (2002) [1964], Ordinary differential equations, Classics in Applied Mathematics, vol. 38, Philadelphia: Society for Industrial and Applied Mathematics). To solve this equation, initial conditions must be set:

\[s(t_0) = s_0\]

For this example, we consider the Van der Pol equation as an example of a time-independent problem.

Solving the Van der Pol time-independent problem

The Van der Pol problem describes the position over time of an oscillator with non-linear damping:

\[\frac{d^2 x(t)}{dt^2} - \mu (1-x^2(t)) \frac{dx(t)}{dt} + x(t) = 0\]

where \(x(t)\) is the position coordinate at time \(t\), and \(\mu\) is the stiffness parameter.

To solve the Van der Pol problem with GEMSEO, we first need to model this second-order equation as a first-order equation. Let \(y = \frac{dx}{dt}\) and \(s = \begin{pmatrix}x\\y\end{pmatrix}\). Then the Van der Pol problem can be rewritten:

\[\begin{split}\frac{ds(t)}{dt} = f(t, s(t)) = \begin{pmatrix} y(t) \\ \mu (1-x^2(t)) y(t) - x(t) \end{pmatrix}\end{split}\]

Step 1 : Defining the problem

mu = 5


def evaluate_f(time, state):
    return state[1], mu * state[1] * (1 - state[0] ** 2) - state[0]


initial_state = array([2, -2 / 3])
initial_time = 0.0
final_time = 50.0
ode_problem = ODEProblem(evaluate_f, initial_state, initial_time, final_time)

By default, the Jacobian of the problem is approximated using the finite difference method. However, it is possible to define an explicit expression of the Jacobian and pass it to the problem. In the case of the Van der Pol problem, this would be:

def evaluate_jac(time, state):
    jac = zeros((2, 2))
    jac[1, 0] = -mu * 2 * state[1] * state[0] - 1
    jac[0, 1] = 1
    jac[1, 1] = mu * (1 - state[0] * state[0])
    return jac


ode_problem = ODEProblem(
    evaluate_f, initial_state, initial_time, final_time, jac=evaluate_jac
)

Step 2: Solving the ODE problem

Whether the Jacobian is specified or not, once the problem is defined, the ODE solver is called on the ODEProblem by using the ODESolversFactory:

ODESolversFactory().execute(ode_problem)

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/scipy/integrate/_ivp/ivp.py:616: UserWarning: The following arguments have no effect for a chosen solver: `min_step`, `jac`.
  solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)

