This dissertation studies quantum invariants of knots and links, particularly the colored Jones polynomials, and their relationships with classical invariants like the hyperbolic volume and the A-polynomial. We consider the volume conjecture that relates the Kashaev invariant, a specialization of the colored Jones polynomial at a specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that relates the colored Jones polynomial and the A-polynomial of a knot. We establish the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and confirm the volume conjecture for some cables of knots.

<div> <div> <div> <p>Resonant orbits in a multi-body environment have been investigated in the past to aid the understanding of perceived chaotic behavior in the solar system. The invariant manifolds associated with resonant orbits have also been recently incorporated into the design of trajectories requiring reduced maneuver costs. Poincaré sections are now also extensively utilized in the search for novel, maneuver-free trajectories in various systems. This investigation employs dynamical systems techniques in the computation and characterization of resonant orbits in the higher-fidelity Circular Restricted Three-Body model. Differential corrections and numerical methods are widely leveraged in this analysis in the determination of orbits corresponding to different resonance ratios. The versatility of resonant orbits in the design of low cost trajectories to support exploration for several planet-moon systems is demonstrated. The efficacy of the resonant orbits is illustrated via transfer trajectory design in the Earth-Moon, Saturn-Titan, and the Mars-Deimos systems. Lastly, Poincaré sections associated with different resonance ratios are incorporated into the search for natural, maneuver-free trajectories in the Saturn-Titan system. To that end, homoclinic and heteroclinic trajectories are constructed. Additionally, chains of periodic orbits that mimic the geometries for two different resonant ratios are examined, i.e., periodic orbits that cycle between different resonances are determined. The tools and techniques demonstrated in this investigation are useful for the design of trajectories in several different systems within the CR3BP. </p> </div> </div> </div>

In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW

Neste trabalho estudamos problemas de reação difusão em domínios unidimensionais que surgem de materiais compostos e obtemos resultados comparando os fluxos do problema original e do problema limite quando a difusão fica muito grande em partes do domínio. Provamos que os autovalores e autofunções do operador linear ilimitado associado à equação limite têm a propriedade de Sturm Liouville e provamos que as soluções do problema de reação difusão têm a propriedade do decrescimento do número de zeros ao longo do tempo. Estes resultados são usados para provar que as variedades instável e estável de pontos de equilíbrios são genericamente transversais e que o fluxo no atrator para o problema de reação difusão é genericamente estruturalmente estável. Estes fatos permitem obter a equivalência topológica dos fluxos restritos aos atratores dos problemas original e seu problema limite. / In this work we study some reaction-difusion problems in one dimensional domains that arise from composite materials. We obtain some results comparing the flux of the original problem and the flux of the limit problem when the difusion becomes large on parts of the physical domain. We prove that the eigenvalues and eigenfunctions of the linear unbounded operator associated with the equation have the Sturm Liouville property and also that the solutions of the reaction difusion equation have the property that the zeros do not increase with time. These results are used to obtain that the stable and unstable manifolds of equilibrium points are generically transversal and that the flux on the attractor for the reaction difusion problem is generically structurally stable. Using this we are able to prove the topological equivalence of the fluxs restricted to the attractors of the original and the limit problem.

La memòria consta de dues parts. En la primera d'elles s'estudien les òrbites homoclíniques i heteroclíniques associades als punts d'equilibri triangulars del problema restringit circular i pla per valors del paràmetre de masses compresos entre 0.1 i 0.5, es donen resultats referents a la seva forma i nombre. En la segona part òrbita halo al voltant del punt l1 del sistema terra-sol, utilitzant les idees geomètriques que proporciona la teoria dels sistemes dinàmics. Es comença l'estudi per models senzills a fi de veure l'essencial de la geometria del problema i la influencia de la lluna, per finalitzar utilitzant el model de sistema solar real donat per les efemèrides del JPL.

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures.

