A Suevery Of Numerical Approaches With Conserved Constants Of Motion And Dynamics Of Newtonian Core-Shell Systems

Chaos, as a universal phenomenon in nonlinear systems, displays some featuresincluding exponentially sensitive dependence on initial conditions, the stochasticityand determinstricity. It has been interested in many fields. Especially the study ofchaos has been widely popular in celestial mechanics. The main factors to truly detectchaos deal with the choice of good numerical methods and suitable chaos indicators.Although classical high order numerical methods give the numerical solutionsthe high accuracy, they are full of artificial dissipation so that some intrinsiccharacteristics of dynamic systems, such as constants of motion, can not be preserved.Symplectic integrators, which can preserve the energy constant and the symplecticstructure, may remedy the defect. So it is regarded as an adaptive method to study thelong term qualitative evolution of Hamiltonlan systems. However, symplecticintegrators hold a lower accuracy and can not preserve other integrals of motionexcept the integral of energy. In addition, its application is limited to Hamiltoniansystems. In view of these facts, Baumgarte's method called the stabilization ofdifferential equation and Nacozy's manifold correction method called the poststabilization are still focused on and developed widely at the present. The idea ofBaumgarte's method is that a traditional numerical method is used to solve thedifferential equations by adding a control term relevant to some invariants so thatthe numedal solutions become much closer to the true solutions. Unlike Baumgarte'smethod, the post stabilization method is carried out as follows: First, a numericalsolution is obtained by solving the differential equations of motion with a certaintraditional numerical integration; then, a stabilizing term is added to the numericalsolution to adjust it to the integral surface. To do the next round of integration,we takeadjusted numerical solution as the initial conditions. A detailed discussion onBaumgatre's stabilization method and Chin's post-stabilization method is carded outtogether with a numerical comparison. The numerical accuracy can be increased andthe numerical stability can be improved by combining the traditional numericalmethod with either of the two stabilization methods. Assuming the optimal stabilizingparameter, the accuracy of the stabilization of Baumgarte is not equivalent to that ofthe post stabilization in general, and it is not certain which one is the more accurate.The stabilization method makes the right function of numerical integration much more complicated. This leads to the complexity of program and the computationalcost. On the other hand, there is the trouble of having to choose the optimal stabilizingparameter. In this sense, we recommend to use the post stabilization scheme. However,it should be noted that, in comparision with the traditional integrator with nostabilization, the step size of the post stabilization method must not be taken too large.Besides the good numerical methods, the study of chaos needs reliable indicatorsfor the identification of chaos. They involve Poincare sections, Lyapunovcharacteristic exponents (LCEs)and fast Lyapunov indicators (FLIs), etc. Each ofthem has its advantages and shortcomings. Poincare sections can display the phasespace structure of dynamical systems visually. As an emphasis, it is most valid onlywhen the number of dimensions of phase space minus the number of constrains is notlarger than three. For a system of many dimensions, perhaps LCEs are a good choice.In general, it takes a very long time of calculation in order to get a reliable value ofLCE. Froeschle's FLIs are superior to LCEs because they are rapid and sensitive toidentify chaos from regular. Wu et al. extended this idea and proposed the FLI withtwo nearby trajectories instead of the variational equation method in 2006. Obviouslythis indicator is easy to treat complicated dynamical systems. In the present thesis, weshall point out other metrics of this indicator. In detail, we do think that it is a betteridea to apply this indicator and pseudo high order symplectic integrators toHamiltonian systems, which can be split a main component and a smaller one. Toshow this clearly, we consider Newtonian core-shell systems, which could account formassive circumstellar dust shells and rings around certain types of star remnants. Thiskind of systems are the limiting cases of the relativistic models when two conditionsof weak field and slow motion of test particles in the vacuum between the core'shorizon and the shell satisfy. They belong to a class of Hamilton with a primary partand a minor one, so we employ the FLI with two nearby trajectories of Wu et al.matched with pseudo eighth order symplectic integrator. Numerical results suggestthat it is very powerful to adopt this tool to describe not only the evolution ofdynamical features from regular motion to chaos with the variation of somedynamical parameters, but also the global structure of phase space for the systems.