ODEResult(time_vector=array([0.00000000e+00, 3.74532749e-02, 1.03428167e-01, 1.72202666e-01,
       2.49134295e-01, 3.37974639e-01, 4.45392319e-01, 5.81153227e-01,
       7.61409032e-01, 1.00893026e+00, 1.31816564e+00, 1.63105754e+00,
       1.92924608e+00, 2.24281802e+00, 2.59456065e+00, 2.97621895e+00,
       3.39050042e+00, 3.86048412e+00, 4.44478615e+00, 4.69135185e+00,
       4.84571211e+00, 5.00007236e+00, 5.07314736e+00, 5.14622236e+00,
       5.20628412e+00, 5.26634588e+00, 5.31863072e+00, 5.37091555e+00,
       5.42944041e+00, 5.50218147e+00, 5.59077871e+00, 5.70214103e+00,
       5.84768592e+00, 6.04941392e+00, 6.35882583e+00, 6.63377208e+00,
       6.90871832e+00, 7.18731601e+00, 7.47587710e+00, 7.78663249e+00,
       8.12193118e+00, 8.47939066e+00, 8.86396069e+00, 9.28743242e+00,
       9.77435635e+00, 1.02491719e+01, 1.05013670e+01, 1.06512736e+01,
       1.08011802e+01, 1.08771177e+01, 1.09530552e+01, 1.10132681e+01,
       1.10734810e+01, 1.11253580e+01, 1.11772350e+01, 1.12362346e+01,
       1.13093179e+01, 1.13984316e+01, 1.15105216e+01, 1.16572100e+01,
       1.18608945e+01, 1.21748138e+01, 1.24467397e+01, 1.27186657e+01,
       1.29981260e+01, 1.32897352e+01, 1.36018848e+01, 1.39369708e+01,
       1.42948273e+01, 1.46805863e+01, 1.51056174e+01, 1.55953182e+01,
       1.60614872e+01, 1.63105618e+01, 1.64584845e+01, 1.66064072e+01,
       1.66787060e+01, 1.67510048e+01, 1.68130053e+01, 1.68750058e+01,
       1.69273591e+01, 1.69797124e+01, 1.70381842e+01, 1.71108942e+01,
       1.71994403e+01, 1.73107277e+01, 1.74561489e+01, 1.76576578e+01,
       1.79665784e+01, 1.82419129e+01, 1.85172475e+01, 1.87958242e+01,
       1.90841554e+01, 1.93948765e+01, 1.97303185e+01, 2.00878750e+01,
       2.04725021e+01, 2.08960759e+01, 2.13831962e+01, 2.18573634e+01,
       2.21093633e+01, 2.22591224e+01, 2.24088815e+01, 2.24848402e+01,
       2.25607989e+01, 2.26209580e+01, 2.26811171e+01, 2.27329936e+01,
       2.27848702e+01, 2.28438721e+01, 2.29169574e+01, 2.30060740e+01,
       2.31181680e+01, 2.32648621e+01, 2.34685561e+01, 2.37824982e+01,
       2.40544125e+01, 2.43263269e+01, 2.46057957e+01, 2.48974259e+01,
       2.52095892e+01, 2.55446813e+01, 2.59025480e+01, 2.62883237e+01,
       2.67133785e+01, 2.72031227e+01, 2.76691667e+01, 2.79181865e+01,
       2.80660802e+01, 2.82139738e+01, 2.82864685e+01, 2.83589632e+01,
       2.84208246e+01, 2.84826860e+01, 2.85350130e+01, 2.85873400e+01,
       2.86458433e+01, 2.87185763e+01, 2.88071568e+01, 2.89184928e+01,
       2.90639904e+01, 2.92656299e+01, 2.95748477e+01, 2.98499848e+01,
       3.01251219e+01, 3.04037521e+01, 3.06922812e+01, 3.10030948e+01,
       3.13385224e+01, 3.16961024e+01, 3.20808051e+01, 3.25044774e+01,
       3.29917723e+01, 3.34653659e+01, 3.37171885e+01, 3.38668170e+01,
       3.40164454e+01, 3.40921259e+01, 3.41678063e+01, 3.42281267e+01,
       3.42884472e+01, 3.43403581e+01, 3.43922691e+01, 3.44512341e+01,
       3.45242937e+01, 3.46133708e+01, 3.47254090e+01, 3.48720149e+01,
       3.50755574e+01, 3.53891472e+01, 3.56613041e+01, 3.59334609e+01,
       3.62128649e+01, 3.65042628e+01, 3.68163278e+01, 3.71514459e+01,
       3.75092906e+01, 3.78949860e+01, 3.83199373e+01, 3.88094972e+01,
       3.92760755e+01, 3.95253273e+01, 3.96733450e+01, 3.98213626e+01,
       3.98930355e+01, 3.99647084e+01, 4.00271482e+01, 4.00895880e+01,
       4.01420268e+01, 4.01944656e+01, 4.02528341e+01, 4.03254689e+01,
       4.04139021e+01, 4.05250303e+01, 4.06702014e+01, 4.08712824e+01,
       4.11792328e+01, 4.14552096e+01, 4.17311864e+01, 4.20095911e+01,
       4.22972773e+01, 4.26076948e+01, 4.29431828e+01, 4.33006627e+01,
       4.36850438e+01, 4.41082958e+01, 4.45948478e+01, 4.50709142e+01,
       4.53234712e+01, 4.54736619e+01, 4.56238526e+01, 4.56989836e+01,
       4.57741146e+01, 4.58349486e+01, 4.58957825e+01, 4.59477578e+01,
       4.59997331e+01, 4.60586192e+01, 4.61316207e+01, 4.62206109e+01,
       4.63325270e+01, 4.64789422e+01, 4.66821594e+01, 4.69949923e+01,
       4.72676485e+01, 4.75403046e+01, 4.78195481e+01, 4.81104158e+01,
       4.84222296e+01, 4.87573635e+01, 4.91151157e+01, 4.95005786e+01,
       4.99252199e+01, 5.00000000e+01]), state_vector=array([[ 2.        ,  1.97967957,  1.95774631,  1.94308024,  1.93023242,
         1.91689798,  1.90127548,  1.88148097,  1.85471391,  1.81683912,
         1.7674537 ,  1.71475964,  1.6615001 ,  1.60161846,  1.52847272,
         1.43943516,  1.3256495 ,  1.15704393,  0.76442896,  0.32727557,
        -0.31175092, -1.40669805, -1.7830636 , -1.95322441, -2.00372775,
        -2.01952524, -2.02129221, -2.01828892, -2.01245847, -2.00373298,
        -1.99229352, -1.97745799, -1.95768762, -1.92970087, -1.88539809,
        -1.84447489, -1.80189035, -1.75680714, -1.70773982, -1.65170545,
        -1.58669355, -1.51069031, -1.41827814, -1.29674842, -1.10699929,
        -0.76489516, -0.31306239,  0.31498705,  1.38039866,  1.77914236,
         1.95543329,  2.00447559,  2.01959313,  2.02109742,  2.01802987,
         2.01210952,  2.00331643,  1.99179557,  1.97685256,  1.95691435,
         1.92863369,  1.88362919,  1.8430957 ,  1.80093013,  1.75565891,
         1.70599636,  1.64958478,  1.58443578,  1.50809415,  1.41497353,
         1.2921288 ,  1.09862316,  0.75519633,  0.29778774, -0.33465201,
        -1.38930815, -1.77053756, -1.94734159, -2.00247818, -2.01931568,
        -2.02112164, -2.01812688, -2.01230701, -2.00358771, -1.99215508,
        -1.97732845, -1.957573  , -1.92961428, -1.88537973, -1.84439696,
        -1.80174778, -1.75666125, -1.70762607, -1.65159047, -1.58653994,
        -1.51049806, -1.41804162, -1.29642152, -1.10641002, -0.76425327,
