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ItemNew Numerical And Computational Methods Leveraging Dynamical Systems Theory For Multi-Body Astrodynamics(Georgia Institute of Technology, 2022-06-01) Kumar, BhanuMany proposed interplanetary space missions, including Europa Lander and Dragonfly, involve trajectory design in environments where multiple large bodies exert gravitational influence on the spacecraft, such as the Jovian and Saturnian systems as well as cislunar space. In these contexts, an analysis based on the mathematical theory of dynamical systems provides both better insight as well as new tools to use for the mission design compared to classic two-body Keplerian methods. Indeed, a rich variety of dynamical phenomena manifest themselves in such systems, including libration point dynamics, stable and unstable mean-motion resonances, and chaos. To understand the previously mentioned dynamical behaviors, invariant manifolds such as periodic orbits, quasi-periodic invariant tori, and stable/unstable manifolds are the major objects whose interactions govern the local and global dynamics of relevant celestial systems. This work is focused on the development of numerical methodologies for computing such invariant manifolds and investigating their interactions. In Chapter 2, after a study of persistence of mean-motion resonances in the planar circular restricted 3-body problem (PCRTBP), techniques for computing the stable/unstable manifolds attached to resonant periodic orbits and heteroclinics corresponding to resonance transitions are presented. Chapter 3 focuses on the development of accurate and efficient parameterization methods for numerical calculation of whiskered quasi-periodic tori and their attached stable/unstable manifolds, for periodically-forced PCRTBP models. As part of this, a method for Levi- Civita regularization of such periodically-forced systems is introduced. Finally, Chapter 4 presents methods for combining the previously mentioned parameterizations with knowledge of the objects’ internal dynamics, collision detection algorithms, and GPU computing to very rapidly compute propellant-free heteroclinic connecting trajectories between them, even in higher dimensional models. Such heteroclinics are key to the generation of chaos and large scale transport in astrodynamical systems.
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ItemDomains of analyticity and Gevrey estimates of tori in weakly dissipative systems(Georgia Institute of Technology, 2021-12-07) Perez Bustamante, AdrianWe consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing. In the first part, we study a family of dissipative standard maps of the cylinder for which the dissipation is a function of a small complex parameter of perturbation. We compute perturbative expansions formally in the parameter of perturbation and use them to estimate the shape of the domains of analyticity of invariant circles as functions of the parameter of perturbation. We also give evidence that the functions might belong to a Gevrey class. The numerical computations we perform support conjectures on the shape of the domains of analyticity. In the second part, we consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. We prove that the formal expansions for the quasiperiodic solutions and the adjustable parameter satisfy Gevrey estimates. To prove this result we introduce a novel method that might be of interest beyond the problem considered in this work.
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ItemFast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations(Georgia Institute of Technology, 2021-08-04) Yao, YianIn this thesis, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2-dimensional maps. The 2-dimensional maps we are considering are motivated by oscillators subject to periodic perturbation. The algorithm is based on solving an invariance equation using a quasi-Newton method, and the algorithm works irrespective of whether the dynamics on the invariant circle conjugates to a rotation or is phase-locked, and thus we expect only finite regularity on the invariant circle but analytic on the stable manifolds. The thesis is divided into the following two parts: In the first part, we derive our quasi-Newton algorithm and prove that starting from an initial guess that satisfies the invariance equation very approximately, the algorithm converges quadratically to a true solution which is close to the initial guess. The proof of the convergence is based on an abstract Nash-Moser Implicit Function Theorem specially tailored for this problem. In the second part, we discuss some implementation details regarding our algorithm and implemented it on the dissipative standard map. We follow different continuation paths along the perturbation and drift parameters and explore the "bundle merging" scenario when the hyperbolicity of the map losses due to the increase of the perturbation. For non-resonant eigenvalues, we also generalize the algorithm to 3-dimension and implemented it on the 3-D Fattened Arnold Family.
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ItemPersistence of Invariant Objects under Delay Perturbations(Georgia Institute of Technology, 2021-07-30) Yang, JiaqiIn this dissertation, we investigate functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs. Despite the singular nature of the perturbations, when the perturbations are small, we get that some invariant objects of the unperturbed equations persist and depend on the parameters with high regularity. The results apply to equations with state-dependent delay perturbations and equations arising in electrodynamics. This is based on joint work with Joan Gimeno and Rafael de la Llave.