(original by Dario Izzo - extended by Ekin Ozturk)
Building upon the notebook here, we show the use of desolver for numerically integrating the system of differential equations $\dot{\mathbf y} = \mathbf f(\mathbf y)$:
which describe, in non dimensional units, the Keplerian motion of a mass point object around some primary body. We show how we can build a high order Taylor map (HOTM, indicated with $\mathcal M$) representing the final state of the system at the time $T$ as a function of the initial conditions.
In other words, we build a polinomial representation of the relation $\mathbf y(T) = \mathbf f(\mathbf y(0), T)$. Writing the initial conditions as $\mathbf y(0) = \overline {\mathbf y}(0) + \mathbf {dy}$, our HOTM will be written as:
$$ \mathbf y(T) = \mathcal M(\mathbf {dy}) $$and will be valid in a neighbourhood of $\overline {\mathbf y}(0)$.
In [1]:
%matplotlib inline
from matplotlib import pyplot as plt
import os
import numpy as np
os.environ['DES_BACKEND'] = 'numpy'
import desolver as de
import desolver.backend as D
from desolver.backend import gdual_double as gdual
In [2]:
T = 1e-3
@de.rhs_prettifier(equ_repr="[vr, -1/r**2 + r*vt**2, vt, -2*vt*vr/r]", md_repr=r"""$$
\begin{array}{l}
\dot r = v_r \\
\dot v_r = - \frac 1{r^2} + r v_\theta^2\\
\dot \theta = v_\theta \\
\dot v_\theta = -2 \frac{v_\theta v_r}{r}
\end{array}
$$""")
def eom_kep_polar(t,y,**kwargs):
return D.array([y[1], - 1 / y[0] / y[0] + y[0] * y[3]*y[3], y[3], -2*y[3]*y[1]/y[0] - T])
eom_kep_polar
Out[2]:
In [3]:
# The initial conditions
ic = [1.,0.1,0.,1.]
In [4]:
D.set_float_fmt('float64')
float_integration = de.OdeSystem(eom_kep_polar, y0=ic, dense_output=False, t=(0, 5.), dt=0.01, rtol=1e-12, atol=1e-12, constants=dict())
float_integration.set_method("RK45")
float_integration.integrate(eta=True)
In [5]:
# Here we transform from polar to cartesian coordinates
# to then plot
y = float_integration.y
cx = [it[0]*np.sin(it[2]) for it in y.astype(np.float64)]
cy = [it[0]*np.cos(it[2]) for it in y.astype(np.float64)]
plt.plot(cx,cy)
plt.title("Orbit resulting from the chosen initial conditions")
plt.xlabel("x")
plt.ylabel("y")
Out[5]:
In [6]:
# Order of the Taylor Map. If we have 4 variables the number of terms in the Taylor expansion in 329 at order 7
order = 5
# We now define the initial conditions as gdual (not float)
ic_g = [gdual(ic[0], "r", order), gdual(ic[1], "vr", order), gdual(ic[2], "t", order), gdual(ic[3], "vt", order)]
In [7]:
import time
start_time = time.time()
D.set_float_fmt('gdual_double')
gdual_integration = de.OdeSystem(eom_kep_polar, y0=ic_g, dense_output=False, t=(0, 5.), dt=0.01, rtol=1e-12, atol=1e-12, constants=dict())
gdual_integration.set_method("RK45")
gdual_integration.integrate(eta=True)
print("--- %s seconds ---" % (time.time() - start_time))
In [8]:
# We extract the last point
yf = gdual_integration.y[-1]
# And unpack it into some convinient names
rf,vrf,tf,vtf = yf
# We compute the final cartesian components
xf = rf * D.sin(tf)
yf = rf * D.cos(tf)
# Note that you can get the latex representation of the gdual
print(xf._repr_latex_())
print("xf (latex):")
xf
Out[8]:
In [9]:
# We can extract the value of the polinomial when $\mathbf {dy} = 0$
print("Final x from the gdual integration", xf.constant_cf)
print("Final y from the gdual integration", yf.constant_cf)
# And check its indeed the result of the 'reference' trajectory (the lineariation point)
print("\nFinal x from the float integration", cx[-1])
print("Final y from the float integration", cy[-1])
In [10]:
# Let us now visualize the Taylor map by creating a grid of perturbations on the initial conditions and
# evaluating the map for those values
Npoints = 10 # 10000 points
epsilon = 1e-3
grid = np.arange(-epsilon,epsilon,2*epsilon/Npoints)
nxf = [0] * len(grid)**4
nyf = [0] * len(grid)**4
i=0
import time
start_time = time.time()
for dr in grid:
for dt in grid:
for dvr in grid:
for dvt in grid:
nxf[i] = xf.evaluate({"dr":dr, "dt":dt, "dvr":dvr,"dvt":dvt})
nyf[i] = yf.evaluate({"dr":dr, "dt":dt, "dvr":dvr,"dvt":dvt})
i = i+1
print("--- %s seconds ---" % (time.time() - start_time))
In [11]:
f, axarr = plt.subplots(1,3,figsize=(15,5))
# Normal plot of the final map
axarr[0].plot(nxf,nyf,'.')