        -0.31201359,  0.31628606,  1.38088686,  1.77947218,  1.95556873,
         2.00449083,  2.01958186,  2.02108544,  2.01801763,  2.01209688,
         2.00330337,  1.99178199,  1.97683827,  1.95689902,  1.92861664,
         1.88360813,  1.84307559,  1.800911  ,  1.75563747,  1.70597007,
         1.64955415,  1.58440132,  1.50805357,  1.41492229,  1.29205718,
         1.09849233,  0.75503441,  0.2975422 , -0.33498128, -1.38948247,
        -1.77136104, -1.94792497, -2.00260855, -2.019325  , -2.02111299,
        -2.0181137 , -2.0122878 , -2.00356396, -1.99212591, -1.97729214,
        -1.95752556, -1.92954733, -1.88526671, -1.84430948, -1.80168776,
        -1.75658958, -1.70751595, -1.65145576, -1.58639673, -1.51033379,
        -1.41783264, -1.29612984, -1.10588414, -0.76367549, -0.31107458,
         0.31745544,  1.38133954,  1.77869893,  1.95493428,  2.0043413 ,
         2.01956479,  2.02108905,  2.01802607,  2.0121123 ,  2.00332388,
         1.99180863,  1.97687302,  1.95694653,  1.92868663,  1.88373246,
         1.8431679 ,  1.80096886,  1.75570879,  1.70608593,  1.6496962 ,
         1.58455019,  1.5082239 ,  1.41513991,  1.29236226,  1.09904929,
         0.75572133,  0.29858578, -0.33358466, -1.38874819, -1.76789103,
        -1.94545094, -2.00205344, -2.01928362, -2.02114867, -2.01816893,
        -2.01236884, -2.00366444, -1.99224954, -1.97744627, -1.95772729,
        -1.92983242, -1.88574821, -1.84468243, -1.80194396, -1.7568951 ,
        -1.70798512, -1.65203007, -1.58700751, -1.51103419, -1.41872343,
        -1.29737256, -1.10812146, -0.76610186, -0.31505007,  0.3125453 ,
         1.37952317,  1.77574881,  1.95320115,  2.00401128,  2.019591  ,
         2.0211568 ,  2.01810407,  2.01220512,  2.00342814,  1.99192682,
         1.97700963,  1.95711174,  1.92890163,  1.88406686,  1.84343944,
         1.80117292,  1.75594906,  1.7064321 ,  1.65011259,  1.58499882,
         1.50874298,  1.41579976,  1.29328562,  1.26847016],
       [-0.66666667, -0.4404801 , -0.25499374, -0.18348972, -0.15577052,
        -0.14663307, -0.14512263, -0.14681405, -0.1503292 , -0.15589715,
        -0.16395156, -0.17351351, -0.18431529, -0.1983367 , -0.21879763,
        -0.25007257, -0.30446515, -0.43225986, -1.14032821, -2.78253762,
        -5.87925401, -6.63373715, -3.5842388 , -1.32636396, -0.47084575,
        -0.10911561,  0.0237915 ,  0.08316757,  0.11186977,  0.12564939,
         0.13153381,  0.13451371,  0.13704778,  0.14041924,  0.14612705,
         0.15185351,  0.15830476,  0.16576471,  0.17477948,  0.18645538,
         0.2022157 ,  0.22442633,  0.25878581,  0.32169987,  0.48568998,
         1.13780434,  2.8468042 ,  5.89223803,  6.76190828,  3.62624714,
         1.28934678,  0.45330457,  0.10148692, -0.0265402 , -0.08410449,
        -0.11240577, -0.12588692, -0.13165005, -0.13459634, -0.13713991,
        -0.14055012, -0.14636989, -0.1520532 , -0.15844717, -0.16596041,
        -0.17512287, -0.18693224, -0.2028109 , -0.22526976, -0.26019893,
        -0.32460754, -0.49548543, -1.16465928, -2.91764946, -5.97922066,
        -6.7172771 , -3.7227132 , -1.41443196, -0.49147887, -0.11023472,
         0.02339507,  0.08304243,  0.11180886,  0.12563523,  0.13153937,
         0.13452591,  0.13706069,  0.14042976,  0.14612878,  0.15186477,
         0.15832829,  0.16579056,  0.17480129,  0.18648071,  0.20225624,
         0.22448875,  0.25888647,  0.32190419,  0.48637077,  1.13956614,
         2.85164903,  5.89802219,  6.75943137,  3.62232651,  1.28702372,
         0.45277044,  0.10146589, -0.02654838, -0.08410798, -0.11240837,
        -0.12588882, -0.13165169, -0.13459798, -0.13714169, -0.14055221,
        -0.14637274, -0.15205612, -0.15845015, -0.16596413, -0.17512803,
        -0.18693912, -0.20282002, -0.225283  , -0.26022095, -0.32465297,
        -0.49564047, -1.16511186, -2.9187932 , -5.98066631, -6.71640819,
        -3.713417  , -1.40536156, -0.48886154, -0.10971734,  0.02358231,
         0.08310671,  0.1118454 ,  0.12565115,  0.13154686,  0.13453101,
         0.13706636,  0.14043797,  0.14614426,  0.15187742,  0.15833714,
         0.16580274,  0.17482294,  0.18651094,  0.2022939 ,  0.22454195,
         0.25897547,  0.32208668,  0.48697938,  1.14115384,  2.85598921,
         5.90322458,  6.75714114,  3.63108751,  1.29695583,  0.45569309,
         0.10207002, -0.02633207, -0.08403553, -0.11236757, -0.12587153,
        -0.13164395, -0.13459293, -0.13713599, -0.14054362, -0.14635557,
        -0.15204274, -0.15844171, -0.16595202, -0.17510512, -0.18690706,
        -0.20278068, -0.22522748, -0.26012744, -0.32445955, -0.49498091,
        -1.16319281, -2.91393311, -5.97453203, -6.72006816, -3.75249154,
        -1.44372935, -0.49996318, -0.1119336 ,  0.02277964,  0.0828308 ,
         0.11168861,  0.12558286,  0.13151484,  0.13450929,  0.13704223,
         0.14040303,  0.14607836,  0.15182347,  0.15829937,  0.16575084,
         0.17473074,  0.18638213,  0.20213336,  0.22431526,  0.25859647,
         0.32131034,  0.484397  ,  1.13449837,  2.83763038,  5.88135126,
         6.76637197,  3.6653422 ,  1.32503616,  0.46332751,  0.10328245,
        -0.02589419, -0.0838847 , -0.11227937, -0.1258301 , -0.13162136,
        -0.13457531, -0.13711644, -0.14051725, -0.14630971, -0.15200339,
        -0.15841099, -0.16591083, -0.17503693, -0.18681334, -0.20266214,
        -0.22505838, -0.25984428, -0.32387533, -0.34017088]]), n_func_evaluations=1730, n_jac_evaluations=0, solver_message='The solver successfully reached the end of the integration interval.', is_converged=True, solver_options={'first_step': None, 'max_step': inf, 'rtol': 0.001, 'atol': 1e-06, 'min_step': 0}, solver_name='RK45')

