Jet transport

In this section we describe the jet transport capabilities included in TaylorIntegration.jl. Jet transport is a tool that allows the propagation under the flow of a small neighborhood in phase space around a given initial condition, instead of propagating a single initial condition only.

To compute the propagation of $\mathbf{x}_0 + \delta \mathbf{x}$, where $\delta \mathbf{x}$ are independent small displacements in phase space around the initial condition $\mathbf{x}_0$, one has to solve high-order variational equations. The idea is to treat $\mathbf{x}_0 + \delta \mathbf{x}$ as a truncated polynomial in the $\delta \mathbf{x}$ variables. The maximum order of this polynomial has to be fixed in advance.

Jet transport works in general with any ordinary ODE solver, provided the chosen solver supports computations using multi-variate polynomial algebra.

A simple example

Following D. Pérez-Palau et al [1], let us consider the differential equations for the harmonic oscillator:

\[\begin{eqnarray*} \dot{x} & = & y, \\ \dot{y} & = & -x, \end{eqnarray*}\]

with the initial condition $\mathbf{x}_0=[x_0, y_0]^T$. We illustrate jet transport techniques using Euler's method

\[\begin{equation*} \mathbf{x}_{n+1} = \mathbf{x}_n + h \mathbf{f}(\mathbf{x}_n). \end{equation*}\]

Instead of considering the initial conditions $\mathbf{x}_0$, we consider the time evolution of the polynomial

\[\begin{equation*} P_{0,\mathbf{x}_0}(\delta\mathbf{x}) = [x_0+\delta x, y_0 + \delta y]^T, \end{equation*}\]

where $\delta x$ and $\delta y$ are small displacements. Below we concentrate in polynomials of order 1 in $\delta x$ and $\delta y$; since the equations of motion of the harmonic oscillator are linear, there are no higher order terms.

Using Euler's method we obtain

\[\begin{eqnarray*} \mathbf{x}_1 & = & \left( \begin{array}{c} x_0 + h y_0 \\ y_0 - h x_0 \end{array} \right) + \left( \begin{array}{cc} 1 & h \\ -h & 1 \end{array} \right) \left( \begin{array}{c} \delta x\\ \delta y \end{array} \right). \\ \mathbf{x}_2 & = & \left( \begin{array}{c} 1-h^2 x_0 + 2 h y_0 \\ 1-h^2 y_0 - 2 h x_0 \end{array} \right) + \left( \begin{array}{cc} 1-h^2 & 2 h \\ -2 h & 1-h^2 \end{array} \right) \left( \begin{array}{c} \delta x\\ \delta y \end{array} \right). \end{eqnarray*}\]

The first terms in the expressions for $\mathbf{x}_1$ and $\mathbf{x}_2$ above correspond to the result of an Euler integration step using the initial conditions only. The other terms are the (linear) corrections which involve the small displacements $\delta x$ and $\delta y$.

In general, for differential equations involving non-linear terms, the resulting expansions in $\delta x$ and $\delta y$ will reflect aspects of the non-linearities of the ODEs. Clearly, jet transport techniques allow to address stability properties beyond the linear case, though memory constraints may play a role. See this example illustrating the implementation for the simple pendulum, and this one illustrating the construction of a Poincaré map with Jet transport techniques.

References

[1] D. Pérez-Palau, Josep J. Masdemont, Gerard Gómez, 2015, Celest. Mech. Dyn. Astron. 123, 239.