Study On Periodic Orbits Near Small Bodies And Entropy Analysis

Small bodies exist widely in the solar system and are interesting for scientific research and engineering exploration.It is the complex gravitational fields that lead to abundant nonlinear dynamical characteristics and challenges on orbital dynamics in the vicinity of small bodies.Shape irregularity of small bodies is one of the factors that cause complex gravitational fields.In this dissertation,the entropy of shape is introduced to describe the shape irregularity of small bodies,and the entropy of frequency is proposed to analyze quasi-periodic orbits around small bodies quantitatively.Besides,periodic orbits with the solar perturbation and its searching method,multiple bifurcations,and the mutation of orbital stability during the continuation of periodic orbits are also studied in this dissertation.In order to compare the shapes of irregular small bodies with spheres quantitatively using a single indicator,the entropy of shape is defined.The entropies of shape of 19 small bodies' polyhedron models are calculated.The comparison between the entropy of shape and the moment of inertia indicator shows that the entropy of shape describes the similarity between small bodies and spheres more precisely than the moment of inertia indicator.Thus irregular small bodies can be classified according to their entropies of shape.In the studies of solar perturbation on periodic orbits around small bodies,the variations in solar perturbation effect in the vicinity of small bodies and solar perturbation on periodic orbits are analyzed first.After that,the hierarchical grid search method of periodic orbits is extended with solar perturbation in this dissertation;12 families of largescale periodic orbits around 433 Eros are generated and analyzed as the result.Finally,searching efficiency is promoted with the application of dipole model in the process of selecting possible initial conditions of periodic orbits in the hierarchical grid search method.In this dissertation,multiple bifurcations and the phenomenon of orbital stability alternating during the continuation of periodic orbits are given and validated in the studies of continuation around 433 Eros;this study extends the theory of bifurcation of continuation of periodic orbits.It is also confirmed that the alternating of the orbits' shape corresponds to the mutation of slope on the map of Jacobi's integral vs.orbital periods.Since orbital stability alternating also exists in the vicinity of 243 Ida,the motions of Dactyl around 243 Ida are analyzed according to the observational data.Dactyl is considered on a quasi-periodic orbit around 243 Ida because the Poincaré section of its motion meets the map of the perturbation of the periodic orbits in the stable zone.The entropy of frequency is introduced to analyze the periodicity of orbits in the vicinity of small bodies quantitatively to improve the preciseness of evaluation on the study of quasi-periodic orbits.This indicator is applied to orbits around 243 Ida and 6489 Golevka as examples.Theoretical differences of entropies of frequency between any orbits are rigorously derived.Numerical experiments in the Hénon-Heiles system and the circular restricted three-body system also prove that the entropy of frequency distinguishes periodic,quasi-periodic,and chaotic motions in dynamical systems more precisely than the orthogonal fast Lyapunov indicator.