By default, the Runge-Kutta method of order 4(5) ("RK45") is used, but other algorithms can be applied by specifying the option algo_name in ODESolversFactory().execute(). See more information on available algorithms in the SciPy documentation.

Step 3: Examining the results

The convergence of the algorithm can be known by examining ODEProblem.is_converged and ODEProblem.solver_message.

The solution of the ODEProblem on the user-specified time interval can be accessed through the vectors ODEProblem.states and ODEProblem.times.

plt.plot(ode_problem.result.time_vector, ode_problem.result.state_vector[0])
plt.plot(ode_problem.result.time_vector, ode_problem.result.state_vector[1])
plt.show()

Shortcut

The class VanDerPol is available in the package gemseo.problems.ode, so it just needs to be imported to be used.

ode_problem = VanDerPol()
ODESolversFactory().execute(ode_problem)

ODEResult(time_vector=array([0.00000000e+00, 3.02889880e-04, 6.05779761e-04, 9.19543822e-04,
       1.23318623e-03, 1.54831722e-03, 1.86699348e-03, 2.19411325e-03,
       2.53977274e-03, 2.92128804e-03, 3.36329597e-03, 3.89893117e-03,
       4.57707708e-03, 5.47709067e-03, 6.70631653e-03, 7.94054333e-03,
       9.04797379e-03, 1.01554042e-02, 1.12779072e-02, 1.23918108e-02,
       1.34830214e-02, 1.45644470e-02, 1.56611247e-02, 1.67810456e-02,
       1.79042147e-02, 1.90056269e-02, 2.00859937e-02, 2.11712234e-02,
       2.22825463e-02, 2.34107825e-02, 2.45253095e-02, 2.56111902e-02,
       2.66876034e-02, 2.77856208e-02, 2.89120756e-02, 3.00392570e-02,
       3.11371929e-02, 3.22106958e-02, 3.32937591e-02, 3.44105558e-02,
       3.55457617e-02, 3.66602394e-02, 3.77390477e-02, 3.88096880e-02,
       3.99102151e-02, 4.10450773e-02, 4.21765437e-02, 4.32693318e-02,
       4.43343877e-02, 4.54156305e-02, 4.65399445e-02, 4.76834857e-02,
       4.87968566e-02, 4.98664641e-02, 5.09306177e-02, 5.20352809e-02,
       5.31808038e-02, 5.43166166e-02, 5.54020844e-02, 5.64569315e-02,
       5.75370351e-02, 5.86714478e-02, 5.98247635e-02, 6.09354745e-02,
       6.19933612e-02, 6.30504461e-02, 6.41614238e-02, 6.53202405e-02,
       6.64601279e-02, 6.75355546e-02, 6.85783247e-02, 6.96584071e-02,
       7.08060804e-02, 7.19705807e-02, 7.30764881e-02, 7.41197994e-02,
       7.51694766e-02, 7.62895609e-02, 7.74646149e-02, 7.86078353e-02,
       7.96699841e-02, 8.06988362e-02, 8.17805312e-02, 8.29451628e-02,
       8.41220623e-02, 8.52204063e-02, 8.62461238e-02, 8.72884232e-02,
       8.84209746e-02, 8.96153316e-02, 9.07605637e-02, 9.18058661e-02,
       9.28191974e-02, 9.39046480e-02, 9.50902543e-02, 9.62803445e-02,
       9.73678845e-02, 9.83731466e-02, 9.94085726e-02, 1.00557278e-01,
       1.01773764e-01, 1.02798733e-01, 1.03823703e-01, 1.04962272e-01,
       1.06182108e-01, 1.07213960e-01, 1.08245812e-01, 1.09384272e-01,
       1.10595952e-01, 1.11630655e-01, 1.12665358e-01, 1.13801753e-01,
       1.15008298e-01, 1.16159033e-01, 1.17197135e-01, 1.18199537e-01,
       1.19282660e-01, 1.20478679e-01, 1.21680064e-01, 1.22765892e-01,
       1.23759737e-01, 1.24786971e-01, 1.25940650e-01, 1.27171198e-01,
       1.28187667e-01, 1.29204136e-01, 1.30346597e-01, 1.31579975e-01,
       1.32604268e-01, 1.33628561e-01, 1.34770773e-01, 1.35994674e-01,
       1.37022340e-01, 1.38050006e-01, 1.39189846e-01, 1.40407685e-01,
       1.41439273e-01, 1.42470861e-01, 1.43608963e-01, 1.44820679e-01,
       1.45855726e-01, 1.46890774e-01, 1.48027163e-01, 1.49233347e-01,
       1.50383847e-01, 1.51422148e-01, 1.52424966e-01, 1.53508296e-01,
       1.54704072e-01, 1.55905059e-01, 1.56990873e-01, 1.57985096e-01,
       1.59012711e-01, 1.60166370e-01, 1.61396492e-01, 1.62413188e-01,
       1.63429884e-01, 1.64572263e-01, 1.65805341e-01, 1.66829869e-01,
       1.67854398e-01, 1.68996551e-01, 1.70220165e-01, 1.71248045e-01,
       1.72275925e-01, 1.73415711e-01, 1.74633292e-01, 1.75665082e-01,
       1.76696871e-01, 1.77834926e-01, 1.79046408e-01, 1.80081644e-01,
       1.81116880e-01, 1.82253227e-01, 1.83459197e-01, 1.84609618e-01,
       1.85648098e-01, 1.86651170e-01, 1.87734584e-01, 1.88930184e-01,
       1.90130976e-01, 1.91216856e-01, 1.92211341e-01, 1.93239157e-01,
       1.94392754e-01, 1.95622627e-01, 1.96639524e-01, 1.97656421e-01,
       1.98798757e-01, 2.00031603e-01, 2.01056313e-01, 2.02081023e-01,
       2.03223138e-01, 2.04446546e-01, 2.05474597e-01, 2.06502647e-01,
       2.07642401e-01, 2.08859796e-01, 2.09891745e-01, 2.10923695e-01,
       2.12061722e-01, 2.13273035e-01, 2.14308422e-01, 2.15343809e-01,
       2.16480133e-01, 2.17685949e-01, 2.18836318e-01, 2.19874941e-01,
       2.20878215e-01, 2.21961700e-01, 2.23157176e-01, 2.24357828e-01,
       2.25443766e-01, 2.26438458e-01, 2.27466434e-01, 2.28619992e-01,
       2.29849685e-01, 2.30866742e-01, 2.31883798e-01, 2.33026111e-01,
       2.34258788e-01, 2.35283645e-01, 2.36308501e-01, 2.37450594e-01,
       2.38673855e-01, 2.39702043e-01, 2.40730231e-01, 2.41869968e-01,
       2.43087230e-01, 2.44119309e-01, 2.45151388e-01, 2.46289401e-01,
       2.47500595e-01, 2.48536105e-01, 2.49571615e-01, 2.50707929e-01,
       2.51913636e-01, 2.53063974e-01, 2.54102714e-01, 2.55106148e-01,
       2.56189695e-01, 2.57385086e-01, 2.58585640e-01, 2.59671630e-01,
       2.60666486e-01, 2.61694591e-01, 2.62848129e-01, 2.64077694e-01,
       2.65094880e-01, 2.66112065e-01, 2.67254368e-01, 2.68486926e-01,
       2.69511901e-01, 2.70536876e-01, 2.71678961e-01, 2.72902118e-01,
       2.73930419e-01, 2.74958719e-01, 2.76098451e-01, 2.77315621e-01,
