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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/223483
Flow map parameterization methods for invariant tori in Hamiltonian systems
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[eng] Given a dynamic system, it is important to identify the invariant objects that organize long-term behavior, as well as their dynamic connections. Both in theory and in applications. The objective of this thesis is to advance in the development of Kolmogorov-Arnold-Moser (KAM) type techniques within the framework of the parameterization method and its application to problems of celestial mechanics.
We have developed KAM iterative schemes for the calculation of partially hyperbolic invariant torus and their invariant bundles in quasiperiodic Hamiltonian systems. We look for invariant bulls and bundles under adequate time-1 maps, which allow us to reduce the dimension of the bull to be calculated by one. The computational cost of manipulating functions grows exponentially with the number of variables in the parameterization. Therefore, reduction by flow maps is computationally advantageous, although it requires numerical integration. However, this integration can be easily parallelized. If the parameterization is approximated with N Fourier coefficients, the iterative step requires O(N) of storage and O(N log N) operations, in contrast to standard Newtonian methods, which need O(N^2) of storage and O(N^3) operations. This gain in efficiency comes from the geometric properties of phase space (i.e., symplectic geometry), systems (symplectic accuracy), torus (isotropy, Lagrangianity), as well as dynamical properties (reducibility). In particular, the reducibility of the linearized dynamics around the torus to a triangular matrix by blocks is known as automatic reducibility and is an important property both in theory and in applications. The algorithms have been implemented and applied to the Three-Body Elliptic Restricted Problem (ERTBP) to compute an extensive set of non-resonant three-dimensional invariant torus along with their invariant bundles.
From these results, we have obtained an a posteriori theorem for partially hyperbolic invariant bulls and their rank 1 invariant bundles in quasiperiodic Hamiltonian systems. The approach followed allows the theorem to be applied to autonomous, periodic and quasiperiodic Hamiltonian systems, and constitutes the demonstration of the convergence of methods based on flow maps. In addition, we simultaneously obtain both stable and unstable bundles, providing a clear geometric view of the tangent space to the torus. The proof is based on geometric properties of a symplectic nature, which hold approximately when the parameterizations approximately satisfy their equations of invariance. We have obtained geometric lemmas that control error in the KAM iterative process. The new error in the invariance equations is controlled with explicit constants, which requires a
careful treatment of the loss of analyticity at each iterative step. The demonstration concludes by obtaining convergence conditions for the KAM iterative process. The a posteriori theorem obtained allows computer-aided proofs to be carried out.
Partially hyperbolic bulls have associated stable and unstable varieties, whiskers, where dynamics converge exponentially fast in the future and in the past, respectively. The stable and unstable bundles with the linear approximations of these varieties. We have also developed KAM schemes to compute high-order Fourier-Taylor expansions of whiskers in autonomous and quasiperiodic Hamiltonian systems. Unlike order-to-order methods, which first calculate the torus and its bundles before calculating the whiskers on an order-by-order basis, the approach followed simultaneously computes both the torus and the whiskers using the same KAM iterative method. This unified framework improves the efficiency of whisker calculation by doubling the number of correct terms in expansion in each iteration. The algorithms have been applied to the calculation of high-order expansions of partially hyperbolic non-resonant invariant torus in the circular and three-body elliptical constrained problems.
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FERNÁNDEZ-MORA, Álvaro. Flow map parameterization methods for invariant tori in Hamiltonian systems. [consulta: 30 de novembre de 2025]. [Disponible a: https://hdl.handle.net/2445/223483]