Astrodynamics

The Koopman Operator (KO) can solve for Halo orbits. The figure reports two sets of 30 different orbits calculated through a KO using basis functions of order 5, around L1 and L2. The figure includes the projection of the orbits in the three reference planes and an isometric three-dimensional view of the trajectories. The black point in the picture represents the secondary celestial body, which is Earth in the selected Sun–Earth system. As the figure shows, the Koopman solution achieves a robust representation of the orbits, the accuracy of which will be assessed later. The periodicity is given by an accurate selection of the initial conditions, which have been found through the implementation of a differential corrector in such a way that the proposed solution describes the same orbits for multiple revolutions. The KO is able to accurately represent orbits around both libration points.

The eigenfunction analysis and the study of the minimum points (minima) are important when seeking stability and periodic orbits in a chaotic application, such as the CRTBP. The KO orbits are able to maintain their periodicity due to the eigendecomposition of the dynamics, where the system is diagonalized and each eigenfunction is studied in terms of magnitude and shape. Therefore, if desired, unstable contributions from eigenfunctions connected to unstable eigenvalues can be separated. The figure reports the first 24 eigenfunctions from the third-order KO solution. The six-dimensional functions are represented with just two dimensions in the (x; vy) plane, because this information sufficiently describes each Lyapunov orbit. The eigenfunctions are centered at the L1 point, meaning that position (0, 0, 0) corresponds to L1 instead of the primary celestial body. Examining a gray-scale gradient, the figure shows the locations of minima in bright white. These curves represent equilibrium conditions, or more precisely, their approximation through a third-order KO. Therefore, when we include the initial conditions in the (x; vy) plane as white dots, their locations settle close to the minimum lines. Thus, the Koopman approximation is correctly approximating the dynamics of the system in terms of its eigenfunctions. However, the accuracy of the approximation decreases as the initial condition becomes larger; the farthest white points are not as well approximated as the white points near the origin. Moreover, the location of the minima is not unique. Many eigenfunctions show well-marked bright lines in the bottom part of the plane, for low velocities. These initial conditions are still connected to periodic orbits. The resulting pathway leaves the Lagrangian L1 point and starts orbiting around the main celestial body with a periodic behavior. Therefore, even if the eigenfunctions become less accurate the farther they are from the equilibrium point, their minima are still able to connect to different kinds of periodic orbits. Note also that the figure reports only initial conditions for orbits connected to negative x and positive vy. However, thanks to the symmetric properties of the CRTBP, the domain of the eigenfunctions can be widened to support retrograde orbits and all the remaining periodic orbits from axial and central symmetries.

A spectral study can also be performmed using KO. The figure compares the eigenvalues, computed using the Galerkin method, from the third-order and sixth-order Koopman matrices. In this particular application, the eigenvalues of the third-order solution are a subset of the sixth-order approximation. Thus, the sixth-order solution incorporates information of previous orders and enhances it with additional nonlinear functions. As such, the location of the high-order eigenvalues is at the edge of the diamond-shaped figure. By looking at the intensity of every singlemarker, it can be noted how eigenvalues closer to the origin and the imaginary axis are repeated multiple times. Indeed, it is possible to know in advance the position of the unperturbed eigenvalues, for any given order, by the simple composition of the eigenvalues that have already been calculated for lower orders. Therefore, position and repetitiveness are explained in terms of statics and combinatorics (the theory of combinations), where all the new eigenvalues are found as a linear combination of the original ones. Moreover, the linear approximation near the L1 is known analytically, aswell as its eigenvalues,which have been portrayed in the figure as circles. The linear part of the KO solution matches the linear contribution of the true dynamics. The six linear eigenvalues are perfectly represented, and the high-order approximations evolve from the latter to best represent the nonlinear terms of the system of ODEs.

The frequencies and spectral behavior of the system provided by the KO can also be used to assess the convergence of the methodology. This can be done by the study of theKoopman modes, that is, the projection of the observables in the operator.The figure presents the intensity of the Koopman modes of the system. Particularly, it shows how the eigenvalues with the highest modulus are the least influential in the solution. However, the small contributions from the modes connected to large eigenvalues are crucial to achieving an accurate approximation of the dynamics, and they make the difference in accuracy when comparing the order 3 solution with the one of order 6. Therefore, many terms other than the linear ones are nonzero, with influence that decreases as the order of the eigenfunctions increases. This also implies that the Koopman modes decay with larger frequency values, leading to a possible truncation of the expansion for eigenfunctions with particularly high frequencies.

The figure shows, with different line styles, the accuracy of the approximation on each orbit for different orders of the KO. As expected, as the halo orbits become larger, the position error increases consequently. Therefore, for any given order, the position error convergence analysis describes a set of curves. In that regard, order 3, represented by a continuous line, is the least-accurate solution, while order 6 is the most accurate. However, it is worth noticing how, for each orbit, the gain in accuracy obtained by increasing the KO order behaves differently depending on the initial condition. The sixth-order KO solution is two orders of magnitude more accurate than the third order for orbits close to L1. However, as the initial condition of the halo orbit moves away from the equilibrium point, the gain in accuracy is reduced, settling around a single order of magnitude. Indeed, for very large initial conditions, increasing the order of the basis functions to over a given value stops being beneficial because the gain in performance does not repay the higher computational burden. This aspect is connected with the Richardson approximation applied for the polynomial representation of the dynamics, which fails to provide an accurate approximation of the system for initial conditions far from the equilibrium point. Consequently, it can be noted that for each set of curves, at different orders, there is a well-marked separation in the bottom part of the graph, while curves from different orders start to get closer as the orbit becomes larger. This aspect has been highlighted in the right figure, where the mean position error  for each orbit is compared with the size of the halo orbit and the maximum order of the basis functions used to represent the KO. Thus, orbits with the furthest initial conditions from the L1 point generate mean position errors that range in value by up to one order of magnitude. However, as we get closer to L1, the mean position error for each KO solution order considered decreases with a different slope, making the accuracy gain more substantial as we increase the order of basis functions used. Additionally, it can also be observed that there is a region very close to L1 where the error seems to reach convergence and stabilize for six order basis functions. Thus, for halo orbits close to L1, an increase of the KO order leads to a substantial improvement in accuracy and in the performance of the approximation of the system.