       2.78347806e-01, 2.79379991e-01, 2.80518003e-01, 2.81729115e-01,
       2.82764725e-01, 2.83800336e-01, 2.84936651e-01, 2.86142285e-01,
       2.87292608e-01, 2.88331443e-01, 2.89335007e-01, 2.90418609e-01,
       2.91613943e-01, 2.92814432e-01, 2.93900469e-01, 2.94895458e-01,
       2.95923669e-01, 2.97077199e-01, 2.98306675e-01, 2.99323967e-01,
       3.00341258e-01, 3.01483562e-01, 3.02716038e-01, 3.03741111e-01,
       3.04766184e-01, 3.05908271e-01, 3.07131358e-01, 3.08159751e-01,
       3.09188145e-01, 3.10327882e-01, 3.11544990e-01, 3.12577264e-01,
       3.13609537e-01, 3.14747556e-01, 3.15958613e-01, 3.16994308e-01,
       3.18030004e-01, 3.19166327e-01, 3.20371913e-01, 3.21522233e-01,
       3.22561150e-01, 3.23564819e-01, 3.24648472e-01, 3.25843771e-01,
       3.27044218e-01, 3.28130300e-01, 3.29125397e-01, 3.30153696e-01,
       3.31307228e-01, 3.32536646e-01, 3.33554026e-01, 3.34571405e-01,
       3.35713717e-01, 3.36946140e-01, 3.37971295e-01, 3.38996450e-01,
       3.40138547e-01, 3.41361589e-01, 3.42390061e-01, 3.43418533e-01,
       3.44558282e-01, 3.45775352e-01, 3.46807701e-01, 3.47840049e-01,
       3.48978081e-01, 3.50189106e-01, 3.51224873e-01, 3.52260640e-01,
       3.53396978e-01, 3.54602537e-01, 3.55752863e-01, 3.56791849e-01,
       3.57795606e-01, 3.58879305e-01, 3.60074586e-01, 3.61275010e-01,
       3.62361134e-01, 3.63356320e-01, 3.64384694e-01, 3.65538237e-01,
       3.66767619e-01, 3.67785073e-01, 3.68802527e-01, 3.69944854e-01,
       3.71177245e-01, 3.72202470e-01, 3.73227695e-01, 3.74369808e-01,
       3.75592825e-01, 3.76621364e-01, 3.77649903e-01, 3.78789669e-01,
       3.80006719e-01, 3.81039132e-01, 3.82071545e-01, 3.83209596e-01,
       3.84420604e-01, 3.85456434e-01, 3.86492263e-01, 3.87628621e-01,
       3.88834168e-01, 3.89984506e-01, 3.91023552e-01, 3.92027384e-01,
       3.93111126e-01, 3.94306402e-01, 3.95506817e-01, 3.96592980e-01,
       3.97588243e-01, 3.98616681e-01, 3.99770239e-01, 4.00999603e-01,
       4.02017121e-01, 4.03034640e-01, 4.04176986e-01, 4.05409362e-01,
       4.06434648e-01, 4.07459933e-01, 4.08602067e-01, 4.09825073e-01,
       4.10853671e-01, 4.11882269e-01, 4.13022057e-01, 4.14239099e-01,
       4.15271569e-01, 4.16304039e-01, 4.17442112e-01, 4.18653117e-01,
       4.19689002e-01, 4.20724886e-01, 4.21861267e-01, 4.23066813e-01,
       4.24217169e-01, 4.25256269e-01, 4.26260165e-01, 4.27343947e-01,
       4.28539228e-01, 4.29739645e-01, 4.30825846e-01, 4.31821174e-01,
       4.32849669e-01, 4.34003248e-01, 4.35232606e-01, 4.36250181e-01,
       4.37267756e-01, 4.38410126e-01, 4.39642498e-01, 4.40667838e-01,
       4.41693177e-01, 4.42835334e-01, 4.44058341e-01, 4.45086992e-01,
       4.46115642e-01, 4.47255454e-01, 4.48472501e-01, 4.49505021e-01,
       4.50537542e-01, 4.51675640e-01, 4.52886651e-01, 4.53922585e-01,
       4.54958519e-01, 4.56094926e-01, 4.57300480e-01, 4.58450858e-01,
       4.59490007e-01, 4.60493959e-01, 4.61577780e-01, 4.62773073e-01,
       4.63973500e-01, 4.65059738e-01, 4.66055122e-01, 4.67083668e-01,
       4.68237271e-01, 4.69466634e-01, 4.70484259e-01, 4.71501885e-01,
       4.72644280e-01, 4.73876659e-01, 4.74902047e-01, 4.75927435e-01,
       4.77069619e-01, 4.78292635e-01, 4.79321333e-01, 4.80350030e-01,
       4.81489870e-01, 4.82706927e-01, 4.83739494e-01, 4.84772061e-01,
       4.85910187e-01, 4.87121211e-01, 4.88157191e-01, 4.89193170e-01,
       4.90329605e-01, 4.91535173e-01, 4.92685577e-01, 4.93724770e-01,
       4.94728772e-01, 4.95812630e-01, 4.97007942e-01, 4.98208385e-01,
       4.99294659e-01, 5.00000000e-01]), state_vector=array([[ 2.00000000e+00,  1.99986735e+00,  1.99981373e+00,
         1.99979150e+00,  1.99978270e+00,  1.99977912e+00,
         1.99977759e+00,  1.99977685e+00,  1.99977642e+00,
         1.99977610e+00,  1.99977577e+00,  1.99977541e+00,
         1.99977496e+00,  1.99977436e+00,  1.99977354e+00,
         1.99977271e+00,  1.99977198e+00,  1.99977124e+00,
         1.99977049e+00,  1.99976975e+00,  1.99976902e+00,
         1.99976830e+00,  1.99976757e+00,  1.99976682e+00,
         1.99976607e+00,  1.99976534e+00,  1.99976462e+00,
         1.99976389e+00,  1.99976315e+00,  1.99976240e+00,
         1.99976166e+00,  1.99976093e+00,  1.99976021e+00,
         1.99975948e+00,  1.99975873e+00,  1.99975798e+00,
         1.99975725e+00,  1.99975653e+00,  1.99975581e+00,
         1.99975506e+00,  1.99975431e+00,  1.99975356e+00,
         1.99975284e+00,  1.99975213e+00,  1.99975140e+00,
         1.99975064e+00,  1.99974989e+00,  1.99974916e+00,
         1.99974845e+00,  1.99974773e+00,  1.99974698e+00,
         1.99974621e+00,  1.99974547e+00,  1.99974476e+00,
         1.99974405e+00,  1.99974331e+00,  1.99974255e+00,
         1.99974179e+00,  1.99974107e+00,  1.99974036e+00,
         1.99973964e+00,  1.99973889e+00,  1.99973812e+00,
         1.99973738e+00,  1.99973667e+00,  1.99973597e+00,
         1.99973523e+00,  1.99973445e+00,  1.99973369e+00,
         1.99973298e+00,  1.99973228e+00,  1.99973156e+00,
         1.99973079e+00,  1.99973002e+00,  1.99972928e+00,
         1.99972858e+00,  1.99972788e+00,  1.99972714e+00,
         1.99972635e+00,  1.99972559e+00,  1.99972488e+00,
         1.99972420e+00,  1.99972348e+00,  1.99972270e+00,
         1.99972192e+00,  1.99972118e+00,  1.99972050e+00,
         1.99971980e+00,  1.99971905e+00,  1.99971825e+00,
         1.99971749e+00,  1.99971679e+00,  1.99971612e+00,
         1.99971539e+00,  1.99971460e+00,  1.99971381e+00,