Neste trabalho estudamos problemas de reação difusão em domínios unidimensionais que surgem de materiais compostos e obtemos resultados comparando os fluxos do problema original e do problema limite quando a difusão fica muito grande em partes do domínio. Provamos que os autovalores e autofunções do operador linear ilimitado associado à equação limite têm a propriedade de Sturm Liouville e provamos que as soluções do problema de reação difusão têm a propriedade do decrescimento do número de zeros ao longo do tempo. Estes resultados são usados para provar que as variedades instável e estável de pontos de equilíbrios são genericamente transversais e que o fluxo no atrator para o problema de reação difusão é genericamente estruturalmente estável. Estes fatos permitem obter a equivalência topológica dos fluxos restritos aos atratores dos problemas original e seu problema limite. / In this work we study some reaction-difusion problems in one dimensional domains that arise from composite materials. We obtain some results comparing the flux of the original problem and the flux of the limit problem when the difusion becomes large on parts of the physical domain. We prove that the eigenvalues and eigenfunctions of the linear unbounded operator associated with the equation have the Sturm Liouville property and also that the solutions of the reaction difusion equation have the property that the zeros do not increase with time. These results are used to obtain that the stable and unstable manifolds of equilibrium points are generically transversal and that the flux on the attractor for the reaction difusion problem is generically structurally stable. Using this we are able to prove the topological equivalence of the fluxs restricted to the attractors of the original and the limit problem.

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures.

<p>The 9:2 lunar synodic resonant near rectilinear halo orbit (NRHO) is the chosen orbit for the Gateway, a future lunar space station constructed by the National Aeronautics and Space Administration (NASA) as well as several commercial and international partners. Designing trajectories in this sensitive lunar region combined with the absence of a singular systematic methodology to approach mission design poses challenges as researchers attempt to design transfers to and from this nearly stable orbit. This investigation builds on previous research in Poincar\'e mapping strategies to design transfers from the 9:2 NRHO using higher-dimensional maps and maps with non-state variables. First, Poincar\'e maps are applied to planar transfers to demonstrate the utility of hyperplanes and establish that maps with only two or three dimensions are required in the planar problem. However, with the addition of two state variables, the spatial problem presents challenges in visualizing the full state. Higher-dimensional maps utilizing glyphs and color are employed for spatial transfer design involving the 9:2 NRHO. The visualization of all required dimensions on one plot accurately reveals low cost transfers into both a 3:2 planar resonant orbit and an L2 vertical orbit. Next, the application of higher-dimensional maps is extended beyond state variables. Visualizing time-of-flight on a map axis enables the selection of faster transfers. Additionally, glyphs and color depicting angular momentum rather than velocity lead to transfers with nearly tangential maneuvers. Theoretical minimum maneuvers occur at tangential intersections, so these transfers are low cost. Finally, a map displaying several initial and final orbit options, discerned through the inclusion of Jacobi constant on an axis, eliminates the need to recompute a map for each initial and final orbit pair. Thus, computation time is greatly reduced in addition to visualizing more of the design space in one plot. The higher-dimensional mapping strategies investigated are relevant for transfer design or other applications requiring the visualization of several dimensions simultaneously. Overall, this investigation outlines Poincar\'e mapping strategies for transfer scenarios of different design space dimensions and represents initial research into non-state variable mapping methods.</p>

A newly proposed F-class mission by the European Space Agency (ESA) in 2019,Comet Interceptor, aims to dynamically intercept a New Solar System Objectsuch as a Dynamically New Comet (DNC). The Spacecraft will be placed in aperiodic (Halo) orbit around the Sun-Earth L2 Lagrangian point, waiting for furtherinstructions about the passage of a comet or an asteroid, which could well bereached within the stipulated mission constraints.A major part of the detection of these bodies will be owed to the Large SynopticSurvey Telescope (Currently under construction in Chile), which hopes to vastlyincrease the ability to discover a possible target using the catalogue of LongPeriod Comets and a set of its orbits. It is suggested that, in a mission length of&lt;5 years, discoveries and warnings are possible so that optimization of thetrajectory and characterisation of the object are done within the set windows.This thesis is aimed at facilitating a transfer to a Potentially Hazardous Asteroid(PHA), a subset of the Near-Earth Objects (NEO), as a secondary choice on theoff-chance that the discovered comet could not be reached from the L2 Librationpoint within the mission constraints.The first section of this thesis deals with the selection of a Potentially HazardousAsteroid for our mission from the larger database of the Near-Earth Objects,based on a measure of impact hazard called the Palermo Scale, while the secondsection of the thesis aims to obtain a suitable Halo orbit around L2 through ananalytical construction method. After a desired orbit is found, the invariantmanifolds around the Halo orbit are constructed and analysed in an attempt toreduce the ΔV, where from the spacecraft can intercept the Potentially Hazardous Asteroid through the trajectory demanding the least energy.