Programming

Dynamic Simulation

Previously, we use Euler's Method to simulate the state variables
However,is there any existed tool we can use?

Differential Equation Solver

result = desolve_odeint(x_prime,ics,times,dvars)

x_prime(x)=-2*x+3
timeRange = srange(0,5,0.5)

#The important line!!!!!###################
result = desolve_odeint(x_prime,5,timeRange,x)
################################################

fig1 = list_plot(zip(timeRange,result[:,0]),plotjoined=True)
show(fig1)

The result is better than what you get from Euler's method

Simulation For Multiple State Variables

x_prime(x,y) = -x+5*y
y_prime(y)=-3*y

timeRange = srange(0,15,0.1)
#######################Important##########################
result = desolve_odeint([x_prime,y_prime],[15,15],timeRange,[x,y])
##########################################################

What did we get?

result

array([[ 1.50000000e+01,  1.50000000e+01],
       [ 1.97232809e+01,  1.11122734e+01],
       [ 2.24029277e+01,  8.23217476e+00],
       [ 2.36465943e+01,  6.09854490e+00],
       [ 2.38970195e+01,  4.51791316e+00],
       [ 2.34754788e+01,  3.34695234e+00],
       [ 2.26139028e+01,  2.47948325e+00],
       [ 2.14786126e+01,  1.83684634e+00],
       [ 2.01878475e+01,  1.36076925e+00],
       [ 1.88247005e+01,  1.00808267e+00],
       [ 1.74466556e+01,  7.46806009e-01],
       [ 1.60926131e+01,  5.53247504e-01],
       [ 1.47880565e+01,  4.09855835e-01],
       [ 1.35488475e+01,  3.03628673e-01],
       [ 1.23840065e+01,  2.24933655e-01],
       [ 1.12977460e+01,  1.66634952e-01],
       [ 1.02909517e+01,  1.23446212e-01],
       [ 9.36225700e+00,  9.14512053e-02],
       [ 8.50881983e+00,  6.77487202e-02],
       [ 7.72687879e+00,  5.01894877e-02],
       [ 7.01214915e+00,  3.71812882e-02],
       [ 6.36010104e+00,  2.75445769e-02],
       [ 5.76615200e+00,  2.04055252e-02],
       [ 5.22579733e+00,  1.51167853e-02],
       [ 4.73469557e+00,  1.11987904e-02],
       [ 4.28872176e+00,  8.29626830e-03],
       [ 3.88399779e+00,  6.14602692e-03],
       [ 3.51690670e+00,  4.55308891e-03],
       [ 3.18409576e+00,  3.37301132e-03],
       [ 2.88247208e+00,  2.49878835e-03],
       [ 2.60919322e+00,  1.85114799e-03],
       [ 2.36165472e+00,  1.37136468e-03],
       [ 2.13747588e+00,  1.01593180e-03],
       [ 1.93448474e+00,  7.52619935e-04],
       [ 1.75070279e+00,  5.57554192e-04],
       [ 1.58433001e+00,  4.13045994e-04],
       [ 1.43373045e+00,  3.05991903e-04],
       [ 1.29741843e+00,  2.26684304e-04],
       [ 1.17404569e+00,  1.67931796e-04],
       [ 1.06238933e+00,  1.24406862e-04],
       [ 9.61340631e-01,  9.21628173e-05],
       [ 8.69894766e-01,  6.82758765e-05],
       [ 7.87141330e-01,  5.05799949e-05],
       [ 7.12255668e-01,  3.74705641e-05],
       [ 6.44490944e-01,  2.77588589e-05],
       [ 5.83170904e-01,  2.05642607e-05],
       [ 5.27683287e-01,  1.52343648e-05],
       [ 4.77473826e-01,  1.12855285e-05],
       [ 4.32040811e-01,  8.36083472e-06],
       [ 3.90930121e-01,  6.19459305e-06],
       [ 3.53730745e-01,  4.58977406e-06],
       [ 3.20070699e-01,  3.39927530e-06],
       [ 2.89613341e-01,  2.51727055e-06],
       [ 2.62054020e-01,  1.86508741e-06],
       [ 2.37117048e-01,  1.38175250e-06],
       [ 2.14552946e-01,  1.02304972e-06],
       [ 1.94135955e-01,  7.57021667e-07],
       [ 1.75661785e-01,  5.61807444e-07],
       [ 1.58945585e-01,  4.16604934e-07],
       [ 1.43820084e-01,  3.08764315e-07],
       [ 1.30133920e-01,  2.28671588e-07],
       [ 1.17750134e-01,  1.69424644e-07],
       [ 1.06544796e-01,  1.25519547e-07],
       [ 9.64057701e-02,  9.30038934e-08],
       [ 8.72315868e-02,  6.88931028e-08],
       [ 7.89304324e-02,  5.10386634e-08],
       [ 7.14192300e-02,  3.78092605e-08],
       [ 6.46228076e-02,  2.80102929e-08],
       [ 5.84731462e-02,  2.07504093e-08],
       [ 5.29086994e-02,  1.53723865e-08],
       [ 4.78737776e-02,  1.13881595e-08],
