Abstract
A recently proposed algorithm for the estimate of the threshold above which a certain torus disappears, which combines classical Birkhoff normalization procedure with KAM theory, is reconsidered and improved. This is done by studying the particular case of the forced pendulum Hamiltonian\(H(y,x,t) = \frac{{y^2 }}{2} + \varepsilon [\cos x + \cos (x - t)]\), and considering the golden torus. We find that there exists an optimal orderN of normalization, and the ratio between the estimate thus obtained and the experimental value turns out to be ∼19. A possible explanation of such result is also suggested.
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Celletti, A., Giorgilli, A. On the numerical optimization of KAM estimates by classical perturbation theory. Z. angew. Math. Phys. 39, 743–747 (1988). https://doi.org/10.1007/BF00948734
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DOI: https://doi.org/10.1007/BF00948734