Fast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations
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In 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.