       [ 4.33179902e-02,  8.43658871e-09],
       [ 3.91957444e-02,  6.25037374e-09],
       [ 3.54657807e-02,  4.63040889e-09],
       [ 3.20907628e-02,  3.42872054e-09],
       [ 2.90369224e-02,  2.53944467e-09],
       [ 2.62736935e-02,  1.88187902e-09],
       [ 2.37734218e-02,  1.39579407e-09],
       [ 2.15110819e-02,  1.03418040e-09],
       [ 1.94640317e-02,  7.65507009e-10],
       [ 1.76117842e-02,  5.67379342e-10],
       [ 1.59358015e-02,  4.21220235e-10],
       [ 1.44193093e-02,  3.12463360e-10],
       [ 1.30471287e-02,  2.30387958e-10],
       [ 1.18055282e-02,  1.69171585e-10],
       [ 1.06820828e-02,  1.24624306e-10],
       [ 9.66554712e-03,  9.29262863e-11],
       [ 8.74574844e-03,  7.11288701e-11],
       [ 7.91347936e-03,  5.46908108e-11],
       [ 7.16041195e-03,  3.98268840e-11],
       [ 6.47900752e-03,  2.72837689e-11],
       [ 5.86244843e-03,  1.91511072e-11],
       [ 5.30456198e-03,  1.48276362e-11],
       [ 4.79976613e-03,  1.36877722e-11],
       [ 4.34300722e-03,  1.30157120e-11],
       [ 3.92971540e-03,  8.42551944e-12],
       [ 3.55575295e-03,  2.11868419e-12],
       [ 3.21737806e-03, -7.80912700e-13],
       [ 2.91120245e-03, -4.45766173e-13],
       [ 2.63416425e-03,  4.73528128e-12],
       [ 2.38349008e-03,  1.30332551e-11],
       [ 2.15667051e-03,  1.72194268e-11],
       [ 1.95143638e-03,  1.03297342e-11],
       [ 1.76573209e-03, -2.22697956e-12],
       [ 1.59770049e-03, -1.04149790e-11],
       [ 1.44565875e-03, -1.14948380e-11],
       [ 1.30808585e-03, -2.73552627e-12],
       [ 1.18360501e-03,  1.37778064e-11],
       [ 1.07096981e-03,  2.54106485e-11],
       [ 9.69053670e-04,  1.49183783e-11],
       [ 8.76835757e-04, -8.97542268e-12],
       [ 7.93393789e-04, -2.59542636e-11],
       [ 7.17892278e-04, -2.80333224e-11],
       [ 6.49575597e-04, -1.11220252e-11],
       [ 5.87760275e-04,  2.22720075e-11],
       [ 5.31827274e-04,  4.97425319e-11],
       [ 4.81217276e-04,  3.35428952e-11],
       [ 4.35423368e-04, -1.37987561e-11],
       [ 3.93987399e-04, -5.10585013e-11],
       [ 3.56494537e-04, -5.62533096e-11],
       [ 3.22569446e-04, -2.43431287e-11],
       [ 2.91872772e-04,  4.03646736e-11],
       [ 2.64097167e-04,  9.81670730e-11],
       [ 2.38965045e-04,  7.18074257e-11],
       [ 2.16224679e-04, -2.27904466e-11],
       [ 1.95648355e-04, -1.03835668e-10],
       [ 1.77030002e-04, -1.15768154e-10],
       [ 1.60183197e-04, -5.41115713e-11],
       [ 1.44939454e-04,  7.24680120e-11],
       [ 1.31146283e-04,  1.93282185e-10],
       [ 1.18666125e-04,  1.54382302e-10],
       [ 1.07373628e-04,  1.10536321e-10],
       [ 9.71557052e-05,  8.85208431e-11],
       [ 8.79101739e-05,  5.79619864e-11],
       [ 7.95444246e-05,  4.78603510e-11],
       [ 7.19747993e-05,  3.31780518e-11],
       [ 6.51255102e-05,  1.64728938e-11],
       [ 5.89279471e-05,  1.68381406e-11],
       [ 5.33201593e-05,  1.80503622e-11],
       [ 4.82460206e-05,  1.54651329e-11],
       [ 4.36547692e-05,  9.73997987e-12],
       [ 3.95003912e-05,  7.00330671e-12],
       [ 3.57414829e-05,  5.87011903e-12],
       [ 3.23408871e-05,  5.17938004e-12],
       [ 2.92636831e-05,  3.57929680e-12],
       [ 2.64798655e-05,  1.87952843e-12],
       [ 2.39603263e-05,  8.37303582e-13],
       [ 2.16809491e-05,  3.77004501e-13],
       [ 1.96181727e-05,  3.83968253e-13],
       [ 1.77516775e-05,  4.83719903e-13]])

We get an array!!!

So..........which is our desired state variable?

# Plot x v.s. time!!!
fig1 = list_plot(zip(timeRange,result[:,0]),color='red',plotjoined=True,legend_label='X Variable')
fig2 = list_plot(zip(timeRange,result[:,1]),plotjoined=True,legend_label='Y Variable')
show(fig1+fig2)

fig4=list_plot(zip(result[:,0],result[:,1]),plotjoined=True)

Plotting Vector Field

v(x,y) = [-x+5*y, -3*y]
fig5= plot_vector_field(v, (x,-20,25), (y,0,20), axes_labels=["X", "Y"])
show(fig5)


show(fig4+fig5)