Dynamical Systems Theory

Nonlinear dynamical systems theory (DST) offers insights in multi-body regimes, where qualitative information is necessary concerning sets of solutions and their evolution. DST is, of course, a broad subject area with applications to many fields. For application to spacecraft trajectory design, it is helpful to first consider special solutions and invariant manifolds, since this aspect of DST offers immediate insights. Under a GSFC grant, Purdue University investigated various dynamical systems methodologies that now are included in software called Generator. In Generator, different types of solution arcs, some based on dynamical systems theory, are input to a process that differentially corrects the trajectory segments to produce a complete path in a complex dynamical model. A two level iteration scheme is utilized whenever differential corrections are required. This approach produces position continuity and then a velocity continuity for a given trajectory. An understanding of the solution space then forms a basis for computation of a preliminary libration and transfer orbit solution and the end-to-end approximation can then be transferred to a direct targeting method like Swingby for final adjustments for launch window, launch vehicle error analysis, maneuver planning, or higher order modeling.

The geometrical theory of dynamical systems is based in phase space and begins with special solutions that include equilibrium points, periodic orbits, and quasi-periodic motions. Differential manifolds are introduced as the geometrical model for the phase space of dependent variables. An invariant manifold is defined as an n-dimensional surface such that an orbit starting on the surface remains on the surface throughout its dynamical evolution. So, an invariant manifold is a set of orbits that form a surface. Invariant manifolds, in particular stable, unstable, and center manifolds, are key components in the analysis of the phase space. Bounded motions, which include periodic orbits such as halo orbits, exist in the center manifold, as well as transitions from one type of bounded motion to another. Sets of orbits that approach or depart an invariant manifold asymptotically are also invariant manifolds (under certain conditions) and these are the stable and unstable manifolds, respectively, associated with the linear stable and unstable modes.

The periodic halo orbits, as defined in the circular restricted problem, are used as a reference solution for investigating the phase space in this analysis. It is possible to exploit the hyperbolic nature of these orbits by using the associated stable and unstable manifolds to generate transfer trajectories as well as general trajectory arcs in this region of space.

The computation process of the stable and unstable manifolds, shown in Table 1, is associated with particular halo orbit design parameters and is accomplished numerically in a straightforward manner. The procedure is based on the availability of the monodromy matrix (the variational or state transition matrix after one period of motion) associated with the Lissajous orbit. A similar state transition matrix of this sort can be computed using the state equations of motion based on circular three-body restricted motion. This matrix essentially serves to define a discrete linear map of a fixed point in some arbitrary Poincare section. As with any discrete mapping of a fixed point, the characteristics of the local geometry of the phase space can be determined from the eigenvalues and eigenvectors of the monodromy matrix. These are characteristics not only of the fixed point, but also of the Lissajous orbit. The local approximation of the stable and unstable manifolds involves calculating the eigenvectors of the monodromy matrix that are associated with the stable and unstable eigenvalues. This approximation can be propagated to any point along the halo orbit using the state transition matrix.

Table 1 - Dynamical System Approach Segments

Utility	Input	Output
Phase (Generic Orbit)	User Data	Control Angles For Lissajous
Lissajous	Universe And User Data	Patch Point And Lissajous Orbit
Monodromy (Periodic Orbit)	Universe And Lissajous Output	Fixed Points And Stable And Unstable Manifold Approximations
Manifold	Universe And Monodromy Output	1-Dimensional Manifold
Transfer	Universe, User Selected Patch Points, Manifold Output	Transfer Trajectory From Earth To L1 Or L2

The periodic and quasi-periodic orbits obtained from the Lissajous program within Generator are obtained using the two-level iteration scheme. An analytic approximation is used as an initial guess. The final corrected orbit is continuous in position and velocity, with the exception of micro delta-Vs, within the full ephemeris model. An example of the generated Lissajous orbit for the MAP mission is shown in Figure 1.

FIGURE 1

Using Generator, the 2-dimensional manifold surfaces can be generated and displayed in configuration space. Figure 2 shows the stable manifolds for a direct transfer to the Sun-Earth L₂ point.

FIGURE 2: Stable Manifold