Lyapunov Instability for Discontinuous Differential Equations

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

On the Stability of Periodic Motions of a Two-Body SystemWith Flexible Connection in an Elliptical Orbit

This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third and fourth order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coeffcients of the normalized Hamiltonian. Then a nonlinear stability analysis for two periodic solutions is performed in the third and fourth order resonance cases as well as in the nonresonance case.

A METHOD FOR PREDICTING ENERGY-STABILIZED MOTION OF SPACECRAFT BASED ON DIFFERENTIAL TAYLOR TRANSFORMATIONS

To predict the motion of spacecrafts, a numerical-analytical method for integrating the differential equation of the orbital motion of a spacecraft stabilized by the Baumgart differential method is proposed. The stabilization of the differential equation of motion by the Baumgart method is carried out according to the energy of the spacecraft. Stabilization is carried out to reduce the influence of the Lyapunov instability on the accumulation of numerical errors in the integration of the differential equation, which is effective when conducting a long-term numerical prediction of the motion of spacecraft. Integration of the stabilized equation is based on differential Taylor transformations. Computational schemes with a constant step and an integration order are considered, as well as schemes with adaptation by an integration step and order. For adaptive schemes, the results of forecasting the motion of spacecraft according to the criterion “accuracy-computational complexity» for a given relative error of integration with respect to integration phase variables and spacecraft energy are presented. It is shown that both options require setting various internal adaptation parameters, but they have comparable efficiency. Recommendations are proposed on the use of the developed method for integrating energy-stabilized equations for predicting the motion of spacecraft in the near space in the Greenwich rectangular coordinate system.