axarr[0].plot(cx,cy)
axarr[0].set_title("The map")
# Zoomed plot of the final map (equal axis)
axarr[1].plot(nxf,nyf,'.')
axarr[1].plot(cx,cy)
axarr[1].set_xlim([cx[-1] - 0.1, cx[-1] + 0.1])
axarr[1].set_ylim([cy[-1] - 0.1, cy[-1] + 0.1])
axarr[1].set_title("Zoom")
# Zoomed plot of the final map (unequal axis)
axarr[2].plot(nxf,nyf,'.')
axarr[2].plot(cx,cy)
axarr[2].set_xlim([cx[-1] - 0.01, cx[-1] + 0.01])
axarr[2].set_ylim([cy[-1] - 0.1, cy[-1] + 0.1])
axarr[2].set_title("Stretch")
#axarr[1].set_xlim([cx[-1] - 0.1, cx[-1] + 0.1])
#axarr[1].set_ylim([cy[-1] - 0.1, cy[-1] + 0.1])
Out[11]:
In [12]:
# First we profile the method evaluate (note that you need to call the method 4 times to get the full state)
In [13]:
%timeit xf.evaluate({"dr":epsilon, "dt":epsilon, "dvr":epsilon,"dvt":epsilon})
In [14]:
# Then we profile the Runge-Kutta 4 integrator
In [15]:
%%timeit
D.set_float_fmt('float64')
float_integration = de.OdeSystem(eom_kep_polar, y0=[it + epsilon for it in ic], dense_output=False, t=(0, 5.), dt=0.01, rtol=1e-12, atol=1e-12, constants=dict())
float_integration.set_method("RK45")
float_integration.integrate(eta=False)
In [16]:
# It seems the speedup is 2-3 orders of magnitude, but did we loose precision?
# We plot the error in the final result as computed by the HOTM and by the Runge-Kutta
# as a function of the distance from the original initial conditions
out = []
pert = np.arange(0,0.1,1e-3)
for epsilon in pert:
res_map_xf = xf.evaluate({"dr":epsilon, "dt":epsilon, "dvr":epsilon,"dvt":epsilon})
res_int = de.OdeSystem(eom_kep_polar, y0=[it + epsilon for it in ic], dense_output=False, t=(0, 5.), dt=0.01, rtol=1e-12, atol=1e-12, constants=dict())
res_int.set_method("RK45")
res_int.integrate()
res_int_x = [it.y[0]*np.sin(it.y[2]) for it in res_int]
res_int_xf = res_int_x[-1]
out.append(np.abs(res_map_xf - res_int_xf))
plt.semilogy(pert,out)
plt.title("Error introduced by the use of the polynomial")
plt.xlabel("Perturbation of the initial conditions")
plt.ylabel("Error in estimating the final state (x)")
Out[16]: