Invariant Manifold Tracking for First-Order Nonlinear Hill's Equations

Abstract

Abstract : An approach to provide nonlinear active control for the first-order nonlinear classical Hill's equations is described. Both the linearized and nonlinear Hill's equations are controlled to remain close to specific invariant manifolds defined through the various system Hamiltonians. It is then shown that trajectories similar to the periodic trajectories of the linearized system can be maintained by the nonlinear equations on invariant manifolds defined by the linearized system of equations. Forcing the nonlinear system trajectories onto an invariant manifold of the linearized system, with an appropriate choice of initial conditions, provides a significant reduction in the along-track drift of the first-order nonlinear Hill's equations as compared to the linearized equations. There is also a small drift reduction in the radial coordinate direction. The cross-track position suffers only a slight increase in the maximum amplitude of its oscillation.