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Typescript. / Thesis (Ph. D.)--University of Hawaii, 1967. / Bibliography: leaves [91]-103. / ix, 103 l illus., maps, tables

An introduction to periodic orbit techniques for deterministic dynamical systems is presented. The Farey map is considered as examples of intermittency in one-dimensional maps. The effect of intermittency on the Markov partition is considered. The Gauss map is shown to be related to the farey map by a simple transformation of trajectories. A method of calculating periodic orbits in the thermostated Lorentz gas is derived. This method relies on minimising the action from the Hamiltonian description of the Lorentz gas, as well as the construction of a generating partition of the phase space. This method is employed to examine a range of bifurcation processes in the Lorentz gas. A novel construction of the Sinai billiard is performed by using symmetry arguments to reduce two particles in a hard walled box to the square Sinai billiard. Infinite families of periodic orbits are found, even at the lowest order, due to the intermittency of the system. The contribution of these orbits is examined and found to be tractable at the lowest order. The number of orbits grows too quickly for consideration of any other terms in the periodic orbit expansion. A simple stochastic model for the diffusion in the Lorentz gas was constructed. The model produced a diffusion coefficient that was a remarkably good fit to more precise numerical calculations. This is a significant improvement to the Machta-Zwanzig approximation for the diffusion coefficient. We outline a general approach to constructing stochastic models of deterministic dynamical systems. This method should allow for calculations to be performed in more complicated systems.

Thesis (MScEng)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: This study investigates the theory of precision satellite orbit propagation and satellite lifetime prediction and lead to the development of two necessary software tools for analysis in these fields. Precision propagation was achieved through the implementation of Cowell's method of special perturbations, considering perturbations due to a 70x70 asymmetrical gravity field, atmospheric drag, Luni-Solar attraction and Solar radiation pressure. The satellite's perturbed equations of motion were integrated utilizing a seveneighth order Runge-Kutta-Fehlberg numerical integration procedure, limiting error propagation by employing adaptive step size control. The MSlS-90 atmospheric density model, providing for diurnal and semi-annual variations, was employed to determine atmospheric density. Care was taken in the precision modelling of the motion of the 12000 equator and equinox. Propagation results for this test case proved to be superior to the SGP4 propagator and a commercial package. The long-term effects of Earth oblateness and atmospheric drag on a satellite's orbital elements were investigated and applied to the orbit decay prediction problem. Orbit decay was predicted by integrating the rates of change of the orbital elements due to Earth oblateness and atmospheric drag. A semi-analytical technique involving Runge-Kutta and Gauss-Legendre quadrature was employed in the solution process. Relevant software was developed to implement the decay theory. Optimum drag coefficients, estimated from drag analysis using precision propagation, were used in decay prediction. Two test cases of observed decayed satellites were used to evaluate the theory. Results for both test cases indicated that the theory fitted observational data well within acceptable limits. / AFRIKAANSE OPSOMMING: 'n Ondersoek is gedoen oor die teorie van presiesie satelliet-wentelbaan vooruitskatting en satelliet-wentelbaanleeftyd afskatting en het gelei tot die ontwikkeling van twee analiseprogramme vir gebruik in hierdie vakgebiede. Presiesie vooruitskatting is bereik deur die gebruik van Cowell se metode van spesiale perturbasies, wat die invloed van 'n nie-simmetriese 70x070 gravitasieveld, atmosferiese sleur, Son-Maan aantrekkingskragte en druk van sonradiasie, in ag neem. Die satelliet se versteurde bewegingsvergelykings is numeries ge-ïntegreer deur gebruik te maak van die sewe-agste orde Runge-Kutta- Fehlberg metode wat fout-voortplanting inhibeer deur gebruik te maak van 'n aanpasbare integrasiestaplengte. Die MSIS-90 atmosferies model, wat voorsiening maak vir dag-nag en half-jaarlikse atmosferiese variasies, is gebruik vir die berekening van atmosferiese digtheid. Sorg is gedra by die presiesie modellering van die beweging van die J2000 ekwator en ekwinokse. Resultate vir hierdie toetsgeval toon meer voortreflik te wees as die SPG4 - en 'n kommersieël-beskikbare vooruitskatter. Die langtermyn effekte van aard-afplatting en atmosferiese sleur op wentelbaanleeftyd is ondersoek en toegepas op die wentelbaanverval-afskattingsprobleem. Wentelbaanverval is bereken deur die integrasie van die tydsafgeleides van die wentelbaanelement onder invloed van aard-afplatting en atmosferiese sleur. Vir die doel is 'n semi-analitiese tegniek, wat gebruik maak van Gauss-Legendre kwadratuur en Runge-Kutta numeriese integrasie, gebruik gemaak. Nodige rekenaar programmatuur is ontwi kkeI om die vervalteorie te implimenteer. Optimale sleur-koëffisiënte is afgeskat deur van presiesie wentelbaananalise gebruik te maak. Twee gevallestudies van bekende vervalde satelliete is gebruik om die vervalteorie te evalueer. Resultate vir beide gevallestudies toon aan dat eksperimentele resultate werklike vervaltye binne aanvaarbare limiete navolg.

This thesis involves the analysis of four classes of nonlinear oscillators. We investigate a damped planar pendulum subject to vertical sinusoidal displacement of appropriate amplitude and frequency, a Hamiltonian planar pendulum with support point oscillating in the vertical direction, a forced spherical pendulum as a constrained dynamical system and a spinning double pendulum with the two masses oscillating in transversal planes. The motivation for this research was to understand and determine the fundamental dynamical properties of the four model systems. For this purpose analytical and numerical tools have been employed. Linearization, phase portraits, Poincare sections, basins of attraction, KAM theory, Lyapunov exponents and normal form theory have been considered as examples. For the damped planar pendulum a rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position becomes asymptotically stable. Furthermore, using numerical tools, the dynamics of the system far from its equilibrium points is systematically investigated. For the undamped and parametrically perturbed planar pendulum, we use KAM type arguments to rigorously prove the stability of the equilibrium point corresponding to the upside-down position. For the spherical pendulum a numerical framework is developed, which allows orbits to explore the entire sphere. We show that the qualitative change in the Poincare sections from regular to chaotic behaviour is in excellent qualitative agreement with corresponding computations of the Lyapunov exponents. Finally we study the dynamics of the spinning double pendulum by using normal form theory. We have identified the regions in physical parameter space where a codimension-two singularity occurs. An algorithm for the Cushman-Sanders normal form is constructed and analyzed. A representative model for the truncated normal form is presented.

This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two.

The main purpose of this thesis is to develop a search strategy for space debris that are in the geosynchronous orbit (GEO) region. The search strategy is not an effort to find the object initially but rather if found one time to aid in finding it again within a small time frame. This was a request from NASA Johnson Space Center Orbital Debris Program Office through the MODEST, Michigan Orbital Debris Survey Telescope, program. A single definitive search pattern was not found, however depending on the COEs of the orbit specific search strategy can be employed. These search strategies are far from perfect and can be improved upon with more rigorous testing as well as a larger data sample. Another goal is to look for correlation between the orbital parameters and the errors in the predicted right ascension (RA) and the declination (DEC). This was accomplished by varying the different orbital parameters by ±10% individually while holding the other parameters constant. This showed some correlation existed between some parameters and their errors, in particular there was correlation between a variation in right ascension of ascending node (RAAN) and the value of RAAN itself. The correlation found was that with the higher the value of RAAN the larger the RMS error.

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