         1.99971308e+00,  1.99971241e+00,  1.99971172e+00,
         1.99971096e+00,  1.99971014e+00,  1.99970946e+00,
         1.99970878e+00,  1.99970802e+00,  1.99970721e+00,
         1.99970652e+00,  1.99970583e+00,  1.99970507e+00,
         1.99970426e+00,  1.99970357e+00,  1.99970288e+00,
         1.99970212e+00,  1.99970132e+00,  1.99970055e+00,
         1.99969986e+00,  1.99969919e+00,  1.99969847e+00,
         1.99969767e+00,  1.99969687e+00,  1.99969615e+00,
         1.99969548e+00,  1.99969480e+00,  1.99969403e+00,
         1.99969321e+00,  1.99969253e+00,  1.99969185e+00,
         1.99969109e+00,  1.99969027e+00,  1.99968959e+00,
         1.99968890e+00,  1.99968814e+00,  1.99968733e+00,
         1.99968664e+00,  1.99968595e+00,  1.99968519e+00,
         1.99968438e+00,  1.99968369e+00,  1.99968301e+00,
         1.99968225e+00,  1.99968144e+00,  1.99968075e+00,
         1.99968006e+00,  1.99967930e+00,  1.99967850e+00,
         1.99967773e+00,  1.99967704e+00,  1.99967637e+00,
         1.99967565e+00,  1.99967485e+00,  1.99967405e+00,
         1.99967332e+00,  1.99967266e+00,  1.99967198e+00,
         1.99967121e+00,  1.99967039e+00,  1.99966971e+00,
         1.99966903e+00,  1.99966827e+00,  1.99966745e+00,
         1.99966676e+00,  1.99966608e+00,  1.99966532e+00,
         1.99966450e+00,  1.99966382e+00,  1.99966313e+00,
         1.99966237e+00,  1.99966156e+00,  1.99966087e+00,
         1.99966018e+00,  1.99965942e+00,  1.99965862e+00,
         1.99965793e+00,  1.99965724e+00,  1.99965648e+00,
         1.99965567e+00,  1.99965491e+00,  1.99965421e+00,
         1.99965354e+00,  1.99965282e+00,  1.99965202e+00,
         1.99965122e+00,  1.99965050e+00,  1.99964984e+00,
         1.99964915e+00,  1.99964838e+00,  1.99964756e+00,
         1.99964688e+00,  1.99964621e+00,  1.99964544e+00,
         1.99964462e+00,  1.99964394e+00,  1.99964325e+00,
         1.99964249e+00,  1.99964168e+00,  1.99964099e+00,
         1.99964031e+00,  1.99963955e+00,  1.99963873e+00,
         1.99963805e+00,  1.99963736e+00,  1.99963660e+00,
         1.99963579e+00,  1.99963510e+00,  1.99963441e+00,
         1.99963365e+00,  1.99963285e+00,  1.99963208e+00,
         1.99963139e+00,  1.99963072e+00,  1.99963000e+00,
         1.99962920e+00,  1.99962840e+00,  1.99962768e+00,
         1.99962701e+00,  1.99962633e+00,  1.99962556e+00,
         1.99962474e+00,  1.99962406e+00,  1.99962338e+00,
         1.99962262e+00,  1.99962180e+00,  1.99962111e+00,
         1.99962043e+00,  1.99961967e+00,  1.99961885e+00,
         1.99961817e+00,  1.99961748e+00,  1.99961672e+00,
         1.99961591e+00,  1.99961522e+00,  1.99961453e+00,
         1.99961377e+00,  1.99961297e+00,  1.99961228e+00,
         1.99961158e+00,  1.99961083e+00,  1.99961002e+00,
         1.99960926e+00,  1.99960856e+00,  1.99960789e+00,
         1.99960717e+00,  1.99960637e+00,  1.99960557e+00,
         1.99960485e+00,  1.99960419e+00,  1.99960350e+00,
         1.99960273e+00,  1.99960191e+00,  1.99960123e+00,
         1.99960055e+00,  1.99959979e+00,  1.99959897e+00,
         1.99959829e+00,  1.99959760e+00,  1.99959684e+00,
         1.99959603e+00,  1.99959534e+00,  1.99959465e+00,
         1.99959389e+00,  1.99959308e+00,  1.99959239e+00,
         1.99959171e+00,  1.99959095e+00,  1.99959014e+00,
         1.99958945e+00,  1.99958876e+00,  1.99958800e+00,
         1.99958720e+00,  1.99958643e+00,  1.99958574e+00,
         1.99958507e+00,  1.99958434e+00,  1.99958355e+00,
         1.99958275e+00,  1.99958202e+00,  1.99958136e+00,
         1.99958067e+00,  1.99957990e+00,  1.99957908e+00,
         1.99957841e+00,  1.99957773e+00,  1.99957697e+00,
         1.99957614e+00,  1.99957546e+00,  1.99957478e+00,
         1.99957401e+00,  1.99957320e+00,  1.99957251e+00,
         1.99957183e+00,  1.99957107e+00,  1.99957026e+00,
         1.99956957e+00,  1.99956888e+00,  1.99956812e+00,
         1.99956731e+00,  1.99956662e+00,  1.99956593e+00,
         1.99956517e+00,  1.99956437e+00,  1.99956360e+00,
         1.99956291e+00,  1.99956224e+00,  1.99956152e+00,
         1.99956072e+00,  1.99955992e+00,  1.99955919e+00,
         1.99955853e+00,  1.99955784e+00,  1.99955708e+00,
         1.99955626e+00,  1.99955558e+00,  1.99955490e+00,
         1.99955414e+00,  1.99955332e+00,  1.99955263e+00,
         1.99955195e+00,  1.99955119e+00,  1.99955037e+00,
         1.99954968e+00,  1.99954900e+00,  1.99954824e+00,
         1.99954743e+00,  1.99954674e+00,  1.99954605e+00,
         1.99954529e+00,  1.99954448e+00,  1.99954379e+00,
         1.99954310e+00,  1.99954234e+00,  1.99954154e+00,
         1.99954077e+00,  1.99954008e+00,  1.99953941e+00,
         1.99953869e+00,  1.99953789e+00,  1.99953709e+00,
         1.99953637e+00,  1.99953570e+00,  1.99953502e+00,
         1.99953425e+00,  1.99953343e+00,  1.99953275e+00,
         1.99953207e+00,  1.99953131e+00,  1.99953049e+00,
         1.99952980e+00,  1.99952912e+00,  1.99952836e+00,
         1.99952754e+00,  1.99952685e+00,  1.99952617e+00,
         1.99952541e+00,  1.99952460e+00,  1.99952391e+00,
         1.99952322e+00,  1.99952246e+00,  1.99952165e+00,
         1.99952096e+00,  1.99952027e+00,  1.99951951e+00,
         1.99951871e+00,  1.99951794e+00,  1.99951725e+00,
         1.99951658e+00,  1.99951586e+00,  1.99951506e+00,
         1.99951426e+00,  1.99951354e+00,  1.99951287e+00,
         1.99951219e+00,  1.99951142e+00,  1.99951060e+00,
         1.99950992e+00,  1.99950924e+00,  1.99950848e+00,
         1.99950766e+00,  1.99950697e+00,  1.99950629e+00,
