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Jet transport: the simple pendulum

In this example we illustrate the use of jet transport techniques in TaylorIntegration.jl for the simple pendulum. We propagate a neighborhood $U_0$ around an initial condition $q_0$ parametrized by the sum $q_0+\xi$, where $q_0=(x_0,p_0)$ represents the coordinates of the initial condition in phase space, and $\xi=(\xi_1,\xi_2)$ represents an small variation with respect to this initial condition. We re-interpret each component of the sum $q_0+\xi$ as a multivariate polynomial in the variables $\xi_1$ and $\xi_2$; below, the maximum order of the multivariate polynomial is fixed at 8. We propagate these multivariate polynomials in time using Taylor's method.

The simple pendulum is defined by the Hamiltonian $H(x, p) = \frac{1}{2}p^2-\cos x$; the corresponding equations of motion are given by

\[\begin{aligned} \dot{x} & = p, \\ \dot{p} & = -\sin x. \end{aligned}\]

We integrate this problem for a neighborhood $U_0$ around the initial condition $q_0 = (x(t_0), p(t_0)) = (x_0, p_0)$. For concreteness we take $p_0=0$ and choose $x_0$ such that the pendulum librates; that is, we will choose a numerical value for the energy $E=H(x_0,p_0)=-\cos x_0$ such that the pendulum's motion in phase space is "below" (inside) the region bounded by the separatrix. In this case, the libration period $T$ of the pendulum is

\[T=\frac{4}{\sqrt{2}}\int_0^{x_0}\frac{dx}{\sqrt{\cos x_0-\cos x}},\]

which can be expressed in terms of the complete elliptic integral of the first kind, $K$:

\[T=4K(\sin(x_0/2)).\]

The Hamiltonian for the simple pendulum is:

H(x) = 0.5x[2]^2-cos(x[1])

The equations of motion are:

function pendulum!(dx, x, p, t)
    dx[1] = x[2]
    dx[2] = -sin(x[1])
end

We define the TaylorN variables necessary to perform the jet transport; varorder represents the maximum order of expansion in the variations $\xi$.

const varorder = 8
using TaylorIntegration
ξ = set_variables("ξ", numvars=2, order=varorder)
2-element Vector{TaylorN{Float64}}:
  1.0 ξ₁ + 𝒪(‖x‖⁹)
  1.0 ξ₂ + 𝒪(‖x‖⁹)

Note that TaylorSeries.jl is @reexport-ed internally by TaylorIntegration.jl.

The nominal initial condition is:

q0 = [1.3, 0.0]
2-element Vector{Float64}:
 1.3
 0.0

The corresponding initial value of the energy is:

H0 = H(q0)
-0.26749882862458735

The parametrization of the neighborhood $U_0$ is represented by

q0TN = q0 .+ ξ
2-element Vector{TaylorN{Float64}}:
  1.3 + 1.0 ξ₁ + 𝒪(‖x‖⁹)
        1.0 ξ₂ + 𝒪(‖x‖⁹)

To understand how the jet transport technique works, we shall evaluate the Hamiltonian at $q_0+\xi$ in order to obtain the 8-th order Taylor expansion of the Hamiltonian with respect to the variations $\xi$, around the initial condition $q_0$:

H(q0TN)
 - 0.26749882862458735 + 0.963558185417193 ξ₁ + 0.13374941431229367 ξ₁² + 0.5 ξ₂² - 0.1605930309028655 ξ₁³ - 0.011145784526024473 ξ₁⁴ + 0.008029651545143275 ξ₁⁵ + 0.0003715261508674824 ξ₁⁶ - 0.00019118217964626843 ξ₁⁷ - 6.634395551205043e-6 ξ₁⁸ + 𝒪(‖x‖⁹)

Note that the 0-th order term of the expression above is equal to the value H(q0), as expected.

Below, we set some parameters for the Taylor integration. We use a method of taylorinteg which returns the solution at t0, t0+integstep, t0+2integstep,...,tmax, where t0 and tmax are the initial and final times of integration, whereas integstep is a time interval chosen by the user; we use the variable tv = t0:integstep:tmax for this purpose and choose integstep as $\frac{T}{8}$.

order = 28     #the order of the Taylor expansion wrt time
abstol = 1e-20 #the absolute tolerance of the integration
using Elliptic # we use Elliptic.jl to evaluate the elliptic integral K
T = 4*Elliptic.K(sin(q0[1]/2)^2) #the libration period
t0 = 0.0        #the initial time
tmax = 6T       #the final time
integstep = T/8 #the time interval between successive evaluations of the solution vector

We perform the Taylor integration using the initial condition x0TN, during 6 periods of the pendulum (i.e., $6T$), exploiting multiple dispatch:

tv = t0:integstep:tmax # the times at which the solution will be evaluated
xv = taylorinteg(pendulum!, q0TN, tv, order, abstol)

The integration above works for any initial neighborhood $U_0$ around the nominal initial condition $q_0$, provided it is sufficiently small.

We will consider the particular case where $U_0$ is a disk of radius $r = 0.05$, centered at $q_0$; that is $U_0=\{ q_0+\xi:\xi=(r\cos\phi,r\sin\phi); \phi\in[0,2\pi) \}$ for a given radius $r>0$. We will denote by $U_t$ the propagation of the initial neighborhood $U_0$ evaluated at time $t$. Also, we denote by $q(t)$ the coordinates of the nominal solution at time $t$: $q(t)=(x(t),p(t))$. Likewise, we will denote the propagation at time $t$ of a given initial variation $\xi_0$ by $\xi(t)$. Then, we can compute the propagation of the boundary $\partial U_t$ of the neighborhood $U_t$.

polar2cart(r, ϕ) = [r*cos(ϕ), r*sin(ϕ)] # convert radius r and angle ϕ to cartesian coordinates
r = 0.05 #the radius of the neighborhood
ϕ = 0.0:0.1:(2π+0.1) #the values of the angle
ξv = polar2cart.(r, ϕ)

We evaluate the jet at $\partial U_{x(t)}$ (the boundary of $U_{x(t)}$) at each value of the solution vector xv; we organize these values such that we can plot them later:

xjet_plot = map(λ->λ.(ξv), xv[:,1])
pjet_plot = map(λ->λ.(ξv), xv[:,2])

Above, we have exploited the fact that Array{TaylorN{Float64}} variables are callable objects. Now, we evaluate the jet at the nominal solution, which corresponds to $\xi=(0,0)$, at each value of the solution vector xv:

x_nom = xv[:,1]()
p_nom = xv[:,2]()

Finally, we shall plot the nominal solution (black dots), as well as the evolution of the neighborhood $U_0$ (in colors), each $\frac{1}{8}$th of a period $T$. The initial condition corresponds to the black dot situated at $q_0=(1.3,0)$

using Plots
plot( xjet_plot, pjet_plot,
    xaxis=("x",), yaxis=("p",),
    title="Simple pendulum phase space",
    leg=false, aspect_ratio=1.0
)
scatter!( x_nom, p_nom,
    color=:black,
    m=(1,2.8,stroke(0))
)
Example block output