Method of an Analysis of the Nonlinear Monotonic Simultaneous Stability of Two Orientation Angles of an Asymmetric Spacecraft in the Martian Atmosphere

One of the most relevant and practically important directions in the development of the modern mathematical theory of stability is the development of effective methods for analyzing the nonlinear stability for solutions of dynamical systems with several degrees of freedom. The aim of this paper is to develop a new qualitative method for analyzing the nonlinear monotonic simultaneous stability of the solutions of a dynamical system of equations describing the motion of a solid body with two angles of orientation. The method is based on the application of the classical mathematical investigation method for the function of one independent variable and combinatorics. The method assumes that two analyzed functions describing the evolution of two orientation angles are defined and are twice continuously differentiable over the considered interval of change of the independent variable. The main result of the paper is the determination of the number of all different cases of nonlinear monotone simultaneous stability. The paper shows how, based on the application of this method, it is possible to solve a practically significant problem of qualitative analysis of the simultaneous nonlinear monotonic stability of two orientation angles of an asymmetric spacecraft performing an uncontrolled rotation in the Martian atmosphere. In conclusion, it should be noted that the presented method can be used to analyze the nonlinear monotonic simultaneous stability of solutions in various continuous dynamic systems.