         1.99950553e+00,  1.99950471e+00,  1.99950402e+00,
         1.99950334e+00,  1.99950258e+00,  1.99950177e+00,
         1.99950108e+00,  1.99950039e+00,  1.99949963e+00,
         1.99949882e+00,  1.99949813e+00,  1.99949744e+00,
         1.99949668e+00,  1.99949588e+00,  1.99949511e+00,
         1.99949442e+00,  1.99949375e+00,  1.99949303e+00,
         1.99949223e+00,  1.99949143e+00,  1.99949070e+00,
         1.99949004e+00,  1.99948935e+00,  1.99948858e+00,
         1.99948776e+00,  1.99948709e+00,  1.99948641e+00,
         1.99948565e+00,  1.99948482e+00,  1.99948414e+00,
         1.99948346e+00,  1.99948269e+00,  1.99948188e+00,
         1.99948119e+00,  1.99948051e+00,  1.99947975e+00,
         1.99947893e+00,  1.99947825e+00,  1.99947756e+00,
         1.99947680e+00,  1.99947599e+00,  1.99947530e+00,
         1.99947461e+00,  1.99947385e+00,  1.99947305e+00,
         1.99947228e+00,  1.99947159e+00,  1.99947092e+00,
         1.99947019e+00,  1.99946940e+00,  1.99946860e+00,
         1.99946787e+00,  1.99946721e+00,  1.99946652e+00,
         1.99946575e+00,  1.99946493e+00,  1.99946425e+00,
         1.99946357e+00,  1.99946281e+00,  1.99946199e+00,
         1.99946131e+00,  1.99946062e+00,  1.99945986e+00,
         1.99945905e+00,  1.99945836e+00,  1.99945767e+00,
         1.99945691e+00,  1.99945610e+00,  1.99945541e+00,
         1.99945472e+00,  1.99945396e+00,  1.99945316e+00,
         1.99945247e+00,  1.99945177e+00,  1.99945102e+00,
         1.99945021e+00,  1.99944945e+00,  1.99944875e+00,
         1.99944808e+00,  1.99944736e+00,  1.99944656e+00,
         1.99944576e+00,  1.99944504e+00,  1.99944457e+00],
       [-6.66543343e-01, -2.69245437e-01, -1.09008060e-01,
        -4.29772973e-02, -1.71965922e-02, -7.09608393e-03,
        -3.14120535e-03, -1.59546970e-03, -9.96661476e-04,
        -7.72199602e-04, -6.95079133e-04, -6.72729664e-04,
        -6.67819834e-04, -6.67129569e-04, -6.67455464e-04,
        -6.68128773e-04, -6.68165750e-04, -6.68203720e-04,
        -6.68367415e-04, -6.68469335e-04, -6.68368264e-04,
        -6.68195270e-04, -6.68151200e-04, -6.68287350e-04,
        -6.68466445e-04, -6.68456565e-04, -6.68265360e-04,
        -6.68131808e-04, -6.68198255e-04, -6.68410285e-04,
        -6.68520439e-04, -6.68369384e-04, -6.68158280e-04,
        -6.68125207e-04, -6.68311119e-04, -6.68529816e-04,
        -6.68486830e-04, -6.68236701e-04, -6.68090506e-04,
        -6.68195775e-04, -6.68468540e-04, -6.68581756e-04,
        -6.68363138e-04, -6.68110707e-04, -6.68096601e-04,
        -6.68346656e-04, -6.68611332e-04, -6.68516926e-04,
        -6.68195878e-04, -6.68041683e-04, -6.68198519e-04,
        -6.68547576e-04, -6.68654482e-04, -6.68346502e-04,
        -6.68051610e-04, -6.68066456e-04, -6.68398770e-04,
        -6.68715925e-04, -6.68544388e-04, -6.68140827e-04,
        -6.67985722e-04, -6.68209542e-04, -6.68654594e-04,
        -6.68739322e-04, -6.68316130e-04, -6.67980802e-04,
        -6.68036409e-04, -6.68473943e-04, -6.68849468e-04,
        -6.68565707e-04, -6.68070285e-04, -6.67923751e-04,
        -6.68232863e-04, -6.68798990e-04, -6.68835975e-04,
        -6.68268954e-04, -6.67899299e-04, -6.68008693e-04,
        -6.68580642e-04, -6.69018388e-04, -6.68576497e-04,
        -6.67984586e-04, -6.67857800e-04, -6.68273471e-04,
        -6.68992398e-04, -6.68942459e-04, -6.68203312e-04,
        -6.67809774e-04, -6.67986023e-04, -6.68729233e-04,
        -6.69228555e-04, -6.68572297e-04, -6.67886535e-04,
        -6.67790817e-04, -6.68336914e-04, -6.69247651e-04,
        -6.69054511e-04, -6.68120368e-04, -6.67716714e-04,
        -6.67971234e-04, -6.68930742e-04, -6.68191961e-04,
        -6.67712960e-04, -6.67905349e-04, -6.68842729e-04,
        -6.68186870e-04, -6.67744891e-04, -6.67943171e-04,
        -6.68826081e-04, -6.68197847e-04, -6.67767664e-04,
        -6.67957820e-04, -6.68797902e-04, -6.69386975e-04,
        -6.68616319e-04, -6.67852448e-04, -6.67748646e-04,
        -6.68345841e-04, -6.69403409e-04, -6.69175944e-04,
        -6.68110510e-04, -6.67673357e-04, -6.67941824e-04,
        -6.69020860e-04, -6.68187538e-04, -6.67672485e-04,
        -6.67875516e-04, -6.68918082e-04, -6.68182315e-04,
        -6.67707151e-04, -6.67916721e-04, -6.68896633e-04,
        -6.68194405e-04, -6.67732240e-04, -6.67933346e-04,
        -6.68862722e-04, -6.68202227e-04, -6.67757860e-04,
        -6.67953839e-04, -6.68835138e-04, -6.68210444e-04,
        -6.67781934e-04, -6.67972170e-04, -6.68809417e-04,
        -6.69394436e-04, -6.68627027e-04, -6.67866203e-04,
        -6.67763183e-04, -6.68359358e-04, -6.69411707e-04,
        -6.69184006e-04, -6.68123158e-04, -6.67687666e-04,
        -6.67956324e-04, -6.69031940e-04, -6.68200717e-04,
        -6.67686396e-04, -6.67889067e-04, -6.68928916e-04,
        -6.68195346e-04, -6.67721057e-04, -6.67930375e-04,
        -6.68907739e-04, -6.68207438e-04, -6.67746070e-04,
        -6.67946904e-04, -6.68873959e-04, -6.68215235e-04,
        -6.67771637e-04, -6.67967352e-04, -6.68846508e-04,
        -6.68223435e-04, -6.67795659e-04, -6.67985634e-04,
        -6.68820898e-04, -6.69403996e-04, -6.68638994e-04,
        -6.67879760e-04, -6.67776909e-04, -6.68372086e-04,
        -6.69421357e-04, -6.69194354e-04, -6.68136221e-04,
        -6.67701397e-04, -6.67969813e-04, -6.69043013e-04,
        -6.68213691e-04, -6.67700057e-04, -6.67902530e-04,
        -6.68940283e-04, -6.68208318e-04, -6.67734665e-04,
        -6.67943775e-04, -6.68919214e-04, -6.68220388e-04,
