Topological Dynamical System In The Highly Irregular Gravity Field

This thesis investigates the topological dynamical system in the highly irregulargravity field, including the different novel forms of the dynamical equations of aparticle orbiting highly irregular-shaped celestial bodies, order and chaos in a rotatingplane-symmetric potential field, orbits and manifolds near the equilibrium points of arotating highly irregularly celestial body, as well as the global periodic orbits, thestability of orbits and the structure of submanifolds in the potential field of rotatinghighly irregular-shaped celestial bodies.The different novel forms of the dynamical equations of a particle orbiting arotating irregular celestial body and the effective potential, the Jacobi integral, etc. ondifferent manifolds are presented. Nine new forms of the dynamical equations of aparticle orbiting a rotating irregular celestial body are presented, and the classicalform of the dynamical equations has also been found. The dynamical equations withthe potential and the effective potential in scalar form in the arbitrary body-fixedframe and the special body-fixed frame are presented and discussed. Moreover, thesimplified forms of the effective potential and the Jacobi integral have been derived.The dynamical equation in coefficient-matrix form has been derived. Other forms ofthe dynamical equations near the asteroid are presented and discussed, including theLagrange form, the Hamilton form, the symplectic form, the Poisson form, thePoisson-bracket form, the cohomology form, and the dynamical equations on theK hler manifold and another complex manifold. Novel forms of the effectivepotential and the Jacobi integral are also presented. The dynamical equations in scalarform and coefficient-matrix form can aid in the study of the dynamical system, thebifurcation, and the chaotic motion of the orbital dynamics of a particle near arotating irregular celestial body. The dynamical equations of a particle near a rotatingirregular celestial body are presented on several manifolds, including the symplecticmanifold, the Poisson manifold, and complex manifolds, which may lead to novelmethods of studying the motion of a particle in the potential field of a rotatingirregular celestial body. Periodic orbits, manifolds and chaos in a rotating plane-symmetric potential fieldare studied; it is found that the dynamical behaviour near the equilibrium point iscompletely determined by the structure of the submanifolds and subspaces near theequilibrium point. The non-degenerate equilibrium points are classified into twelvecases. The necessary and sufficient conditions for linearly stable, non-resonantunstable and resonant equilibrium points are established. Furthermore, it is found thata resonant equilibrium point is a Hopf bifurcation point, which leads to chaoticmotion near the resonant equilibrium point; the appearance and disappearance ofperiodic-orbit families are found near resonant equilibrium points with parametricvariation. In addition, it is discovered that the number of periodic-orbit familiesdepends on the structure of the submanifolds. The theory developed here is lastlyapplied to two particular cases, the rotating homogeneous cube and the circularrestricted three-body problem.The orbits and manifolds near the equilibrium points of a rotating highlyirregularly celestial body are discussed. The linearised equations of motion relative tothe equilibrium points in the gravitational field of a rotating body, the characteristicequation and the stable conditions of the equilibrium points are derived and discussed.First, a new metric is presented to link the orbit and the geodesic of the smoothmanifold. Then, using the eigenvalues of the characteristic equation, the equilibriumpoints are classified into8cases. A theorem is presented and proved to describe thestructure of the submanifold as well as the stable and unstable behaviours of amassless test particle near the equilibrium points. The linearly stable, thenon-resonant unstable, and the resonant equilibrium points as well the dynamicallaws around them are discussed. There are three families of periodic orbits and fourfamilies of quasi-periodic orbits near the linearly stable equilibrium point. For thenon-resonant unstable equilibrium points, there are four relevant cases; for theperiodic orbit and the quasi-periodic orbit, the structures of the submanifold and thesubspace near the equilibrium points are studied for each case. For the resonantequilibrium points, the dimension of the resonant manifold is greater than4, and wefind at least one family of periodic orbits near the resonant equilibrium points. As anapplication of the theory developed here, relevant orbits for the asteroids216 Kleopatra,1620Geographos,4769Castalia and6489Golevka are studied.Moreover, the distribution of characteristic multipliers, the stability of orbits,periodic orbits and the structure of submanifolds in the potential field of rotatinghighly irregular-shaped celestial bodies are also studied. The topological structure ofsubmanifolds for the orbits in the potential field of a rotating highly irregular-shapedcelestial body is discovered that it can be classified into34different cases, including6ordinary cases,3collisional cases,3degenerate real saddle cases,7periodic cases,7period-doubling cases,1periodic and collisional case,1periodic and degeneratereal saddle case,1period-doubling and collisional case,1period-doubling anddegenerate real saddle case as well as4periodic and period-doubling cases. It isfound that the different distribution of characteristic multipliers fixes the structure ofsubmanifolds, the types of orbits, the dynamical behavior and the phase diagram ofthe motion. Classifications and properties for each case are presented. The theorydeveloped here is applied for the asteroids6489Golevka and243Ida to find thedynamical behaviour around these irregular-shaped bodies.