        -6.67759630e-04, -6.67960262e-04, -6.68885554e-04,
        -6.68228170e-04, -6.67785154e-04, -6.67980669e-04,
        -6.68858199e-04, -6.68236357e-04, -6.67809138e-04,
        -6.67998915e-04, -6.68832674e-04, -6.69414314e-04,
        -6.68651139e-04, -6.67893112e-04, -6.67790389e-04,
        -6.68384808e-04, -6.69431744e-04, -6.69205270e-04,
        -6.68149197e-04, -6.67714882e-04, -6.67983113e-04,
        -6.69054480e-04, -6.68226600e-04, -6.67713488e-04,
        -6.67915810e-04, -6.68951972e-04, -6.68221225e-04,
        -6.67748056e-04, -6.67957007e-04, -6.68930984e-04,
        -6.68233279e-04, -6.67772986e-04, -6.67973463e-04,
        -6.68897416e-04, -6.68241050e-04, -6.67798477e-04,
        -6.67993839e-04, -6.68870134e-04, -6.68249226e-04,
        -6.67822432e-04, -6.68012059e-04, -6.68844673e-04,
        -6.69425207e-04, -6.68663418e-04, -6.67906309e-04,
        -6.67803684e-04, -6.68397527e-04, -6.69442689e-04,
        -6.69216616e-04, -6.68162108e-04, -6.67728180e-04,
        -6.67996271e-04, -6.69066243e-04, -6.68239460e-04,
        -6.67726745e-04, -6.67928953e-04, -6.68963905e-04,
        -6.68234084e-04, -6.67761283e-04, -6.67970114e-04,
        -6.68942979e-04, -6.68246126e-04, -6.67786185e-04,
        -6.67986545e-04, -6.68909480e-04, -6.68253889e-04,
        -6.67811653e-04, -6.68006898e-04, -6.68882254e-04,
        -6.68262057e-04, -6.67835585e-04, -6.68025097e-04,
        -6.68856841e-04, -6.69436534e-04, -6.68675800e-04,
        -6.67919388e-04, -6.67816837e-04, -6.68410242e-04,
        -6.69454055e-04, -6.69228288e-04, -6.68174971e-04,
        -6.67741337e-04, -6.68009321e-04, -6.69078232e-04,
        -6.68252283e-04, -6.67739871e-04, -6.67941991e-04,
        -6.68976023e-04, -6.68246907e-04, -6.67774385e-04,
        -6.67983125e-04, -6.68955144e-04, -6.68258939e-04,
        -6.67799267e-04, -6.67999538e-04, -6.68921698e-04,
        -6.68266696e-04, -6.67824716e-04, -6.68019873e-04,
        -6.68894514e-04, -6.68274858e-04, -6.67848631e-04,
        -6.68038057e-04, -6.68869139e-04, -6.69448191e-04,
        -6.68688259e-04, -6.67932379e-04, -6.67829884e-04,
        -6.68422955e-04, -6.69465742e-04, -6.69240208e-04,
        -6.68187796e-04, -6.67754386e-04, -6.68022289e-04,
        -6.69090393e-04, -6.68265079e-04, -6.67752897e-04,
        -6.67954950e-04, -6.68988282e-04, -6.68259702e-04,
        -6.67787393e-04, -6.67996063e-04, -6.68967438e-04,
        -6.68271727e-04, -6.67812260e-04, -6.68012462e-04,
        -6.68934033e-04, -6.68279479e-04, -6.67837694e-04,
        -6.68032784e-04, -6.68906881e-04, -6.68287636e-04,
        -6.67861596e-04, -6.68050956e-04, -6.68881534e-04,
        -6.69460099e-04, -6.68700778e-04, -6.67945302e-04,
        -6.67842850e-04, -6.68435667e-04, -6.69477673e-04,
        -6.69252316e-04, -6.68200594e-04, -6.67767353e-04,
        -6.68035195e-04, -6.69102684e-04, -6.68277854e-04,
        -6.67765846e-04, -6.67967850e-04, -6.69000648e-04,
        -6.68272476e-04, -6.67800330e-04, -6.68008946e-04,
        -6.68979832e-04, -6.68284496e-04, -6.67825184e-04,
        -6.68025335e-04, -6.68946458e-04, -6.68292245e-04,
        -6.67850608e-04, -6.68045647e-04, -6.68919330e-04,
        -6.68300399e-04, -6.67874500e-04, -6.68063810e-04,
        -6.68894005e-04, -6.69472198e-04, -6.68713343e-04,
        -6.67958173e-04, -6.67855755e-04, -6.68448379e-04,
        -6.69489790e-04, -6.69264568e-04, -6.68213370e-04,
        -6.67780259e-04, -6.68048054e-04, -6.69115076e-04,
        -6.68290613e-04, -6.67778738e-04, -6.67980703e-04,
        -6.69013097e-04, -6.68285235e-04, -6.67813212e-04,
        -6.68021788e-04, -6.68992302e-04, -6.68297251e-04,
        -6.67838057e-04, -6.68038169e-04, -6.68958951e-04,
        -6.68304997e-04, -6.67863472e-04, -6.68058473e-04,
        -6.68931842e-04, -6.68313148e-04, -6.67887358e-04,
        -6.68076629e-04, -6.68906533e-04, -6.69484444e-04,
        -6.68725942e-04, -6.67971006e-04, -6.67868612e-04,
        -6.68461090e-04, -6.69502049e-04, -6.69276929e-04,
        -6.68226131e-04, -6.67793117e-04, -6.68060877e-04,
        -6.69127544e-04, -6.68303361e-04, -6.67791586e-04,
        -6.67993523e-04, -6.69025609e-04, -6.68297983e-04,
        -6.67826052e-04, -6.68034599e-04, -6.69004829e-04,
        -6.68309996e-04, -6.67850891e-04, -6.68050974e-04,
        -6.68971496e-04, -6.68317740e-04, -6.67876300e-04,
        -6.68071271e-04, -6.68944402e-04, -6.68325889e-04,
        -6.67900180e-04, -6.68089423e-04, -6.68919105e-04,
        -6.69496802e-04, -6.68738569e-04, -6.67983810e-04,
        -6.67881435e-04, -6.68473802e-04, -6.69514417e-04,
        -6.69289375e-04, -6.68238880e-04, -6.67805940e-04,
        -6.68073674e-04, -6.69140071e-04, -6.68316100e-04,
        -6.67804401e-04, -6.68006316e-04, -6.69038169e-04,
        -6.68310722e-04, -6.67838861e-04, -6.68047385e-04,
        -6.69017402e-04, -6.68322733e-04, -6.67863695e-04,
        -6.68063756e-04, -6.68984082e-04, -6.68330475e-04,
        -6.67889100e-04, -6.68084049e-04, -6.68956998e-04,
        -6.68338623e-04, -6.67912975e-04, -6.68102196e-04,
        -6.68931711e-04, -6.69509245e-04, -6.68751216e-04,
        -6.67996591e-04, -6.67894230e-04, -6.68486514e-04,
        -6.69526868e-04, -6.69301886e-04, -6.67382536e-04]]), n_func_evaluations=3134, n_jac_evaluations=0, solver_message='The solver successfully reached the end of the integration interval.', is_converged=True, solver_options={'first_step': None, 'max_step': inf, 'rtol': 0.001, 'atol': 1e-06, 'min_step': 0}, solver_name='RK45')